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 22.4, Problem 3E
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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).
Give a linear time algorithm via pseudo code that takes as input a directed acyclic graph
G (V, E) and two vertices u and v, that returns the number of simple paths from u to v in G.
Your algorithm needs only to count the simple paths, not list them. Explain why your code
runs in linear time.
6.
Give an algorithm that determines whether or not a given undirected graph contains a
cycle. Your algorithm should run in time O(n) and independent of m.
Chapter 22 Solutions
Introduction to Algorithms
Ch. 22.1 - Prob. 1ECh. 22.1 - Prob. 2ECh. 22.1 - Prob. 3ECh. 22.1 - Prob. 4ECh. 22.1 - Prob. 5ECh. 22.1 - Prob. 6ECh. 22.1 - Prob. 7ECh. 22.1 - Prob. 8ECh. 22.2 - Prob. 1ECh. 22.2 - Prob. 2E
Ch. 22.2 - Prob. 3ECh. 22.2 - Prob. 4ECh. 22.2 - Prob. 5ECh. 22.2 - Prob. 6ECh. 22.2 - Prob. 7ECh. 22.2 - Prob. 8ECh. 22.2 - Prob. 9ECh. 22.3 - Prob. 1ECh. 22.3 - Prob. 2ECh. 22.3 - Prob. 3ECh. 22.3 - Prob. 4ECh. 22.3 - Prob. 5ECh. 22.3 - Prob. 6ECh. 22.3 - Prob. 7ECh. 22.3 - Prob. 8ECh. 22.3 - Prob. 9ECh. 22.3 - Prob. 10ECh. 22.3 - Prob. 11ECh. 22.3 - Prob. 12ECh. 22.3 - Prob. 13ECh. 22.4 - Prob. 1ECh. 22.4 - Prob. 2ECh. 22.4 - Prob. 3ECh. 22.4 - Prob. 4ECh. 22.4 - Prob. 5ECh. 22.5 - Prob. 1ECh. 22.5 - Prob. 2ECh. 22.5 - Prob. 3ECh. 22.5 - Prob. 4ECh. 22.5 - Prob. 5ECh. 22.5 - Prob. 6ECh. 22.5 - Prob. 7ECh. 22 - Prob. 1PCh. 22 - Prob. 2PCh. 22 - Prob. 3PCh. 22 - Prob. 4P
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- You are given a weighted, undirected graph G = (V, E) which is guaranteed to be connected. Design an algorithm which runs in O(V E + V 2 log V ) time and determines which of the edges appear in all minimum spanning trees of G. Do not write the code, give steps and methods. Explain the steps of algorithm, and the logic behind these steps in plain Englisharrow_forwardSuppose are you given an undirected graph G = (V, E) along with three distinct designated vertices u, v, and w. Describe and analyze a polynomial time algorithm that determines whether or not there is a simple path from u to w that passes through v. [Hint: By definition, each vertex of G must appear in the path at most once.]arrow_forwardGive an algorithm to detect whether a given undirected graph contains a cycle. If the graph contains a cycle, then your algorithm should output one. (It should not output all cycles in the graph, just one of them.) The running time of your algorithm should be O(m+n) for a graph with n nodes and m edges.arrow_forward
- We recollect that Kruskal's Algorithm is used to fi nd the minimum spanning tree in a weighted graph. Given a weighted undirected graph G = (V , E, W), with n vertices/nodes, the algorithm will fi rst sort the edges in E according to their weights. It will then select (n-1) edges with smallest weights that do not form a cycle. (A cycle in a graph is a path along the edges of a graph that starts at a node and ends at the same node after visiting at least one other node and not traversing any of the edges more than once.) Use Kruskal's Algorithm to nd the weight of the minimum spanning tree for the following graph.arrow_forward36. Let G be a simple graph on n vertices and has k components. Then the number m of edges of G satisfies n-k ≤m if G is a null graph. This statement is A. sometimes true B. always true C. never true D. Neither true nor falsearrow_forward5. Given an undirected graph with n vertices and m edges, find an O(n+m) time algorithm that determines whether it is possible to color all the vertices red and blue such that every edge is between a red vertex and blue vertex. If such a coloring exists, your algorithm should produce one.arrow_forward
- Let G D .V; E/ be a connected, undirected graph. Give an O.V C E/-time algorithm to compute a path in G that traverses each edge in E exactly once in eachdirection. Describe how you can find your way out of a maze if you are given alarge supply of penniesarrow_forwardGiven the following Graphs: Graph A: В 12c 1 4. E F G- H 4 J K 3 Graph B: Graph B is the undirected version of Graph A. 3.arrow_forwardWe are given an undirected graph, that is connected. We're also given that each edge is associated with a positive weight. Now, we want to make this graph acyclic by removing some edges. Find this set of edges by designing an algorithm. Note: this edge set should have the smallest total weight out of other potential edge set solutions. The algorithm should run in O((m + n)log n) time.arrow_forward
- ou are given a directed graph G = (V, E) and two vertices s and t. Moreover, each edge of this graph is colored either blue or red. Your goal is to find whether there is at least one path from s to t such that all red edges in this path appear after all blue edges (the path may not contain any blue edges or any red edges, but if it has both types of edges, all red edges should appear after all blue edges). Design and analyze an algorithm for solving this problem in O(n + m) time.arrow_forwardYou are organizing a programming competition, where contestants implement Dijkstra's algorithm. Given adirected graph G = (V, E) with integer-weight edges and a starting vertex s ∈ V , their programs are supposedto output triplets (v, v.d, v.π) for each vertex v ∈ V . Design an O(V +E) time algorithm that takes as inputthe original graph G in both adjacency matrix (G.M) and adjacency list (G.Adj) representations, startingvertex s, and the output of a contestant's program (given as an array A of triplets), and returns whetherA is the correct output for G. Write down the pseudocode for your algorithm, explain why it correctlyveri es the output, and analyze your algorithm's running time. You may assume that all edge weights of the input graph provided to the contestantsare nonnegative and A (the output of their programs) is in the valid format, i.e., you don't need to verifythat A is actually an array of triplets, with v and v.π being valid vertices and v.d being an integer.Can you…arrow_forwardSuppose we represent a graph G = (V,E) as an adjacency matrix. Give a simple Implementation via pseudo code of Prim's algorithm for this case that runs in 0(V²) time. Explain why your code the running time has the upper bound as 0(v²).arrow_forward
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