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 22.1, Problem 3E
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Consider 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.
Create an algorithm that, given a directed graph g = (v e) and a distinguished vertex s v, finds the shortest path from s to v for each v v. If g contains n vertices and e edges, your method must execute in o(n + e) time.
A graph G = <V, E> is given where V = {A, B, C, D, E, F, G, H, I}, and
E = {(A, B, 50), (A, C, 30), (B, E, 100), (B, D, 30), (C, I, 100), (D,E, 150),
(D, H, 40), (E, F, 40), (F, G, 200) , (G, I, 80)}
Given the nodes represent the cities and weights the distances, solve the traveling salesman problem starting with city C using the nearest neighbor algorithm.
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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- Consider 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…arrow_forwardGiven the following example of UAG graphs: i)- In Java, give implementation to find the shortest path for graph 1 2 12 9 10 3 5 \21 14 15 8. 4arrow_forwardthink about a search problem on a graph G = (N, E, C), where N represents nodes, E represents edges between nodes, and the weight of an edge e ∈ E is denoted by C(e), where C(e) > 1 for all e ∈ E. We have a heuristic h that calculates the smallest number of edges from a starting state to a goal state. Now, imagine removing edges from the graph while leaving the heuristic values the same. The question is whether the heuristic remains admissible and consistent after this change. and prove themarrow_forward
- 1. Suppose that you have a graph G = may have negative cycles in it. We call a path P "simple" if it includes no repeated (V, E) with weight function w : E R. The graph vertices. Prove that either the shortest simple path P(s, t) exists for any s, t e V or there is at least one edge e e E such that w(e) = -0. 2. Provide an algorithm that finds the shortest simple path from s to t that works if no simple path from s to t includes negative edges. Does your algorithm work when you allow non-simple paths?arrow_forwardGiven an undirected graph G = <V,E>, a vertex cover is a subset of vertices S V such that for each edge (u,v) belongs to E, either u S or v S or both. The Vertex Cover Problem is to find minimum size of the set S. Consider the following algorithm to Vertex Cover Problem: (1) Initialize the result as {} (2) Consider a set of all edges in given graph. Let the set be E’. (3) Do following while E’ is not empty ...a) Pick an arbitrary edge (u,v) from set E’ and add u and v to result ...b) Remove all edges from E which are either incident on u or v. (4) Return result. It claim that this algorithm is exact for undirected connected graphs. Is this claim True or False? Justify the answer.arrow_forwardBe G=(V, E)a connected graph and u, vEV. The distance Come in u and v, denoted by d(u, v), is the length of the shortest path between u'and v, Meanwhile he width from G, denoted as A(G), is the greatest distance between two of its vertices. a) Show that if A(G) 24 then A(G) <2. b) Show that if G has a cut vertex and A(G) = 2, then Ġhas a vertex with no neighbors.arrow_forward
- In this problem you will design an algorithm that takes as input a directed acyclic graph G = (V,E) and two vertices s and t, and returns the number of simple paths from s tot in G. For example, the directed acyclic graph below contains exactly four simple paths from vertex p to vertex v: pov, poryv, posryv, and psryv. Notice: your algorithm needs only to count the simple paths, not list them. m y W Design a recursive backtracking (brute-force) algorithm that determines the number of paths from s to t. Write down the pseudocode of your algorithm and prove its correctness, i.e., convince us that it works beyond any doubt. (Hint: using induction.).arrow_forwardLet G = (V, E) be a directed graph, and let wv be the weight of vertex v for every v ∈ V . We say that a directed edgee = (u, v) is d-covered by a multi-set (a set that can contain elements more than one time) of vertices S if either u isin S at least once, or v is in S at least twice. The weight of a multi-set of vertices S is the sum of the weights of thevertices (where vertices that appear more than once, appear in the sum more than once).1. Write an IP that finds the multi-set S that d-cover all edges, and minimizes the weight.2. Write an LP that relaxes the IP.3. Describe a rounding scheme that guarantees a 2-approximation to the best multi-setarrow_forwardConsider a weighted, directed graph G with n vertices and m edges that have integer weights. A graph walk is a sequence of not-necessarily-distinct vertices v1, v2, ... , Vk such that each pair of consecutive vertices Vi, Vi+1 are connected by an edge. This is similar to a path, except a walk can have repeated vertices and edges. The length of a walk in a weighted graph is the sum of the weights of the edges in the walk. Let s, t be given vertices in the graph, and L be a positive integer. We are interested counting the number of walks from s to t of length exactly L. Assume all the edge weights are positive. Describe an algorithm that computes the number of graph walks from s to t of length exactly L in O((n+ m)L) time. Prove the correctness and analyze the running time. (Hint: Dynamic Programming solution)arrow_forward
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