Analysis of d-ary heaps

a. How would you represent a d-ary heap in an array? Answer this question by:

Giving an expression for J-th-Child(i,j): the index of the j-th child as a function of the index i of the given node, and the child index j within the given node.

Giving an expression for D-Ary-Parent(i): the index of the parent of a node as a function of its index i within the array.

Checking that your solution works by showing that D-Ary-Parent(J-th-Child(i,j)) = i (when you start at node i, and apply your formula to go to a child, and then your other formula to go back to the parent, you must end up back at i).

b. What is the height of a d-ary heap of n elements as a function of n and d? By what factor does this height differ from that of a binary heap of n elements?

c. Give an efficient implementation of Extract-Max in a d-ary max-heap. (Hint: How would you modify the existing code?) Analyze the running time of your implementation in terms of n and d. (Note that d must be part of your expression even if it occurs in a constant term.)

d. Give an efficient implementation of Insert in a d-ary max-heap. Analyze its running time in terms of n and d.

HEAP-ExTRACT-MAx (A) 1 if A.heap-size