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 16.1, Problem 1E
Program Plan Intro
To give a dynamic-
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Determine P(A x B) – (A x B) where A = {a} and B = {1, 2}.
3. We are given a weighted undirected graph G containing exactly 10 cycles. Write
an algorithm to compute the MST of G. Your algorithm should have a runtime of
O(V + E).
Computer Science
Consider the d-Independent Set problem:
Input: an undirected graph G = (V,E) such that every vertex has degree less or equal than d.
Output: The largest Independent Set.
Describe a polynomial time algorithm Athat approximates the optimal solution by a factor α(d). Your must
write the explicit value of α, which may depend on d. Describe your algorithm in words (no pseudocode) and
prove the approximation ratio α you are obtaining. Briefly explain why your algorithm runs in polytime.
Chapter 16 Solutions
Introduction to Algorithms
Ch. 16.1 - Prob. 1ECh. 16.1 - Prob. 2ECh. 16.1 - Prob. 3ECh. 16.1 - Prob. 4ECh. 16.1 - Prob. 5ECh. 16.2 - Prob. 1ECh. 16.2 - Prob. 2ECh. 16.2 - Prob. 3ECh. 16.2 - Prob. 4ECh. 16.2 - Prob. 5E
Ch. 16.2 - Prob. 6ECh. 16.2 - Prob. 7ECh. 16.3 - Prob. 1ECh. 16.3 - Prob. 2ECh. 16.3 - Prob. 3ECh. 16.3 - Prob. 4ECh. 16.3 - Prob. 5ECh. 16.3 - Prob. 6ECh. 16.3 - Prob. 7ECh. 16.3 - Prob. 8ECh. 16.3 - Prob. 9ECh. 16.4 - Prob. 1ECh. 16.4 - Prob. 2ECh. 16.4 - Prob. 3ECh. 16.4 - Prob. 4ECh. 16.4 - Prob. 5ECh. 16.5 - Prob. 1ECh. 16.5 - Prob. 2ECh. 16 - Prob. 1PCh. 16 - Prob. 2PCh. 16 - Prob. 3PCh. 16 - Prob. 4PCh. 16 - Prob. 5P
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- We know that when we have a graph with negative edge costs, Dijkstra’s algorithm is not guaranteed to work. (a) Does Dijkstra’s algorithm ever work when some of the edge costs are negative? Explain why or why not. (b) Find an algorithm that will always find a shortest path between two nodes, under the assumption that at most one edge in the input has a negative weight. Your algorithm should run in time O(m log n), where m is the number of edges and n is the number of nodes. That is, the runnning time should be at most a constant factor slower than Dijkstra’s algorithm. To be clear, your algorithm takes as input (i) a directed graph, G, given in adjacency list form. (ii) a weight function f, which, given two adjacent nodes, v,w, returns the weight of the edge between them. For non-adjacent nodes v,w, you may assume f(v,w) returns +1. (iii) a pair of nodes, s, t. If the input contains a negative cycle, you should find one and output it. Otherwise, if the graph contains at least one…arrow_forwardThe Triangle Vertex Deletion problem is defined as follows: Given: an undirected graph G = (V, E) , with IVI=n, and an integer k>= 0. Is there a set of at most k vertices in G whose deletion results in deleting all triangles in G? (a) Give a simple recursive backtracking algorithm that runs in O(3^k * ( p(n))) where p(n) is a low-degree polynomial corresponding to the time needed to determine whether a certain vertex belongs to a triangle in G. (b) Selecting a vertex that belong to two different triangles can result in a better algorithm. Using this idea, provide an improved algorithm whose running time is O((2.562^n) * p(n)) where 2.652 is the positive root of the equation x^2=x+4arrow_forwardGiven an undirected weighted graph G with n nodes and m edges, and we have used Prim’s algorithm to construct a minimum spanning tree T. Suppose the weight of one of the tree edge ((u, v) ∈ T) is changed from w to w′, design an algorithm to verify whether T is still a minimum spanning tree. Your algorithm should run in O(m) time, and explain why your algorithm is correct. You can assume all the weights are distinct. (Hint: When an edge is removed, nodes of T will break into two groups. Which edge should we choose in the cut of these two groups?)arrow_forward
- 2. Depth-first search can be used to detect cycles in undirected graphs (see lectures). As- suming an adjacency list representation of the graph, depth-first search has complexity (n+m). Explain why the complexity of the cycle detection algorithm by depth-first search is actually O(n) regardless of what m is.arrow_forwardLet G = (V, E) be a DAG, where every edge e = ij and every vertex x have positive weighs w(i,j) and w(x), respectively, associated with them. Design an algorithm for computing a maximum weight path. What is the time complexity of your algorithm? (You must start with the correct definitions, and then write a recurrence relation.)arrow_forwardGiven a sequence A formed by n positive numbers and a positive integer d, we are interested in a distant max-product subsequence (MPS) of A, which is a subsequence of A formed by elements whose indices are at least d units apart and have the maximum product. Describe a dynamic programming algorithm that reports the product of the MPS of A. For example, if A = [2, 10, 12, 9, 1, 3, 5] and d = 2, the output should be 10 × 9 × 5 = 450. In your solution, it suffices to complete the first two steps of the DP algorithm. That is, define subproblems, describe the optimal for a larger subproblem as a function of the optimal solution for smaller subproblems, and write a recursive formula for the optimal value of a subproblem (remember to include the base case). Assume the indices start at 1.arrow_forward
- a) Why can Dijkstra's algorithm not work properly on graphs with negative weighted edges? Explain with example. Implement the greedy approach for coin change algorithm and show your step by step approach to give change of 139 in coin change with {1, 3, 5, 25, 45, 60} unit values. c) Write the procedure for calculating GCD and LCM of the following numbers. Also write the answers at last. 2, 4, 5, 20arrow_forwardConsider the Minimum-Weight-Cycle Problem: Input: A directed weighted graph G (V, E) (where the weight of edge e is w(e)) and an integer k. Output: TRUE if there is a cycle with total weight at most k and FALSE if there is no cycle with total weight at most k. Remember, a cycle is a list of vertices such that each vertex has an edge to the next and the final vertex has an edge to the first vertex. Each vertex can only occur once in the cycle. A vertex with a self-loop forms a cycle by itself. (a) Assume that all edge weights are positive. Give a polynomial-time algorithm for the Minimum-Weight-Cycle Problem. For full credit, you should: - Give a clear description of your algorithm. If you give pseudocode, you should support it with an expla- nation of what the algorithm does. Give the running time of your algorithm in terms of the number of vertices n and the number of edges m. You do not need to prove the correctness of your algorithm or the correctness of your running time analysis.arrow_forwardDesign a dynamic programming algorithm for the bigger-is-smarter elephant problem by comparing it with the longestcommon-subsequence problem. To do this the LCS problem needs to be generalized to have weights on the letters.arrow_forward
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