An algorithm performs $${\left( {\log N} \right)^{1/2}}$$ find operations, N insert operations, $${\left( {\log N} \right)^{1/2}}$$ delete operations, and $${\left( {\log N} \right)^{1/2}}$$decrease-key operations on a set of data items with keys drawn from a linearly ordered set. For a delete operation, a pointer is provided to the record that must be deleted. For the decrease-key operation, a pointer is provided to the record that has its key decreased. Which one of the following data structures is the most suited for the algorithm to use, if the goal is to achieve the best total asymptotic complexity considering all the operations?

Which one of the following is the recurrence equation for the worst case time complexity of the Quicksort algorithm for sorting n ( $$ \ge 2$$ ) numbers? In the recurrence equations given in the options below, c is a constant.

Match the following:

List 1

(P) Prim’s algorithm for minimum spanning tree
(Q) Floyd-Warshall algorithm for all pairs shortest paths
(R) Mergesort
(S) Hamiltonian circuit

List 2

(i) Backtracking
(ii) Greedy method
(iii) Dynamic programming
(iv) Divide and conquer

Consider a max heap, represented by the array: 40, 30, 20, 10, 15, 16, 17, 8, 4.

Array Index	1	2	3	4	5	6	7	8	9
Value	40	30	20	10	15	16	17	8	4

Now consider that a value 35 is inserted into this heap. After insertion, the new heap is