data structures using selection algorithms these algorithms are used to find the minimum and maximum values in linear or sub-linear time graph algorithms heaps can be used as internal traversal data structures this guarantees that runtime is reduced by an order of polynomial heaps are therefore used for implementing prim' minimal spanning tree algorithm and dijkstra' shortest path problem points to remember binary heap is defined as complete binary tree in which every node satisfies the heap property there are two types of binary heapsmax heap and min heap in min heapelements at every node will either be less than or equal to the element at its left and right child similarlyin max heapelements at every node will either be greater than or equal to element at its left and right child binomial tree of order has root node whose children are the root nodes of binomial trees of order - - binomial tree bi of height has nodes binomial heap is collection of binomial trees that satisfy the following propertieso every binomial tree in satisfies the minimum heap property there can be one or zero binomial trees for each order including zero order fibonacci heap is collection of trees fibonacci heaps differ from binomial heapsas they have more relaxed structureallowing improved asymptotic time bounds exercises review questions define binary heap differentiate between min-heap and max-heap compare binary trees with binary heaps explain the steps involved in inserting new value in binary heap with the help of suitable example explain the steps involved in deleting value from binary heap with the help of suitable example discuss the applications of binary heaps form binary max-heap and min-heap from the following sequence of data heaps are excellent data structures to implement priority queues justify this statement define binomial heap draw its structure differentiate among binarybinomialand fibonacci heaps explain the operations performed on fibonacci heap why are fibonacci heaps preferred over binary and binomial heaps analyse the complexity of the algorithm to unite two binomial heaps the running time of the algorithm to find the minimum key in binomial heap is (log ncomment discuss the process of inserting new node in binomial heap explain with the help of an example the algorithm min-extract_binomial-heap(runs in (log ntime where is the number of nodes in justify this statement explain how an existing node is deleted from binomial heap with the help of relevant example explain the process of inserting new node in fibonacci heap write down the algorithm to unite two fibonacci heaps what is the procedure to extract the node with the minimum value from fibonacci heapgive the algorithm and analyse its complexity consider the figure given below and state whether it is heap or not