For a given relation R on the set A, let M stand for the matrix representation of R. Let's pretend there's a set A with n unique items. Therefore, M would be an n-by-n matrix. If R is a rooted (directed) tree, how many 0s and 1s does M have?

1. Suppose that we are given a binary search tree T with distinct keys. We would like to find out the kth smallest and largest keys in T. The value of k is of course smaller than the size of T. Please provide efficient approaches to find these two keys and analyze your methods.

Consider the array t = [1, 2, 3, 4, 5, 8, 0 , 7, 6] of size n = 9, . a) Draw the complete tree representation for t. b) What is the index of the first leaf of the tree in Part a (in level order)? In general, give a formula for the index of the first leaf in the corresponding complete binary tree for an arbitrary array of size n. c) Redraw the tree from Part a after each call to fixheap, in Phase 1 of heapsort. Remember, the final tree obtained will be a maxheap. d) Now, starting with the final tree obtained in Part c, redraw the tree after each call to fixheap in Phase 2 of heap sort. For each tree, only include the elements from index 0 to index right (since the other elements are no longer considered part of the tree). e) For the given array t, how many calls to fixheap were made in Phase 1? How many calls to fixheap were made in Phase 2? f) In general , give a formula for the total number of calls to fixheap in Phase 1, when heapsort is given an arbitrary array of size n. Justify…

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Suppose we consider the special case of IS, WIS where the input graph is a tree T (V, E) with an identified root r ≤ V (there are no other limitations on structure). This means that for every v € V(T), the subtree rooted at v is well-defined - we will refer to this subtree as Tv. = In this section, you are asked to develop a polynomial-time algorithm for solving WIS (and including IS, when weights are all 1). For every v € V, we define the following: Kv,o to be the total weight of the maximum-weight Independent set of subtree T, which does not include v itself. • Kv,1 to be the total weight of the maximum-weight Independent set of subtree T with v belonging to the independent set. D(i) Develop a pair of recurrences which express the value of к,0 (and similarly of ×,₁) in terms of the values of the child nodes of v. Include the "base case" whsn v is a leaf, and then describe how we can exploit the recurrences to design a polynomial-time algorithm to solve WIS for a tree. D(ii) In the…

Let G be an undirected edge-weighted graph with V vertices and O(V2) edges. Describe an efficient algorithm to compute a minimum spanning tree that runs in O(V2) time. You are only allowed to use data structures such as arrays and priority queues. Do not use complex data structures such as Fibonacci heaps. Give a formal proof of correctness for your algorithm. An answer without any formal proof will be considered incomplete. Analyze the running time of the algorithm.

Let T : P −→ L be a function that inputs a perfect binary tree and outputs a list of vertices, defined as follows. B. If X is a binary tree consisting only of a single vertex v, then T (X ) = v, a list with only one entry. R. If X is a binary tree with root v, left subtree Xl and right subtree Xr , then T (X ) equals the list T (Xl ), T (Xr ), v.