Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 14.2, Problem 4E
Program Plan Intro
To describe the implementation of RB-ENUMERATE procedure in
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Tl and T2 are two very large binary trees, with Tl much bigger than T2. Create an algorithm to determine if T2 is a subtree of Tl.A tree T2 is a subtree of Tl if there exists a node n in Tl such that the subtree of n is identical to T2. That is, if you cut off the tree at node n, the two trees would be identical.
You are given a tree of N nodes rooted at node 1 and each node has a value where value of i'th node is A₁.
You're given Q queries, in each query two integers x, k given. Find if possible such that all nodes in the subtree
of node a (including x) has values greater than or equal to k by performing swapping of two nodes any number
of times. You can swap two nodes u and v such that u belongs to subtree of x (including x) and v does not. If
possible then find minimum number of swapping of nodes required otherwise print -1.
Note: Since the queries are independent, the initial tree is retained after each query.
Input format
• The first line contains T denoting the number of test cases. The description of each test case is as follows.
o The first line contains an integer N denoting the number of nodes in tree.
o Next line contains space-separated integers denoting value of each node.
o Each of the following N- 1 lines contain two integers u and v denoting edge between nodes u and
V.
o The…
In java and in O(logn) time
Write a method that balances an existing BST, call it balance(). A BST is balanced if the height of its
left and right sub-trees are different by at most one. Recursively applied. If the tree is balanced, then
searching for keys will take act like binary search and require only logn comparisons. No performance
requirements on your balancing algorithm. (Come up with a way yourself - don't skip to 3.3. That
section is really complicated and meant for the harder case where you need to do it in log time.)
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- (S 10 3 Max Sum= 10 + 6 + 8 = 24 S Max Sum= 2 + 3 + 5 = 10 Write an algorithm that, for a given tree T, calculates the maximum sum achievable across all sets S (where no two elements have a parent/child relationship). You only need to return the maximal sum and not a set of nodes that achieves it (for simplicity). What is the runtime and space complexity of the algorithm? Hint: For any given node n, come up with a recursive solution MAX-SUM(n, include) which calculates the maximum possible sum over a non parent/child subset S over a tree rooted at node n. The parameter 'include' determines if node n should be included in a set S or not (this impacts which of the descendants of n can be used). Note that the maximum sum for the entire tree is given by MAX-SUM(T.root, true) + MAX-SUM(T.root, false). Then, use dynamic programming techniques to avoid solving the same call to MAX-SUM more than once. Memoization can be done by adding two new fields to each node (n.maxSumNodelncluded and…arrow_forwardUsing Java Design an algorithm for the following operations for a binary tree BT, and show the worst-case running times for each implementation:preorderNext(x): return the node visited after node x in a pre-order traversal of BT.postorderNext(x): return the node visited after node x in a post-order traversal of BT.inorderNext(x): return the node visited after node x in an in-order traversal of BT.arrow_forwardLet T be a binary search tree whose keys are all distinct. Suppose T is created by inserting each key into the growing tree. Let x be a leaf node and y be its parent node. Show that if x is the right child of y, then y.key is the largest key in T smaller than x.key. (Hint: Show that the assumption that there is another node z whose key satisfies y.key < z.key < x.key will lead to a contradiction.)arrow_forward
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