set of choices to select from it' often easier to conceptualize greedy algorithm recursivelyand then implement it using iteration for higher performance even if the greedy approach does not yield an optimum solutionit can give insights into the optimum algorithmor serve as heuristic sometimes the correct greedy algorithm is not obvious table top tips for greedy algorithms invariants common approach to designing an efficient algorithm is to use invariants brieflyan invariant is condition that is true during execution of program this condition may be on the values of the variables of the programor on the control logic well-chosen invariant can be used to rule out potential solutions that are suboptimal or dominated by other solutions for examplebinary searchmaintains the invariant that the space of candidate solutions contains all possible solutions as the algorithm executes sorting algorithms nicely illustrate algorithm design using invariants intuitivelyselection sort is based on finding the smallest elementthe next smallest elementetc and moving them to their right place more preciselywe work with successively larger subarrays beginning at index and preserve the invariant that these subarrays are sortedtheir elements are less than or equal to the remaining elementsand the entire array remains permutation of the original array invariants boot camp suppose you were asked to write program that takes as input sorted array and given target value and determines if there are two entries in the array that add up to that value for exampleif the array is - ithen there are entries adding to and to but not to and the brute-force algorithm for this problem consists of pair of nested for loops its complexity is ( )where is the length of the input array faster approach is to add each element of the array to hash tableand test for each element if ewhere is the target valueis present in the hash table while reducing time complexity to ( )this approach requires (nadditional storage for the hash the most efficient approach uses invariantsmaintain subarray that is guaranteed to hold solutionif it exists this subarray is initialized to the entire arrayand iteratively shrunk from one side or the other the shrinking makes use of the sortedness of the array specificallyif the sum of the leftmost and the rightmost elements is less than the targetthen the leftmost element can never be combined with some element to obtain the target similar observation holds for the rightmost element def has_two_sum (at)ij len( while <jif [ia[ =treturn true elif [ia[jti +