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 35.5, Problem 1E
Program Plan Intro
To determine that after the execution of the line
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It was claimed that:(a, b) ≤ (c, d) ⇔ (a < c) ∨ (a = c ∧ b ≤ d) defines a well-ordering on N x N. Show that this is actually the case.
Consider the following problem:L is a sorted list containing n signed integers (n being big enough), for example [-5, -2, -1, 0, 1, 2, 4] (here, n has a value of 7). If L is known to contain the integer 0, how can you find the index of 0 ?
Given a linked list L storing n integers, present an algorithm (either in words or in a pseudocode) that decides whether L contains any 0 or not. The output of your algorithm should be either Yes or No. What is the running time of your algorithm in the worst-case, using O notation?
Chapter 35 Solutions
Introduction to Algorithms
Ch. 35.1 - Prob. 1ECh. 35.1 - Prob. 2ECh. 35.1 - Prob. 3ECh. 35.1 - Prob. 4ECh. 35.1 - Prob. 5ECh. 35.2 - Prob. 1ECh. 35.2 - Prob. 2ECh. 35.2 - Prob. 3ECh. 35.2 - Prob. 4ECh. 35.2 - Prob. 5E
Ch. 35.3 - Prob. 1ECh. 35.3 - Prob. 2ECh. 35.3 - Prob. 3ECh. 35.3 - Prob. 4ECh. 35.3 - Prob. 5ECh. 35.4 - Prob. 1ECh. 35.4 - Prob. 2ECh. 35.4 - Prob. 3ECh. 35.4 - Prob. 4ECh. 35.5 - Prob. 1ECh. 35.5 - Prob. 2ECh. 35.5 - Prob. 3ECh. 35.5 - Prob. 4ECh. 35.5 - Prob. 5ECh. 35 - Prob. 1PCh. 35 - Prob. 2PCh. 35 - Prob. 3PCh. 35 - Prob. 4PCh. 35 - Prob. 5PCh. 35 - Prob. 6PCh. 35 - Prob. 7P
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- Prove Proposition E. To sort an array of N random strings, 3-way string quicksort uses~ 2N ln N character compares, on the average.arrow_forwardIllustrate that via AVL single rotation, any binary search tree T1 can be transformed into another search tree T2 (with the same items).Give an algorithm to perform this transformation using O(N log N) rotation on average. Use c++.arrow_forwardConsider the following problem:L is a sorted list containing n signed integers (n being big enough), for example [-5, -2, -1, 0, 1, 2, 4] (here, nhas a value of 7). If L is known to contain the integer 0, how can you find the index of 0 ?arrow_forward
- Illustrate that via AVL single rotation, any binary search tree T1 can betransformed into another search tree T2 (with the same items) Give an algorithm to perform this transformation using O(N log N) rotation on averagearrow_forwardLet S be a set and let C = (π1, π2,...,πn) be an increasing chain ofpartitions (PART(S), ≤) such that π1 = αS and πn = ωS. Then, the collection HC = ni=1 πi that consists of the blocks of all partitions in the chain is a hierarchy on S.arrow_forwardConsider 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…arrow_forward
- Given a circularly linked list L containing an even number of nodes, describe how to split L into two circularly linked list of half the the size in C++.arrow_forwardFor any given AVL tree T of n elements, we need to design an algorithm that finds elements x in T such that a <= x <= b, where a and b are two input values. Assume that T has m elements x satisfying a <= x <= b. What is the best time complexity for such an algorithm? A. O(m) B. O(m log n) C.O(m + log n) D.O(log m + log n)arrow_forwardProve that any algorithm that finds an element X in a sorted list of N elements requires O (logN) comparisonsarrow_forward
- Inserting a node in an unordered doubly linked list of N nodes. What is the Worst-Case Complexity(in O notation) and When does the worst case happen(under what conditions)?arrow_forwardSuppose we hash elements of a set U of keys into m slots. Show that if |U| (n − 1)m, then there are at least n keys that all hash to the same slot, so that the worst-case searching time for hashing with linked list to resolve collisions is Θ(n).arrow_forwardFor the operation below, provide the worst case running time in terms of n (Big O notation). Briefly justify your answer >>> Using binary search in a sorted singly linked list (as you completed in LAB 4). The implementation for __len__ traverses the list to count each nodearrow_forward
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