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Dynamic programming
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In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions – ideally, using a memory-based data structure. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) The technique of storing solutions to subproblems instead of recomputing them is called "memoization".
Dynamic programming algorithms are often used for optimization. A dynamic programming algorithm will examine the previously solved subproblems and will combine their solutions to give the best solution for the given problem. In comparison, a greedy algorithm treats the solution as some sequence of steps and picks the locally optimal choice at each step. Using a greedy algorithm does not guarantee an optimal solution, because picking locally optimal choices may result in a bad global solution, but it is often faster to calculate. Fortunately, some greedy algorithms (such as Kruskal's or Prim's for minimum spanning trees) are proven to lead to the optimal solution.
For example, in the coin change problem of finding the minimum number of coins of given denominations needed to make a given amount, a dynamic programming algorithm would find an optimal solution for each amount by first finding an optimal solution for each smaller amount and then using these solutions to construct an optimal solution for the larger amount. In contrast, a greedy algorithm might treat the solution as a sequence of coins, starting from the given amount and at each step subtracting the largest possible coin denomination that is less than the current remaining amount. If the coin denominations are 1,4,5,15,20 and the given amount is 23, this greedy algorithm gives a non-optimal solution of 20+1+1+1, while the optimal solution is 15+4+4.
In addition to finding optimal solutions to some problem, dynamic programming can also be used for counting the number of solutions, for example counting the number of ways a certain amount of change can be made from a given collection of coins, or counting the number of optimal solutions to the coin change problem described above.
Dynamic programming is both a mathematical optimization method and a computer programming method. In both contexts it refers to simplifying a complicated problem by breaking it down into simpler sub-problems in a recursive manner. While some decision problems cannot be taken apart this way, decisions that span several points in time do often break apart recursively; Bellman called this the "Principle of Optimality". Likewise, in computer science, a problem that can be solved optimally by breaking it into sub-problems and then recursively finding the optimal solutions to the sub-problems is said to have optimal substructure.
If sub-problems can be nested recursively inside larger problems, so that dynamic programming methods are applicable, then there is a relation between the value of the larger problem and the values of the sub-problems.[1] In the optimization literature this relationship is called the Bellman equation.
Dynamic programming in mathematical optimization[edit]
In terms of mathematical optimization, dynamic programming usually refers to simplifying a decision by breaking it down into a sequence of decision steps over time. This is done by defining a sequence of value functions V1, V2, ..., Vn, with an argument y representing the state of the system at times i from 1 to n. The definition of Vn(y) is the value obtained in state y at the last time n. The values Vi at earlier times i = n −1, n − 2, ..., 2, 1 can be found by working backwards, using a recursive relationship called the Bellman equation. For i = 2, ..., n, Vi−1 at any state y is calculated from Vi by maximizing a simple function (usually the sum) of the gain from a decision at time i − 1 and the function Vi at the new state of the system if this decision is made. Since Vi has already been calculated for the needed states, the above operation yields Vi−1 for those states. Finally, V1 at the initial state of the system is the value of the optimal solution. The optimal values of the decision variables can be recovered, one by one, by tracking back the calculations already performed.
Dynamic programming in bioinformatics[edit]
Dynamic programming is widely used in bioinformatics for the tasks such as sequence alignment, protein folding, RNA structure prediction and protein-DNA binding. The first dynamic programming algorithms for protein-DNA binding were developed in the 1970s independently by Charles DeLisi in USA[2] and Georgii Gurskii and Alexander Zasedatelev in USSR.[3] Recently these algorithms have become very popular in bioinformatics and computational biology, particularly in the studies of nucleosome positioning and transcription factor binding.
Dynamic programming in computer programming[edit]
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There are two key attributes that a problem must have in order for dynamic programming to be applicable: optimal substructure and overlapping sub-problems. If a problem can be solved by combining optimal solutions to non-overlapping sub-problems, the strategy is called "divide and conquer" instead. This is why merge sort and quick sort are not classified as dynamic programming problems.
Optimal substructure means that the solution to a given optimization problem can be obtained by the combination of optimal solutions to its sub-problems. Such optimal substructures are usually described by means of recursion. For example, given a graph G=(V,E), the shortest path p from a vertex u to a vertex v exhibits optimal substructure: take any intermediate vertex w on this shortest path p. If p is truly the shortest path, then it can be split into sub-paths p1 from u to w and p2 from w to v such that these, in turn, are indeed the shortest paths between the corresponding vertices (by the simple cut-and-paste argument described in Introduction to Algorithms). Hence, one can easily formulate the solution for finding shortest paths in a recursive manner, which is what the Bellman–Ford algorithm or the Floyd–Warshall algorithm does.
Overlapping sub-problems means that the space of sub-problems must be small, that is, any recursive algorithm solving the problem should solve the same sub-problems over and over, rather than generating new sub-problems. For example, consider the recursive formulation for generating the Fibonacci series: Fi = Fi−1 + Fi−2, with base case F1 = F2 = 1. Then F43 = F42 + F41, and F42 = F41 + F40. Now F41 is being solved in the recursive sub-trees of both F43 as well as F42. Even though the total number of sub-problems is actually small (only 43 of them), we end up solving the same problems over and over if we adopt a naive recursive solution such as this. Dynamic programming takes account of this fact and solves each sub-problem only once.
This can be achieved in either of two ways:[citation needed]
- Top-down approach: This is the direct fall-out of the recursive formulation of any problem. If the solution to any problem can be formulated recursively using the solution to its sub-problems, and if its sub-problems are overlapping, then one can easily memoize or store the solutions to the sub-problems in a table. Whenever we attempt to solve a new sub-problem, we first check the table to see if it is already solved. If a solution has been recorded, we can use it directly, otherwise we solve the sub-problem and add its solution to the table.
- Bottom-up approach: Once we formulate the solution to a problem recursively as in terms of its sub-problems, we can try reformulating the problem in a bottom-up fashion: try solving the sub-problems first and use their solutions to build-on and arrive at solutions to bigger sub-problems. This is also usually done in a tabular form by iteratively generating solutions to bigger and bigger sub-problems by using the solutions to small sub-problems. For example, if we already know the values of F41 and F40, we can directly calculate the value of F42.
Some programming languages can automatically memoize the result of a function call with a particular set of arguments, in order to speed up call-by-name evaluation (this mechanism is referred to as call-by-need). Some languages make it possible portably (e.g. Scheme, Common Lisp or Perl). Some languages have automatic memoization built in, such as tabled Prolog and J, which supports memoization with the M. adverb.[4] In any case, this is only possible for a referentially transparent function.
Example: Economic optimization[edit]
Optimal consumption and saving[edit]
A mathematical optimization problem that is often used in teaching dynamic programming to economists (because it can be solved by hand[5]) concerns a consumer who lives over the periods and must decide how much to consume and how much to save in each period.
Let be consumption in period t, and assume consumption yields utility as long as the consumer lives. Assume the consumer is impatient, so that he discounts future utility by a factor b each period, where . Let be capital in period t. Assume initial capital is a given amount , and suppose that this period's capital and consumption determine next period's capital as , where A is a positive constant and . Assume capital cannot be negative. Then the consumer's decision problem can be written as follows:
- subject to for all
Written this way, the problem looks complicated, because it involves solving for all the choice variables . (Note that is not a choice variable—the consumer's initial capital is taken as given.)
The dynamic programming approach to solve this problem involves breaking it apart into a sequence of smaller decisions. To do so, we define a sequence of value functions , for which represent the value of having any amount of capital k at each time t. Note that , that is, there is (by assumption) no utility from having capital after death.
The value of any quantity of capital at any previous time can be calculated by backward induction using the Bellman equation. In this problem, for each , the Bellman equation is
- subject to
This problem is much simpler than the one we wrote down before, because it involves only two decision variables, and . Intuitively, instead of choosing his whole lifetime plan at birth, the consumer can take things one step at a time. At time t, his current capital is given, and he only needs to choose current consumption and saving .
To actually solve this problem, we work backwards. For simplicity, the current level of capital is denoted as k. is already known, so using the Bellman equation once we can calculate , and so on until we get to , which is the value of the initial decision problem for the whole lifetime. In other words, once we know , we can calculate , which is the maximum of , where is the choice variable and .
Working backwards, it can be shown that the value function at time is
where each is a constant, and the optimal amount to consume at time is
which can be simplified to
We see that it is optimal to consume a larger fraction of current wealth as one gets older, finally consuming all remaining wealth in period T, the last period of life.
Examples: Computer algorithms[edit]
Dijkstra's algorithm for the shortest path problem[edit]
From a dynamic programming point of view, Dijkstra's algorithm for the shortest path problem is a successive approximation scheme that solves the dynamic programming functional equation for the shortest path problem by the Reaching method.[6][7][8]
In fact, Dijkstra's explanation of the logic behind the algorithm,[9] namely
Problem 2. Find the path of minimum total length between two given nodes and .
We use the fact that, if is a node on the minimal path from to , knowledge of the latter implies the knowledge of the minimal path from to .
is a paraphrasing of Bellman's famous Principle of Optimality in the context of the shortest path problem.
Fibonacci sequence[edit]
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Here is a naïve implementation of a function finding the nth member of the Fibonacci sequence, based directly on the mathematical definition:
function fib(n) if n <= 1 return n return fib(n − 1) + fib(n − 2)
Notice that if we call, say, fib(5)
, we produce a call tree that calls the function on the same value many different times:
fib(4) + fib(3)
(fib(3) + fib(2)) + (fib(2) + fib(1))
((fib(2) + fib(1)) + (fib(1) + fib(0))) + ((fib(1) + fib(0)) + fib(1))
(((fib(1) + fib(0)) + fib(1)) + (fib(1) + fib(0))) + ((fib(1) + fib(0)) + fib(1))
In particular, fib(2)
was calculated three times from scratch. In larger examples, many more values of fib
, or subproblems, are recalculated, leading to an exponential time algorithm.
Now, suppose we have a simple map object, m, which maps each value of fib
that has already been calculated to its result, and we modify our function to use it and update it. The resulting function requires only O(n) time instead of exponential time (but requires O(n) space):
var m := map(0 → 0, 1 → 1) function fib(n) if key n is not in map m m[n] := fib(n − 1) + fib(n − 2) return m[n]
This technique of saving values that have already been calculated is called memoization; this is the top-down approach, since we first break the problem into subproblems and then calculate and store values.
In the bottom-up approach, we calculate the smaller values of fib
first, then build larger values from them. This method also uses O(n) time since it contains a loop that repeats n − 1 times, but it only takes constant (O(1)) space, in contrast to the top-down approach which requires O(n) space to store the map.
function fib(n) if n = 0 return 0 else var previousFib := 0, currentFib := 1 repeat n − 1 times // loop is skipped if n = 1 var newFib := previousFib + currentFib previousFib := currentFib currentFib := newFib return currentFib
In both examples, we only calculate fib(2)
one time, and then use it to calculate both fib(4)
and fib(3)
, instead of computing it every time either of them is evaluated.
Note that the above method actually takes time for large n because addition of two integers with bits each takes time. (The nth fibonacci number has bits.) Also, there is a closed form for the Fibonacci sequence, known as Binet's formula, from which the -th term can be computed in approximately time, which is more efficient than the above dynamic programming technique. However, the simple recurrence directly gives the matrix form that leads to an approximately algorithm by fast matrix exponentiation.
A type of balanced 0–1 matrix[edit]
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Consider the problem of assigning values, either zero or one, to the positions of an n × n matrix, with n even, so that each row and each column contains exactly n / 2 zeros and n / 2 ones. We ask how many different assignments there are for a given . For example, when n = 4, four possible solutions are
There are at least three possible approaches: brute force, backtracking, and dynamic programming.
Brute force consists of checking all assignments of zeros and ones and counting those that have balanced rows and columns (n / 2 zeros and n / 2 ones). As there are possible assignments, this strategy is not practical except maybe up to .
Backtracking for this problem consists of choosing some order of the matrix elements and recursively placing ones or zeros, while checking that in every row and column the number of elements that have not been assigned plus the number of ones or zeros are both at least n / 2. While more sophisticated than brute force, this approach will visit every solution once, making it impractical for n larger than six, since the number of solutions is already 116,963,796,250 for n = 10, as we shall see.
Dynamic programming makes it possible to count the number of solutions without visiting them all. Imagine backtracking values for the first row – what information would we require about the remaining rows, in order to be able to accurately count the solutions obtained for each first row value? We consider k × n boards, where 1 ≤ k ≤ n, whose rows contain zeros and ones. The function f to which memoization is applied maps vectors of n pairs of integers to the number of admissible boards (solutions). There is one pair for each column, and its two components indicate respectively the number of zeros and ones that have yet to be placed in that column. We seek the value of ( arguments or one vector of elements). The process of subproblem creation involves iterating over every one of possible assignments for the top row of the board, and going through every column, subtracting one from the appropriate element of the pair for that column, depending on whether the assignment for the top row contained a zero or a one at that position. If any one of the results is negative, then the assignment is invalid and does not contribute to the set of solutions (recursion stops). Otherwise, we have an assignment for the top row of the k × n board and recursively compute the number of solutions to the remaining (k − 1) × n board, adding the numbers of solutions for every admissible assignment of the top row and returning the sum, which is being memoized. The base case is the trivial subproblem, which occurs for a 1 × n board. The number of solutions for this board is either zero or one, depending on whether the vector is a permutation of n / 2 and n / 2 pairs or not.
For example, in the first two boards shown above the sequences of vectors would be
((2, 2) (2, 2) (2, 2) (2, 2)) ((2, 2) (2, 2) (2, 2) (2, 2)) k = 4 0 1 0 1 0 0 1 1 ((1, 2) (2, 1) (1, 2) (2, 1)) ((1, 2) (1, 2) (2, 1) (2, 1)) k = 3 1 0 1 0 0 0 1 1 ((1, 1) (1, 1) (1, 1) (1, 1)) ((0, 2) (0, 2) (2, 0) (2, 0)) k = 2 0 1 0 1 1 1 0 0 ((0, 1) (1, 0) (0, 1) (1, 0)) ((0, 1) (0, 1) (1, 0) (1, 0)) k = 1 1 0 1 0 1 1 0 0 ((0, 0) (0, 0) (0, 0) (0, 0)) ((0, 0) (0, 0), (0, 0) (0, 0))
The number of solutions (sequence A058527 in the OEIS) is
Links to the MAPLE implementation of the dynamic programming approach may be found among the external links.
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Consider a checkerboard with n × n squares and a cost-function c(i, j) which returns a cost associated with square i, j (i being the row, j being the column). For instance (on a 5 × 5 checkerboard),
5 | 6 | 7 | 4 | 7 | 8 |
4 | 7 | 6 | 1 | 1 | 4 |
3 | 3 | 5 | 7 | 8 | 2 |
2 | – | 6 | 7 | 0 | – |
1 | – | – | *5* | – | – |
1 | 2 | 3 | 4 | 5 |
Thus c(1, 3) = 5
Let us say there was a checker that could start at any square on the first rank (i.e., row) and you wanted to know the shortest path (sum of the costs of the visited squares are at a minimum) to get to the last rank, assuming the checker could move only diagonally left forward, diagonally right forward, or straight forward. That is, a checker on (1,3) can move to (2,2), (2,3) or (2,4).
5 | |||||
4 | |||||
3 | |||||
2 | x | x | x | ||
1 | o | ||||
1 | 2 | 3 | 4 | 5 |
This problem exhibits optimal substructure. That is, the solution to the entire problem relies on solutions to subproblems. Let us define a function q(i, j) as
- q(i, j) = the minimum cost to reach square (i, j).
If we can find the values of this function for all the squares at rank n, we pick the minimum and follow that path backwards to get the shortest path.
Note that q(i, j) is equal to the minimum cost to get to any of the three squares below it (since those are the only squares that can reach it) plus c(i, j). For instance:
5 | |||||
2 | |||||
1 | |||||
1 | 2 | 3 | 4 | 5 |
Now, let us define q(i, j) in somewhat more general terms:
The first line of this equation is there to make the recursive property simpler (when dealing with the edges, so we need only one recursion). The second line says what happens in the last rank, to provide a base case. The third line, the recursion, is the important part. It is similar to the A,B,C,D example. From this definition we can make a straightforward recursive code for q(i, j). In the following pseudocode, n is the size of the board, c(i, j)
is the cost-function, and min()
returns the minimum of a number of values:
function minCost(i, j) if j < 1 or j > n return infinity else if i = 1 return c(i, j) else return min( minCost(i-1, j-1), minCost(i-1, j), minCost(i-1, j+1) ) + c(i, j)
It should be noted that this function only computes the path-cost, not the actual path. We will get to the path soon. This, like the Fibonacci-numbers example, is horribly slow since it wastes time recomputing the same shortest paths over and over. However, we can compute it much faster in a bottom-up fashion if we store path-costs in a two-dimensional array q[i, j]
rather than using a function. This avoids recomputation; before computing the cost of a path, we check the array q[i, j]
to see if the path cost is already there.
We also need to know what the actual shortest path is. To do this, we use another array p[i, j]
, a predecessor array. This array implicitly stores the path to any square s by storing the previous node on the shortest path to s, i.e. the predecessor. To reconstruct the path, we lookup the predecessor of s, then the predecessor of that square, then the predecessor of that square, and so on, until we reach the starting square. Consider the following code:
function computeShortestPathArrays() for x from 1 to n q[1, x] := c(1, x) for y from 1 to n q[y, 0] := infinity q[y, n + 1] := infinity for y from 2 to n for x from 1 to n m := min(q[y-1, x-1], q[y-1, x], q[y-1, x+1]) q[y, x] := m + c(y, x) if m = q[y-1, x-1] p[y, x] := -1 else if m = q[y-1, x] p[y, x] := 0 else p[y, x] := 1
Now the rest is a simple matter of finding the minimum and printing it.
function computeShortestPath() computeShortestPathArrays() minIndex := 1 min := q[n, 1] for i from 2 to n if q[n, i] < min minIndex := i min := q[n, i] printPath(n, minIndex)
function printPath(y, x) print(x) print("<-") if y = 2 print(x + p[y, x]) else printPath(y-1, x + p[y, x])
Sequence alignment[edit]
In genetics, sequence alignment is an important application where dynamic programming is essential.[10] Typically, the problem consists of transforming one sequence into another using edit operations that replace, insert, or remove an element. Each operation has an associated cost, and the goal is to find the sequence of edits with the lowest total cost.
The problem can be stated naturally as a recursion, a sequence A is optimally edited into a sequence B by either:
- inserting the first character of B, and performing an optimal alignment of A and the tail of B
- deleting the first character of A, and performing the optimal alignment of the tail of A and B
- replacing the first character of A with the first character of B, and performing optimal alignments of the tails of A and B.
The partial alignments can be tabulated in a matrix, where cell (i,j) contains the cost of the optimal alignment of A[1..i] to B[1..j]. The cost in cell (i,j) can be calculated by adding the cost of the relevant operations to the cost of its neighboring cells, and selecting the optimum.
Different variants exist, see Smith–Waterman algorithm and Needleman–Wunsch algorithm.
Tower of Hanoi puzzle[edit]
The Tower of Hanoi or Towers of Hanoi is a mathematical game or puzzle. It consists of three rods, and a number of disks of different sizes which can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape.
The objective of the puzzle is to move the entire stack to another rod, obeying the following rules:
- Only one disk may be moved at a time.
- Each move consists of taking the upper disk from one of the rods and sliding it onto another rod, on top of the other disks that may already be present on that rod.
- No disk may be placed on top of a smaller disk.
The dynamic programming solution consists of solving the functional equation
- S(n,h,t) = S(n-1,h, not(h,t)) ; S(1,h,t) ; S(n-1,not(h,t),t)
where n denotes the number of disks to be moved, h denotes the home rod, t denotes the target rod, not(h,t) denotes the third rod (neither h nor t), ";" denotes concatenation, and
- S(n, h, t) := solution to a problem consisting of n disks that are to be moved from rod h to rod t.
Note that for n=1 the problem is trivial, namely S(1,h,t) = "move a disk from rod h to rod t" (there is only one disk left).
The number of moves required by this solution is 2n − 1. If the objective is to maximize the number of moves (without cycling) then the dynamic programming functional equation is slightly more complicated and 3n − 1 moves are required.[11]
Egg dropping puzzle[edit]
The following is a description of the instance of this famous puzzle involving n=2 eggs and a building with H=36 floors:[12]
- Suppose that we wish to know which stories in a 36-story building are safe to drop eggs from, and which will cause the eggs to break on landing (using U.S. English terminology, in which the first floor is at ground level). We make a few assumptions:
- An egg that survives a fall can be used again.
- A broken egg must be discarded.
- The effect of a fall is the same for all eggs.
- If an egg breaks when dropped, then it would break if dropped from a higher window.
- If an egg survives a fall, then it would survive a shorter fall.
- It is not ruled out that the first-floor windows break eggs, nor is it ruled out that eggs can survive the 36th-floor windows.
- If only one egg is available and we wish to be sure of obtaining the right result, the experiment can be carried out in only one way. Drop the egg from the first-floor window; if it survives, drop it from the second-floor window. Continue upward until it breaks. In the worst case, this method may require 36 droppings. Suppose 2 eggs are available. What is the least number of egg-droppings that is guaranteed to work in all cases?
To derive a dynamic programming functional equation for this puzzle, let the state of the dynamic programming model be a pair s = (n,k), where
- n = number of test eggs available, n = 0, 1, 2, 3, ..., N − 1.
- k = number of (consecutive) floors yet to be tested, k = 0, 1, 2, ..., H − 1.
For instance, s = (2,6) indicates that two test eggs are available and 6 (consecutive) floors are yet to be tested. The initial state of the process is s = (N,H) where N denotes the number of test eggs available at the commencement of the experiment. The process terminates either when there are no more test eggs (n = 0) or when k = 0, whichever occurs first. If termination occurs at state s = (0,k) and k > 0, then the test failed.
Now, let
- W(n,k) = minimum number of trials required to identify the value of the critical floor under the worst-case scenario given that the process is in state s = (n,k).
Then it can be shown that[13]
- W(n,k) = 1 + min{max(W(n − 1, x − 1), W(n,k − x)): x = 1, 2, ..., k }
with W(n,0) = 0 for all n > 0 and W(1,k) = k for all k. It is easy to solve this equation iteratively by systematically increasing the values of n and k.
An interactive online facility is available for experimentation with this model as well as with other versions of this puzzle (e.g. when the objective is to minimize the expected value of the number of trials.)[13]
Faster DP solution using a different parametrization[edit]
Notice that the above solution takes time with a DP solution. This can be improved to time by binary searching on the optimal in the above recurrence, since is increasing in while is decreasing in , thus a local minimum of is a global minimum. Also, by storing the optimal for each cell in the DP table and referring to its value for the previous cell, the optimal for each cell can be found in constant time, improving it to time. However, there is an even faster solution that involves a different parametrization of the problem:
Let be the total number of floors such that the eggs break when dropped from the th floor (The example above is equivalent to taking ).
Let be the minimum floor from which the egg must be dropped to be broken.
Let be the maximum number of values of that are distinguishable using tries and eggs.
Then for all .
Let be the floor from which the first egg is dropped in the optimal strategy.
If the first egg broke, is from to and distinguishable using at most tries and eggs.
If the first egg did not break, is from to and distinguishable using tries and eggs.
Therefore, .
Then the problem is equivalent to finding the minimum such that .
To do so, we could compute in order of increasing , which would take time.
Thus, if we separately handle the case of , the algorithm would take time.
But the recurrence relation can in fact be solved, giving , which can be computed in time using the identity for all .
Since for all , we can binary search on to find , giving an algorithm. [14]
Matrix chain multiplication[edit]
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Matrix chain multiplication is a well-known example that demonstrates utility of dynamic programming. For example, engineering applications often have to multiply a chain of matrices. It is not surprising to find matrices of large dimensions, for example 100×100. Therefore, our task is to multiply matrices . As we know from basic linear algebra, matrix multiplication is not commutative, but is associative; and we can multiply only two matrices at a time. So, we can multiply this chain of matrices in many different ways, for example:
- ((A1 × A2) × A3) × ... An
- A1×(((A2×A3)× ... ) × An)
- (A1 × A2) × (A3 × ... An)
and so on. There are numerous ways to multiply this chain of matrices. They will all produce the same final result, however they will take more or less time to compute, based on which particular matrices are multiplied. If matrix A has dimensions m×n and matrix B has dimensions n×q, then matrix C=A×B will have dimensions m×q, and will require m*n*q scalar multiplications (using a simplistic matrix multiplication algorithm for purposes of illustration).
For example, let us multiply matrices A, B and C. Let us assume that their dimensions are m×n, n×p, and p×s, respectively. Matrix A×B×C will be of size m×s and can be calculated in two ways shown below:
- Ax(B×C) This order of matrix multiplication will require nps + mns scalar multiplications.
- (A×B)×C This order of matrix multiplication will require mnp + mps scalar calculations.
Let us assume that m = 10, n = 100, p = 10 and s = 1000. So, the first way to multiply the chain will require 1,000,000 + 1,000,000 calculations. The second way will require only 10,000+100,000 calculations. Obviously, the second way is faster, and we should multiply the matrices using that arrangement of parenthesis.
Therefore, our conclusion is that the order of parenthesis matters, and that our task is to find the optimal order of parenthesis.
At this point, we have several choices, one of which is to design a dynamic programming algorithm that will split the problem into overlapping problems and calculate the optimal arrangement of parenthesis. The dynamic programming solution is presented below.
Let's call m[i,j] the minimum number of scalar multiplications needed to multiply a chain of matrices from matrix i to matrix j (i.e. Ai × .... × Aj, i.e. i<=j). We split the chain at some matrix k, such that i <= k < j, and try to find out which combination produces minimum m[i,j].
The formula is:
if i = j, m[i,j]= 0
if i < j, m[i,j]= min over all possible values of k (m[i,k]+m[k+1,j] + )
where k is changed from i to j − 1.
- is the row dimension of matrix i,
- is the column dimension of matrix k,
- is the column dimension of matrix j.
This formula can be coded as shown below, where input parameter "chain" is the chain of matrices, i.e. :
function OptimalMatrixChainParenthesis(chain)
n = length(chain)
for i = 1, n
m[i,i] = 0 //since it takes no calculations to multiply one matrix
for len = 2, n
for i = 1, n - len + 1
for j = i + 1, len -1
m[i,j] = infinity //so that the first calculation updates
for k = i, j-1
q = m[i, k] + m[k+1, j] +
if q < m[i, j] // the new order of parenthesis is better than what we had
m[i, j] = q //update
s[i, j] = k //record which k to split on, i.e. where to place the parenthesis
So far, we have calculated values for all possible m[i, j], the minimum number of calculations to multiply a chain from matrix i to matrix j, and we have recorded the corresponding "split point"s[i, j]. For example, if we are multiplying chain A1×A2×A3×A4, and it turns out that m[1, 3] = 100 and s[1, 3] = 2, that means that the optimal placement of parenthesis for matrices 1 to 3 is and to multiply those matrices will require 100 scalar calculation.
This algorithm will produce "tables" m[, ] and s[, ] that will have entries for all possible values of i and j. The final solution for the entire chain is m[1, n], with corresponding split at s[1, n]. Unraveling the solution will be recursive, starting from the top and continuing until we reach the base case, i.e. multiplication of single matrices.
Therefore, the next step is to actually split the chain, i.e. to place the parenthesis where they (optimally) belong. For this purpose we could use the following algorithm:
function PrintOptimalParenthesis(s, i, j) if i = j print "A"i else print "(" PrintOptimalParenthesis(s, i, s[i, j]) PrintOptimalParenthesis(s, s[i, j] + 1, j) ")"
Of course, this algorithm is not useful for actual multiplication. This algorithm is just a user-friendly way to see what the result looks like.
To actually multiply the matrices using the proper splits, we need the following algorithm:
function MatrixChainMultiply(chain from 1 to n) // returns the final matrix, i.e. A1×A2×... ×An
OptimalMatrixChainParenthesis(chain from 1 to n) // this will produce s[ . ] and m[ . ] "tables"
OptimalMatrixMultiplication(s, chain from 1 to n) // actually multiply
function OptimalMatrixMultiplication(s, i, j) // returns the result of multiplying a chain of matrices from Ai to Aj in optimal way
if i < j
// keep on splitting the chain and multiplying the matrices in left and right sides
LeftSide = OptimalMatrixMultiplication(s, i, s[i, j])
RightSide = OptimalMatrixMultiplication(s, s[i, j] + 1, j)
return MatrixMultiply(LeftSide, RightSide)
else if i = j
return Ai // matrix at position i
print "error, i <= j must hold"
function MatrixMultiply(A, B) // function that multiplies two matrices
if columns(A) = rows(B)
for i = 1, rows(A)
for j = 1, columns(B)
C[i, j] = 0
for k = 1, columns(A)
C[i, j] = C[i, j] + A[i, k]*B[k, j]
return C
print "error, incompatible dimensions."
The term dynamic programming was originally used in the 1940s by Richard Bellman to describe the process of solving problems where one needs to find the best decisions one after another. By 1953, he refined this to the modern meaning, referring specifically to nesting smaller decision problems inside larger decisions,[15] and the field was thereafter recognized by the IEEE as a systems analysis and engineering topic. Bellman's contribution is remembered in the name of the Bellman equation, a central result of dynamic programming which restates an optimization problem in recursive form.
Bellman explains the reasoning behind the term dynamic programming in his autobiography, Eye of the Hurricane: An Autobiography (1984). He explains:
- "I spent the Fall quarter (of 1950) at RAND. My first task was to find a name for multistage decision processes. An interesting question is, Where did the name, dynamic programming, come from? The 1950s were not good years for mathematical research. We had a very interesting gentleman in Washington named Wilson. He was Secretary of Defense, and he actually had a pathological fear and hatred of the word research. I'm not using the term lightly; I'm using it precisely. His face would suffuse, he would turn red, and he would get violent if people used the term research in his presence. You can imagine how he felt, then, about the term mathematical. The RAND Corporation was employed by the Air Force, and the Air Force had Wilson as its boss, essentially. Hence, I felt I had to do something to shield Wilson and the Air Force from the fact that I was really doing mathematics inside the RAND Corporation. What title, what name, could I choose? In the first place I was interested in planning, in decision making, in thinking. But planning, is not a good word for various reasons. I decided therefore to use the word "programming". I wanted to get across the idea that this was dynamic, this was multistage, this was time-varying. I thought, let's kill two birds with one stone. Let's take a word that has an absolutely precise meaning, namely dynamic, in the classical physical sense. It also has a very interesting property as an adjective, and that it's impossible to use the word dynamic in a pejorative sense. Try thinking of some combination that will possibly give it a pejorative meaning. It's impossible. Thus, I thought dynamic programming was a good name. It was something not even a Congressman could object to. So I used it as an umbrella for my activities."
The word dynamic was chosen by Bellman to capture the time-varying aspect of the problems, and because it sounded impressive.[10] The word programming referred to the use of the method to find an optimal program, in the sense of a military schedule for training or logistics. This usage is the same as that in the phrases linear programming and mathematical programming, a synonym for mathematical optimization.[16]
The above explanation of the origin of the term is lacking. As Russell and Norvig in their book [17] have written, referring to the above story: "This cannot be strictly true, because his first paper using the term (Bellman, 1952) appeared before Wilson became Secretary of Defense in 1953." Also, there is a comment in a speech by Harold J. Kushner, where he remembers Bellman. Quoting Kushner as he speaks of Bellman: "On the other hand, when I asked him the same question, he replied that he was trying to upstage Dantzig's linear programming by adding dynamic. Perhaps both motivations were true."
Algorithms that use dynamic programming[edit]
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- Recurrent solutions to lattice models for protein-DNA binding
- Backward induction as a solution method for finite-horizon discrete-time dynamic optimization problems
- Method of undetermined coefficients can be used to solve the Bellman equation in infinite-horizon, discrete-time, discounted, time-invariant dynamic optimization problems
- Many string algorithms including longest common subsequence, longest increasing subsequence, longest common substring, Levenshtein distance (edit distance)
- Many algorithmic problems on graphs can be solved efficiently for graphs of bounded treewidth or bounded clique-width by using dynamic programming on a tree decomposition of the graph.
- The Cocke–Younger–Kasami (CYK) algorithm which determines whether and how a given string can be generated by a given context-free grammar
- Knuth's word wrapping algorithm that minimizes raggedness when word wrapping text
- The use of transposition tables and refutation tables in computer chess
- The Viterbi algorithm (used for hidden Markov models)
- The Earley algorithm (a type of chart parser)
- The Needleman–Wunsch algorithm and other algorithms used in bioinformatics, including sequence alignment, structural alignment, RNA structure prediction
- Floyd's all-pairs shortest path algorithm
- Optimizing the order for chain matrix multiplication
- Pseudo-polynomial time algorithms for the subset sum, knapsack and partition problems
- The dynamic time warping algorithm for computing the global distance between two time series
- The Selinger (a.k.a. System R) algorithm for relational database query optimization
- De Boor algorithm for evaluating B-spline curves
- Duckworth–Lewis method for resolving the problem when games of cricket are interrupted
- The value iteration method for solving Markov decision processes
- Some graphic image edge following selection methods such as the "magnet" selection tool in Photoshop
- Some methods for solving interval scheduling problems
- Some methods for solving the travelling salesman problem, either exactly (in exponential time) or approximately (e.g. via the bitonic tour)
- Recursive least squares method
- Beat tracking in music information retrieval
- Adaptive-critic training strategy for artificial neural networks
- Stereo algorithms for solving the correspondence problem used in stereo vision
- Seam carving (content-aware image resizing)
- The Bellman–Ford algorithm for finding the shortest distance in a graph
- Some approximate solution methods for the linear search problem
- Kadane's algorithm for the maximum subarray problem
See also[edit]
- Convexity in economics
- Greedy algorithm
- Non-convexity (economics)
- Stochastic programming
- Stochastic dynamic programming
- ^ Cormen, T. H.; Leiserson, C. E.; Rivest, R. L.; Stein, C. (2001), Introduction to Algorithms (2nd ed.), MIT Press & McGraw–Hill, ISBN 0-262-03293-7 . pp. 327–8.
- ^ DeLisi, Biopolymers, 1974, Volume 13, Issue 7, pages 1511–1512, July 1974
- ^ Gurskiĭ GV, Zasedatelev AS, Biofizika, 1978 Sep-Oct;23(5):932-46
- ^ "M. Memo". J Vocabulary. J Software. Retrieved 28 October 2011.
- ^ Stokey et al., 1989, Chap. 1
- ^ Sniedovich, M. (2006), "Dijkstra's algorithm revisited: the dynamic programming connexion" (PDF), Journal of Control and Cybernetics, 35 (3): 599–620. Online version of the paper with interactive computational modules.
- ^ Denardo, E.V. (2003), Dynamic Programming: Models and Applications, Mineola, NY: Dover Publications, ISBN 978-0-486-42810-9
- ^ Sniedovich, M. (2010), Dynamic Programming: Foundations and Principles, Taylor & Francis, ISBN 978-0-8247-4099-3
- ^ Dijkstra 1959, p. 270
- ^ a b Eddy, S. R., What is dynamic programming?, Nature Biotechnology, 22, 909–910 (2004).
- ^ Moshe Sniedovich (2002), "OR/MS Games: 2. The Towers of Hanoi Problem,", INFORMS Transactions on Education, 3 (1): 34–51.
- ^ Konhauser J.D.E., Velleman, D., and Wagon, S. (1996). Which way did the Bicycle Go? Dolciani Mathematical Expositions – No 18. The Mathematical Association of America.
- ^ a b Sniedovich, M. (2003). The joy of egg-dropping in Braunschweig and Hong Kong. INFORMS Transactions on Education, 4(1) 48–64.
- ^ Dean Connable Wills, Connections between combinatorics of permutations and algorithms and geometry
- ^ Stuart Dreyfus. "Richard Bellman on the birth of Dynamical Programming".
- ^ Nocedal, J.; Wright, S. J.: Numerical Optimization, page 9, Springer, 2006..
- ^ Russel, S. and Norvig, P., Artificial Intelligence: A Modern Approach, 3rd edition, Prentice Hall, 2009.
Further reading[edit]
- Adda, Jerome; Cooper, Russell (2003), Dynamic Economics, MIT Press. An accessible introduction to dynamic programming in economics. The link contains sample programs.
- Bellman, Richard (1954), "The theory of dynamic programming", Bulletin of the American Mathematical Society, 60 (6): 503–516, doi:10.1090/S0002-9904-1954-09848-8, MR 0067459. Includes an extensive bibliography of the literature in the area, up to the year 1954.
- Bellman, Richard (1957), Dynamic Programming, Princeton University Press. Dover paperback edition (2003), ISBN 0-486-42809-5.
- Bertsekas, D. P. (2000), Dynamic Programming and Optimal Control (2nd ed.), Athena Scientific, ISBN 1-886529-09-4. In two volumes.
- Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001), Introduction to Algorithms (2nd ed.), MIT Press & McGraw–Hill, ISBN 0-262-03293-7. Especially pp. 323–69.
- Dreyfus, Stuart E.; Law, Averill M. (1977), The Art and Theory of Dynamic Programming, Academic Press, ISBN 978-0-12-221860-6.
- Giegerich, R.; Meyer, C.; Steffen, P. (2004), "A Discipline of Dynamic Programming over Sequence Data" (PDF), Science of Computer Programming, 51 (3): 215–263, doi:10.1016/j.scico.2003.12.005.
- Meyn, Sean (2007), Control Techniques for Complex Networks, Cambridge University Press, ISBN 978-0-521-88441-9.
- S. S. Sritharan (1991), "Dynamic Programming of the Navier-Stokes Equations", Systems and Control Letters, 16(4), 299–307.
- Stokey, Nancy; Lucas, Robert E.; Prescott, Edward (1989), Recursive Methods in Economic Dynamics, Harvard Univ. Press, ISBN 978-0-674-75096-8.
External links[edit]
This article's use of external links may not follow Wikipedia's policies or guidelines. (March 2016) (Learn how and when to remove this template message) |
- An Introduction to Dynamic Programming
- Dynamic Optimization Online Course
- Dyna, a declarative programming language for dynamic programming algorithms
- Wagner, David B., 1995, "Dynamic Programming." An introductory article on dynamic programming in Mathematica.
- Ohio State University: CIS 680: class notes on dynamic programming, by Eitan M. Gurari
- A Tutorial on Dynamic programming
- MIT course on algorithms – Includes a video lecture on DP along with lecture notes
- More DP Notes
- King, Ian, 2002 (1987), "A Simple Introduction to Dynamic Programming in Macroeconomic Models." An introduction to dynamic programming as an important tool in economic theory.
- Dynamic Programming: from novice to advanced A TopCoder.com article by Dumitru on Dynamic Programming
- Algebraic Dynamic Programming – a formalized framework for dynamic programming, including an entry-level course to DP, University of Bielefeld
- Dreyfus, Stuart, "Richard Bellman on the birth of Dynamic Programming."
- Dynamic programming tutorial
- A Gentle Introduction to Dynamic Programming and the Viterbi Algorithm
- Tabled Prolog BProlog and XSB
- Online interactive dynamic programming modules including, shortest path, traveling salesman, knapsack, false coin, egg dropping, bridge and torch, replacement, chained matrix products, and critical path problem.


text:
"Aquilani has a deal with Liverpool for other two years. I was there last week. Within few days we will know the club's intentions. The boy would be happy to come back to Liverpool, but only if he can have the right possibilities. If Ac Milan wanted to redeem it, they should have done before. " These the words of Franco Zavaglia, the agent of the midfielder Alberto Aquilani to Radio Manà Manà Sport.
"Aquilani wants only to play and to show his worth. He has been unlucky during his career and the ankle injury, occurred 3 years ago was not handled well".

subset: commoncrawl


text:
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text:
Best Business Web Sites: Economics
Sponsored by the BRASS Education Committee
Maintained byLinda Hauck
Villanova University
Econ Data & Links
Gives the current value of economic statistic in table format. It provides quick access to a variety of up-to-date statistics. When it was feasible the links in the table were chosen so the user has access to time series or data for other areas and to related data.
Economic Working Papershttp://netec.wustl.edu/local/search.html
NetEc is an international academic effort to improve the communication of economics via electronic media. It consists of: information on printed working papers on BibEc, data about electronic working papers on WoPEc, code for economics and econometrics on CodEc, World Wide Web resources in economics on WebEc, home page papers in economics on HoPEc, jokes about economists and economics on JokEc.
Resources for Economists on the Internethttp://rfe.org
This guide lists more than 900 resources on the Internet to those interested in economics.
International Economic Conditions
Annual Report of the Bank of International Settlements
The BIS's predominant tasks are "to promote the cooperation of central banks and to provide additional facilities for international financial operations". The annual report contains data on industrial and emerging markets.
Demographic Yearbook (United Nations)http://unstats.un.org/unsd/demographic/products/dyb/default.htm
Standardized annual data relating to population and social concerns from nearly every nation in the world.
Monthly Bulletin of Statistics Online (United Nations)http://unstats.un.org/unsd/mbs/default.asp
Current economic statistics from most nations of the world.
Membership in the Organization for Economic Co-operation and Development (OECD) consists of countries committed to democracy and the market economy. The organization is engaged in many programs relating to economic development and world trade. Click on "Statistics" to enter the OECD Statistics Portal to access detailed country data relating to 26 main topics.
Statistical Agencies (International) – U.S. Census Bureauhttp://www.census.gov/main/www/stat_int.html
Information on National Statistical Systems – United Nations
The two resources, above, provide links to national statistical offices worldwide, arranged by region. Most data is offered in the official languages of each nation, with English occasionally given as an alternative language.
World Development Indicatorshttp://www.worldbank.org/data/countrydata/countrydata.html
Time series statistics on national accounts, international transactions, manufacturing, andmonetary indicators are provided for World Bank member countries.
U.S. Economic Indicators
Federal Open Market Committee (FOMC)
The FOMC controls U.S. monetary policy through open market sale of Treasury securities, regulation of the interest rate charged on loans to commercial banks and other depository institutions, and regulation of amount of funds depository institutions must have on hand against depository liabilities. Policy decisions are made in response to economic dynamics across the U.S. Regional economic conditions are assessed by each Federal Reserve Bank District and reported in the Beige Book: Summary of Commentary on Current Economic Conditions.
Federal Reserve Economic Data (FRED)http://research.stlouisfed.org/fred2/
Created by St. Louis Federal Reserve District, FRED provides historical U.S. economic and financial data, including daily U.S. interest rates, monetary and business indicators, exchange rates, balance of payments and regional economic data.
Monthly Labor Reviewhttp://stats.bls.gov/opub/mlr/mlrhome.htm
Contains articles as well as a monthly statistics section on employment and earnings. Covers less detail but a longer range of years than Employment and Earnings.
Statistical Abstract of the United Stateshttp://www.census.gov/compendia/statab/
A well indexed, comprehensive collection of statistics in tabular format. Includes basic statistics for a wide range of economic indicators. The index leads to specific table numbers. Each table lists the source of the statistics for further reference.
Survey of Current Businesshttp://www.bea.gov/scb/index.htm
Includes monthly and quarterly cyclical indicator statistics. Tables of economic indicators cover such statistics as personal income, industrial productivity, business sales and inventories, commodity prices, producer prices, labor force, and earnings. Statistical series usually give monthly statistics for the last year and annual statistics for the previous two years.
Treasury Bulletinhttp://www.fms.treas.gov/bulletin/index.html
This report of federal fiscal operations includes details of federal debt, bonds, notes, market yield, and other federal financial obligations. Positions of major foreign currencies are included, along with a profile of the domestic economy including growth of GDP, federal deficit, and rate of personal savings.
Last updated: 03/28/08
Previously maintained by:
Dennis Smith, University of Pittsburgh, and Elisabeth Leonard, University of California, San Diego
Kevin Harwell, Penn State University
List originally compiled by Jennifer Boettcher, Georgetown University
Sponsored by the BRASS Education Committee


text:
this wiki
For a list of player characters in the Fallout series of games, see player character. |
Narg would like to prove his worthiness to lead the tribe and will let nothing stand in his way.
"— Character selection descriptionNarg is one of three pre-made player characters available during character creation in Fallout 2.
Narg's exceptional physique has made him one of the best hunters in the tribe. Narg's first, and usually only, impulse is to crush anything that he can't figure out. Narg has become quite adept at crushing, and slicing, and dicing. Narg would like to prove his worthiness to lead the tribe and he'll let nothing stand in his way. Narg is a warrior by birth. A mighty constitution, great endurance and lightning-fast reflexes, as well as an excellent knowledge of the Wasteland made him one of the best hunters of the tribe. He has no fear left for the perils or long journeys. Spears and knives are Narg's weapons of choice. He is young, fearless, and determined to prove his loyalty to Arroyo, no matter the cost.
Narg appears only in Fallout 2.


text: What Happened After Raw Tonight?Following Raw tonight, John Cena defeated Ryback in the dark match main event via pinfall after hitting the AA. The Wyatt Family Makes its WWE TV DebutFor those who missed it, Bray Wyatt and the Wyatt Family, consisting of Bray, Luke Harper and Erick Rowan, made their TV debuts on Raw tonight. They came out and attacked Kane following his match against Christian, and Bray cut a promo declaring that "they're here." |


text:
Apexicon is back. Last year's Kickstarter didn't work out, not by a long shot, but the team are back with a reduced target, and a few more months of work and experience under their belts. The game looks like it could be the spiritual sequel to take the place of the actual sequel that Puzzle Quest is weirdly lacking. There are strong differences between the various classes, and special abilities and board control are more important than owning a large dictionary (or internet connection). Essentially, it's turn-based wordplay combat with RPG elements and that is something I would like to play.
Posts Tagged 'actos games'
Letters, Red In Tooth And Claw: Apexicon Kickstarter
Forget Plants vs Zombies and Peggle – I'm still waiting for the next Bookworm Adventures game. Using words as a weapon is (almost) always fun. Why, if Subutai's CLANG had been called SCRIBBLE and been about wielding pens instead of swords, we'd all be happily writing virtual reality letters to one another right now. Apexicon is a Kickstarter for a wordy game and the campaign page that opens with a torrent of namechecks, like a rapper making a comeback and thanking his crew. I say torrent but there are only three names – Puzzle Quest, Scrabble and Bookworm Adventures. Actos Games are hoping to create a fusion of those three.


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Anesthesia |
Female urinary retention after surgery01/23/2006 |
I rececently had abdominoplasty. Trouble with urination began the evening after surgery. Unable to empty bladder. Undetected by surgeon four days after surgery. Also was taking Mepergan for pain. On fifth day urologist was consulted and he diagnosed this as an anesthestia complication and inserted foley catheter. Must wear this for five days. Was the problem caused by the anesthesia or the continued use of Mepergan?
Your problem is known as postoperative urinary retention. Urinary retention is the failure to empty the bladder when it is full of urine. There are a number of things associated with urinary retention after surgery, which is a surprisingly common condition. These include age (older), previous urinary problems, a large amount of fluid received during or after surgery, prostate enlargement (men), site of surgery (hernia), and type of anesthesia (spinal or epidural anesthesia).
In a recent study which used ultrasound to measure the amount of urine in the bladder, the best predictor of urinary retention was the amount of urine in the bladder immediately after surgery. If the full bladder is not emptied this can cause persistent problems because of the prolonged stretching of the bladder (detrusor) muscle. Unfortunately many patients with full bladders after surgery cannot tell that their bladder is full. Ultrasound machines are not usually available (except in research studies!) and recovery room nurses must rely on the patient's symptoms or on their ability to assess the bladder size by feeling the patient's abdomen. This is a very inaccurate method and is why the authors of the study recommend that ultrasound equipment be used routinely in the recovery room.
If urinary retention is recognized in the recovery room, the bladder can be emptied with a urinary catheter. This is a tube placed through the urethra into the bladder. The catheter may be retained for a time or may be used to empty the bladder and then is removed. When surgery is prolonged, or the operation is one in which a lot of fluid must be given or blood is lost, it is usual practice to insert a urinary catheter at the beginning of the procedure. This will help prevent problems of urinary retention in the early recovery period. The decision to place a urinary catheter is usually made by the surgical team and the anesthesiologist.
Another cause of urinary retention is the use of medication with "anticholinergic" properties. Among the many medicines that have this effect is promethazine, one of the drugs contained in Mepergan (a combination of meperidine, a pain killer, and promethazine). While anesthetic factors may have contributed to urinary retention, you can see that this is a not uncommon condition (16% in the study I've quoted), and there are a variety of possible causes, and of methods for detection or early prevention.
I hope you recover soon. For more information I suggest you speak directly with your surgeon and with your anesthesiologist, who is likely to be very glad to respond to your concerns.
Gareth S Kantor, MD Assistant Professor of Anesthesiology School of Medicine Case Western Reserve University |


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- lists
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Common words appear frequently in written and spoken language across many genres from radio to academic journals.
Informal words should be reserved for casual, colloquial communication.
correctional facility
Synonyms for house-correction
correctional facility
More words related to house-correction
noun. correctional facility
Roget's 21st Century Thesaurus, Third Edition Copyright © 2013 by the Philip Lief Group.
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IBM Rational License Key Administrator is a program that helps you enter or import license keys and change your license configuration on the Rational License Key Server. Your organization can track or monitor licenses and evaluate the licenses of various products that organization uses.
QuuSoft Duplicate Cleaner will be the most handy tool to solve your problems. Don't waste your time browsing through directories to find duplicates, leave this to the excellent program. With QuuSoft Duplicate Cleaner, you can free up disk space and permanently destroy the data within the Recycle Bin, removing all traces of previously deleted files.
Duplicate Cleaner Pro can help you save storage space by identifying and eliminating duplicate files. The tool provides three types of scans. First, there is Regular Mode, which finds duplicates mainly comparing the actual contents of the files. Second, there is Image Mode, which groups pictures according to their contents.
Mirrakey provides a simple-to-use Software Licensing Key system that is powerful and flexible. Mirrakey is an ActiveX DLL and supports any development environment that is COM (Component Object Model) compliant. Mirrakey is also flexible enough to allow you to select one or more of three properties (ProductName, SerialNo and UserName) to generate your software's license key.
PELock - is an application that allows you to protect Windows executable files (so called exe-protector). Besides that, PELock allows easy creation of application of evaluation-demo type, which use license key system.
The Key Creator extends the standard .NET licensing mechanism to provide a secure means of licensing .NET components and applications using encrypted license keys. Key Creator uses simple mechanism with proven and unbreakable encryption algorithm to generate the License Key. Key Creator is a License key generator utility.
License4J License Manager can generate license text, floating license text and license keys. The program stores all products, templates, licenses, and settings in a database. The database can be on a local embedded Derby database or on database servers: MySQL, PostgreSQL, and Microsoft SQL Server.
Smaple License is a smart and simple way to make your Software as Copy Protected. This will generate License Keys for your Software based on the individual Machine. The License Keys can be generated along with a Validity date - This enables creation of the Demo version of your Software easy and simple.
This free program lets you find all the duplicated files in your system and delete those that you are not using. It has a nice interface with lots of options that make it easier to find precisely the types of files that are duplicated. The program is free to download and use, but it has some limitations.
Glary Duplicate Cleaner is a program that examines your hard drive for duplicate files. It will deep scan for all types of files takes only one or two clicks and send them to the recycle bin with your permission. You can also filter your duplicate search results by file type.
Windows License Key Dump is a free command-line tool designed to recover the product/serial keys of Windows and other popular applications such as Office, SQL Server, Adobe, Nero and many more. By default it dumps all the recovered license key list to console, but also you can save the Windows License Key list to TEXT file.
This program lets you find all the duplicate files in your system and select those you want to discard. It performs a thorough scanning of your system and detects those duplicated files according to different criteria. Once you have found those files, you can either send them to the recycle bin, save them in a new folder, or "pulverize" them, that is, overwrite them so they cannot be recovered.
Product Key Explorer is intended to retrieve the serial keys of already installed products, which definitely comes very handy if you have lost the original data. This way, you can reinstall these software products using the same key. This tool can recover not only your operating system's key but also those of other products, and luckily, more than 8,000 programs are supported.
Sooner or later, we all end up storing duplicate media files and duplicate documents on our computers inadvertently. Duplicate File Remover is a robust and easy-to-use tool to help us get rid of all those unnecessary files in just 6 easy steps. Its wizard-driven interface is suitable for all kinds of users, and all searches can be easily customized to fit our specific needs and preferences.
Product Key Finder is a free Windows key finder applications that retrieves license key information (CD Keys) from the Windows registry. Product Key Finder currently supports retrieval of 200+ licenses from applications such Microsoft Windows, Microsoft Office and Adobe Acrobat.
Keeping track of possible dupes will not only help you keep your file collection organized, but will also allow you to recover unnecessarily occupied disk space. Duplicate File Detective provides you with all the features you need to accomplish both objectives. Its multi-paneled interface and wizard-driven processes make the entire search for duplicates a simple and rewarding task.
Serial Key Maker is a powerful program that enables you to create secure software license keys. You can create time-limited, demo and non-expiring keys, create multiple keys in one click, quickly encrypt strings using your own private key and lock a license key to a particular machine.
Pepakura Designer allows you to create cutout paper crafts from 3D files in a very easy manner. You only need to open a 3D file and click on the Unfold option. An unfolded cutout model will be generated automatically. The program doesn't have the capabilities for creating 3D models, so you will need another program to do it.


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MathAudio Room EQ VST: - Corrects deficiencies of room acoustics (multipoint compensation). - Corrects acoustic imperfections of speakers. - Optimized for maximum accuracy of the correction. - Corrects both amplitude and phase components of frequency response. - Quells resonance peaks of frequency response while leaving the deep notches to prevent large excursions of the speaker diaphragm. Avoids the muddy sound of conventional linearizing equalizers. - Allows correction of more than two speakers simultaneously by means of loading multiple instances of the plug-in and using different correction files in each instance. - Manually adjustable level of compensation allows one to reach the maximum transparency of the sound. - Works with USB measurement microphones (e.g. MiniDSP UMIK-1) or standard measurement microphones (e.g. NADY CM100 or Dayton Audio EMM-6). - Applies a patented method of frequency response correction.
Price | USD99.95 |
License | Free to try |
File Size | 1.96 MB |
Version | 2.5.6 |
Operating System | Windows 7 Windows Vista Windows XP Windows Server 2008 Windows 10 Windows 8 Windows Windows 2003 |


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Description: Happy Stash Quilts Flip & Fuse Fun! Book by Marcia Harmening. The Tahoe Collection. These patterns use all machine applique.! Use fusible interfacing with fabric to sew applique shapes, then flip pieces right side out and fuse in place. 5 designs include a table runner, placemats and 4 quilts. Each project gives instructions, templates, fabric requirements and photos. 23 pages in a soft cover.
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Education, politics, and anything else that catches my attention.
Wednesday, March 11, 2009
Wednesday Trivia
Grover Cleveland, 1885-1889 and 1893-1897.
Today's question is:
What is the name of Earth's 2nd highest mountain?
3 comments:
- Fritz J. said...
The answer to today's question is K2, and I remember that from the funny papers years ago--Joe Palooka I think it was. Anyhow, shortly after Edmund Hillary climbed Everest the comic strip had its hero climb K2. Then the hero supposedly discovered that K2 was actually a few feet higher than Mt. Everest. Needless to say, K2 is not higher than Everest, but then it was only a comic strip and written for entertainment.
As for yesterday's question, I read that and started trying to remember. Finally I went to bed hoping the answer would come in my sleep. Needless to say, it didn't. Then off and on today the question kept interrupting my thoughts and finally it came to me. The answer was Grover Cleveland. So I quickly opened my browser and dang, you had already posted the answer. Now I know I could have looked it up, but that would have been cheating. Ah well, the history of my life. A day late and a dollar short. - 7:08 PM
- Chanman said...
I believe it is K-9. Don't remember why it doesn't have a given name but you said we couldn't Google for answers.


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Saturday, March 24, 2012
703-5 16th Avenue
Lake Como, NJ
USATF Certified Course
Flat, fast, scenic, lakeside 5K course (closed local roads) through Lake Como, Belmar, & Spring Lake starting and ending at Bar Anticipation
The exciting start of our 2nd Annual De-FEET Breast Cancer 5K Run/Walk!
We would like to take this opportunity to thank everyone who participated in our 2nd Annual De-FEET Breast Cancer 5K Run/Walk!!
Overall, Thank You to everyone who supported our 5K by participating, donating, fundraising, sponsoring, volunteering, helping plan and anyone involved in what was a hugely successful event!
A total of 1134 registrants were part of the event, and well over 1,300 people were on site at the 5K!
It was a truly amazing day with so many participants and spectators, great on-site business vendors, amazing weather, and a fun and special Post-5K Celebration - including an unlimited family & friends BBQ, great music by DJ Funkmaster Dex, individual & team awards, our Survivor Thrive Grant Program ceremony, a special Survivor/Memorial Dedication, and amazing Tricky Tray raffle prizes and an iPad raffle!
We are extremely proud to announce that our breast cancer foundation raised over $35,000 in proceeds from our 2nd Annual De-FEET Breast Cancer 5K, which will help us tremendously with our foundation's programs and projects as we work towards our goal of positively impacting the breast cancer community both locally and beyond.
100% of these proceeds from the 5K will fund our ongoing breast cancer projects & programs at Robert Wood Johnson University Hospital, New Brunswick, help assist breast cancer patients and their families in our community, and in addition help fund Biomedical Research in search of a cure.
At our Post-5K Celebration we were very proud to give grants of $2,500 each to two local breast cancer survivors who are in need as part of our Survivors Thrive Grant Program. Your support will continue to help programs such as this which helps make a difference in our community.
5K Registrants should have received an email from one of our great event photographers (US Candids) that the pictures they took at our 5K are up on their website!
If not, or if you would like to see the pictures again, click on this link below from US Candids and type in 3/24/12 or mama mare and the many pictures they took at our 5K will come up!
You can also search by name or bib # to find your pictures if your bib # is visible in the picture.
You can order any of these photos from the site and based on the amount ordered the photographer will donate to our foundation a portion of the proceeds from the photo sales after their costs - which is amazing!
Click below for all 5K results from our timing company, Best Racing Systems!
Click below for a great video from our videographer, Blunt Object Productions!
We will begin to plan our 3rd Annual De-FEET Breast Cancer 5K Run/Walk shortly, so please check back for updates!
ANY questions regarding our 5K, please contact: email@example.com
Warmest Regards to all of you and THANK YOU everyone for your support always!
Jonathan Arroyo, Krista Kasper and the Mama Mare Breast Cancer Foundation
Jonathan Arroyo
Vice Chair
5K Event Co-Director
Krista Kasper
5K Event Co-Director
Our Proud Sponsors of this Event
2nd Annual De-Feet Breast Cancer 5K Walk/Run
Contact Us | Click Here to Make a Donation
© 2011 The Mama Mare Breast Cancer Foundation. All rights reserved.
7th Annual De-FEET Breast Cancer 5K Run/Walk
"Fight like a Superhero" Dinner & Dance Fundraiser
The Second Annual Bikers for Boobs Motorcycle Run
6th Annual De-FEET Breast Cancer 5K Run/Walk
"Mad Hatter" Dinner & Dance Fundraiser
5th Annual Drink for Pink Pub Crawl
Bikers for Boobs Motorcycle Run
5th Annual De-FEET Breast Cancer 5K Run/Walk
Mardi Gras Celebrating Survivors
3rd Annual Hope to Heal Dinner of Excellence
Golden Bear Chargers Play Against Breast Cancer


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Studio by Buttes Charmont Park
Maksimum Konuk Sayısı2
Yatak odası1
Boyut12 metrekare
Floor No.4
- Mutfak
- Havlu
- İnternet/Kablosuz ağ
Accommodation summary
Adres | Rue François Pinton, 75019 Paris, Fransa |
Mülkiyet türü | Daire |
Referans kodu | 7LLPAEZ5 |
Acceptance rate | 100% |
Response time | A few hours |
Calendar updated | 2 days ago |
The cosy studio includes a double sofa bed, a compact kitchenette, and an office area. The kitchen facilities include a hob and a kettle. There is a private tiled bathroom with a walk-in shower and a hairdryer.
The beautiful Canal Saint-Martin is just a 15-minute walk away. Notre Dame Cathedral can be reached within 20 minutes' journey by Paris Metro.
The area is serviced by great transport connections around Paris by metro and various bus routes.
Ev kuralları
No smoking.No pets.
Please note that the building does not have an elevator.
Ücretler |
Monthly rate for up to 2 guests | TRY 4006 |
Detaylar |
Check-in Saati (Sonrası) | esnek |
Check-in before | esnek |
Check-out Saati (Öncesi) | esnek |
İptal Koşulları |
Bu mülk için yapılan yorumlar
very friendly and welcoming host / perfect accommodation, small but very practically equipped, I'd definitely recommend this studio!
Merci pour tous Claire!
We stayed at Claires appartement for four days and it was perfect for us. All works without any problems. Claire answered very quick to our questions so that we havent had any problems with getting the keys and so on. Therefor we can highly recommend Claire as your host.
The apartment is just perfect for a weekend trip to Paris. Despite being very small (which was how expected it to be), it provided everything needed and it was very clean and neat, which made it easy to feel comfortable. The neighbourhood is quiet and relaxed, we felt perfectly safe and the metro connection is good. Claire was very helpful and flexible regarding check-in and check-out, she was always available, responding very quickly to our requests. I can recommend this apartment strongly and will keep it in mind for our next trip to Paris!
Good location. The apartment was clean and provided with everything you might need. It is very small but perfect for a couple
This appartment is beautiful! In a great zone, new and very confortable. Near the metro and market. Our stayment was grat.
- Kalite güvenceli konaklama
- Her ev sahibi telefonla doğrulanmıştır
- SSL şifrelemesi ile güvenli ödeme


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PISCES : PERCIFORMES : Gobiidae | BONY FISH |
Gobius niger |
Description: This is a large and stout bodied goby with adults reaching a maximum length of approximately 17cm. It is usually greyish-brown in colour with darker and paler mottling and patches. Despite its common name it is not as dark as breeding males of the rock goby (Gobius paganellus). Lines of black sensory papillae are visible on the head. The most distinctive characteristic is the elongate, first dorsal fin, this is most obvious in males.
Habitat: The black goby lives in sandy or muddy areas or on rock that is covered by sediment. It can tolerate reduced salinity and so is also found in estuaries, lagoons and sea loughs. It mainly feeds on small crustaceans, small fish, molluscs and worms.
Distribution: This species is widespread all around the coasts of Britain and Ireland.
Similar Species: The black goby is sometimes confused with the rock goby (Gobius paganellus) however the latter is confined to rocky areas whilst the black goby lives in sandy or muddy areas. The two species can easily be differentiated by the shape of the first dorsal fin which is elongate and triangular in the black goby.
Key Identification Features:
Distribution Map from NBN: Grid map (fast) : Interactive map (slower, requires login to view records) : National Biodiversity Network mapping facility, data for UK.
Picton, B.E. & Morrow, C.C. (2016). Gobius niger (Linnaeus, 1758). [In] Encyclopedia of Marine Life of Britain and Ireland. http://www.habitas.org.uk/marinelife/species.asp?item=ZG7230 Accessed on 2017-03-29 |
Copyright © National Museums of Northern Ireland, 2002-2015 |


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Mylan is back with some serious prog-tech house beats on "Doubt EP", a super-charged three tracker filled with all the wonders of house.
First up is the title track "Doubt", surreal, dark hypnotic beats and percussions carve out a cool rhythmic vibe with a tribal element. Some very nice synth and FX work really adds a new flavour to the sound, but the big focus remains on those percussive grooves. A dark and sinister vocal pops in from time to time and the overall darker theme is complete.
"Replace" is a superbly crafted experience in the frequencies of bass, heart throbbing basses pulsate throughout this track and give it a very distinct vibe. Also on offer here, apart from super cool vocal samples, is another brilliant percussion and drum arrangement with some very funky and cool housey chords keeping this groove ever flowing.
"Again" continues the overall theme of the EP with a darker edge and tech in mind, more low end basses that will have any sub speaker shaking and a real dense thud on the kick will get any floor moving to the beat. Top quality FX work blends in perfectly with sweeping pads and that tense vocal sample, top notch stuff!
17 April 2013 Bonzai Progressive FEATURED RELEASES Bonzai Progressive, Mylan


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World of Warcraft: Mists of Pandaria Developer Interview @ HardwareHeaven.com In this video Kaeyi Dream interviews Scott Mercer and Darren Williams from Blizzard on the latest WoW expansion: Mists of Pandaria. http://www.hardwareheaven.com/review...sept-2012.html |
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CBJ Mailing Address
City Hall |
CBJ Online Contact
Search the SiteLoading
Juneau Police DepartmentQuestions Posted to "Ask a Dispatcher"Q. I would like to make a response to the comments about the cars in the mendenhall parking lot "passing drugs". I, as well as the others who sit there, are not passing drugs nor are we passing money for drugs. I question what people would rather have us teenagers doing, partying or hanging out keeping eachother out of trouble? Most of the time, if we are parked next to eachother and "passing" something in between, it's usually a cd or a cigarette. Thank you very much for your time. Dear Juneau Resident,


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Alienware Arena is giving away Heroes of Newerth (HoN) keys that will score you 100 Goblin coins. Gear out your heroes with the newest and flashiest items in the game.
Oh, did I mention that one lucky player will also win an Alienware M17x laptop that will be painted with HoN art? That's right. After redeeming your key, you will be automatically entered into a sweepstakes for a one-of-a-kind Alienware laptop.
So, if you're looking for some free coin and a chance at a HoN branded Alienware laptop:
Once you register you can reedem your free goblin coins here:
Free Hon coins are in limited amount so hurry up.
More Details about the Alienware laptop you can win:


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Lot 248: Alexis Preller - Revelation
South African and International Art
by Strauss & Co
07 November 2016
Illovo, Johannesburg, South Africa
Alexis Preller (1911-1975) Please Register/Login to access your Invaluable Alerts
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Alexis Preller
signed and dated 45
oil on canvas
59,5 by 70cm
Literature: Esmé Berman and Karel Nel. (2009) Alexis Preller: Africa, the Sun and Shadows, Johannesburg: Shelf Publishing. Illustrated in black and white on page 87.
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Title | Moldova: Law No. 54-XV of 2005 on Combating Extremist Activity |
Publisher | National Legislative Bodies / National Authorities |
Publication Date | 23 May 2005 |
Country | Republic of Moldova |
Topics | Anti-terrorism law | Terrorism |
Cite as | Moldova: Law No. 54-XV of 2005 on Combating Extremist Activity [Republic of Moldova], 23 May 2005, available at: http://www.refworld.org/docid/4c3d797a2.html [accessed 30 March 2017] |
Comments | Official translation by OSCE Office for Democratic Institutions and Human Rights. This is the final draft. |
Disclaimer | This is not a UNHCR publication. UNHCR is not responsible for, nor does it necessarily endorse, its content. Any views expressed are solely those of the author or publisher and do not necessarily reflect those of UNHCR, the United Nations or its Member States. |


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Test Match Series:
Pakistan v Australia
21-07-2010 at Headingley Carnegie, Day 4 of 5 Pakistan beat Australia by 3 wickets
Australia won the toss and decided to bat
Back to top
Back to top
Back to top
Back to top Umpires: I J Gould, R E Koertzen, B C Broad, Nadeem Ghauri Pakistan: Butt (C), Farhat, Ali, U Akmal, Amin, Shoaib Malik, Kamran Akmal (W), Aamer, Gul, Kaneria, Asif Australia: S Watson, Katich, Ponting (C), Clarke, M Hussey, North, Paine (W), M Johnson, Smith, Hilfenhaus, Bollinger
Copyright © 2017 BBC. The BBC is not responsible for the content of external sites. Read more.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.


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Announcement A bewitching boutique and bookstore. * Quality antiques, vintage, crafts, collectibles, out of print and antiquarian books for discerning occultists. * www.SubRosaMagick.com * see my artwork at www.flybynightstudio.blogspot.com 1-860-57-TAROT
A bewitching boutique and bookstore. * Quality antiques, vintage, crafts, collectibles, out of print and antiquarian books for discerning occultists. * www.SubRosaMagick.com * see my artwork at www.flybynightstudio.blogspot.com 1-860-57-TAROT
Preston Mallow on Feb 10, 2017
IT ARRIVED!!! I was so excited to get home from work to see this beauty! Im in absolute love with it! Exactly as picture! Words cant describe how happy i am about this!! :D
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Perfect! Arrived carefully packaged and they were just as described. Love them to "death". Thank you!!!
K G on Jan 24, 2017
Fast shipping! Product just as described. Love it. Thank you so much!
Join me Under the Rose!
The Mystical Meaning Behind SubRosa
by Rev. Alicia Lyon Folberth
The phrase Sub Rosa comes from the Latin, literally 'under the rose,' from the ancient association of the rose with confidentiality. The rose, the queen of flowers, is the symbol of beauty of youth, love, joy, silence, and has long been associated with the Mysteries.
In Egypt and Greece the candidates for initiation into the Mysteries wore roses; and roses were the constant ornaments of the temples during the ceremony of dedication and opening the feasts of the Mysteries. The object of decorating the candidates for initiation with roses is to remind them of that which had been communicated to them as secrets (sub-rosa) they were bound to preserve with inviolable silence. The ritual symbolism of the rose is illustrated on the stone of the Labyrinth of Rhodes (the Island of Flowers). The shape of the rose shows the course followed by initiates, its rings between the rows of petals symbolize the path leading to the center, where the minotaur awaits.
In ancient Egypt, the rose was the emblem of the god Horus, later regarded by the Greeks and Romans as the god of silence. The expression 'Sub Rosa' or 'under the rose' originates from a Greek/Roman misinterpretation of an Egyptian hieroglyphic adopting Horus as a God (along with Isis and Osiris), the Greeks translated his Egyptian name her-pa-khrad to Harpocrates. The Egyptian hieroglyphic showed boy sucking his finger; the Greeks thought the boy had his finger to his lips, in the sign of secrecy. Cupid is said to have given Horus/Harpocrates a rose as a token of gratitude for not betraying the confidence of Cupid's amorous Mother, Venus. Thus the rose became doubly linked with confidentiality through Horus, and as the symbol of beauty, was dedicated to Venus/Aphrodite the goddess of love.
In older folk stories, 'under the rose' is the place where mothers tell their children that they were found, in a secret place. The modern telling of the story suggests an innocuous cabbage substituted for the feminine velvety flower, since the rose's flesh-colored petals were considered to have a 'genital' significance.
The custom developed of hanging a rose over the council table to indicate that all present were sworn to secrecy. Sub rosa, or 'under the rose' has come to mean that which is carried out in secret. The ceilings of Roman banquet rooms were decorated with roses to remind guests that what was spoken sub vino (under the influence of wine) was also sub rosa.
The rose, especially among the Freemasons, is the symbol of silence, which the candidate has faithfully promised and observed. The three Rosettes on a Master Mason's Apron indicate that every Master Mason has thrice been obliged to Fidelity, to Secrecy, and to Silence: Fidelity to the Craft, Secrecy as regards their sacred Secrets, and Silence as to the proceedings of the Lodge, which should never be disclosed to the profane.
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Rev. Alicia Lyon Folberth ☆ Priestess ☆ Witch ☆ Psychic Reader ☆ Teacher ☆ Reiki Master ☆ Artist/Designer ☆ Writer ☆ TV Show Producer ☆ http://www.ebay.com/usr/aliciassubrosamagick http://www.subrosamagick.com http://flybynightstudio.blogspot.com/
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You are here:
Rating(1-10) | Knowledgeability = 10 | Clarity of Response = 10 | Politeness = 10 |
Comment | No Comment |
Bonjour! Hello! Since I am from Quebec, I can answer questions regarding Quebec culture, history, arts (music, literature, ...). I can guide you through the beautiful geography of the province of Quebec. I live in Quebec City. I am your guide, always at your service.
I was the Quebec Guide (About.com).
I teach French as a foreign language and my students (University) want to know more and more about life, culture and our way of living, here, in Quebec City and in the Province of Quebec.
AATF (American Association of Teachers of French)
AQUEFLS (Association des professeurs de français)
Neve Shalom (looking for peace in the Middle East)
Immortel verses (poems)
WEB site : about languages, different cultures, arts ...
B.A in Linguistics
M.A. in Cognitive Sciences
Ph.D in Educational Technology
Awards and Honors
Lauréate des Amériques (TV5 compétition for Teachers of French)
Past/Present Clients
AllExperts (French language)
About.com (Quewbec Guide)
Hebrewonline.com (translation and proof-reading)


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Ruapehu Scenic Helicopter Flights
Explore the North Island's most remarkable topographical and geological features – the three peaks of Mt Ruapehu, Mt Ngauruhoe and Mt Tongariro – with Heliview Ruapehu. Voted 12th nationwide on the AA's 101 Must-Do list. Don't miss this one!
- Adult
- $379.00 - $3,999.00
We fly over the dramatic peaks of the Central Plateau, based from Ohakune, the southern gateway to the World Heritage Tongariro National Park.
See into the live craters of the sprawling Mount Tongariro, the symmetrical cone of Mount Ngauruhoe and the North Island's largest volcano – Mount Ruapehu - and its emerald blue crater lake. Awe at the snow-capped mountains, pristine alpine lakes, untouched native bush, rivers and waterfalls. Spot skiers on Turoa and Whakapapa ski-fields – New Zealand's largest ski resorts, and get a bird's eye view of 'Lord of the Rings' and 'The Hobbit' filming sites.
We offer a range of awe-inspiring flights to explore this stunning region. Or if you're after the exclusive touch, talk to us about your ideas.
Physical address
Ruapehu Scenic Helicopter Flights, 7304 State Highway 4, Raetihi Airfield, Ohakune, Ruapehu, New Zealand.
Opening times
- Business hours:
- By arrangement.
- Months of operation:
- All months of the year
Pricing and Conditions
Adult | $379.00 - $3,999.00 |


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