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 24.3, Problem 8E
Program Plan Intro
To modify the Dijkstra’s
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Let G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Let P be the shortest path between two nodes s, t. Now, suppose we replace each edge weight ℓ(e) withℓ(e)^2, then P is still a shortest path between s and t.
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…
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.
Chapter 24 Solutions
Introduction to Algorithms
Ch. 24.1 - Prob. 1ECh. 24.1 - Prob. 2ECh. 24.1 - Prob. 3ECh. 24.1 - Prob. 4ECh. 24.1 - Prob. 5ECh. 24.1 - Prob. 6ECh. 24.2 - Prob. 1ECh. 24.2 - Prob. 2ECh. 24.2 - Prob. 3ECh. 24.2 - Prob. 4E
Ch. 24.3 - Prob. 1ECh. 24.3 - Prob. 2ECh. 24.3 - Prob. 3ECh. 24.3 - Prob. 4ECh. 24.3 - Prob. 5ECh. 24.3 - Prob. 6ECh. 24.3 - Prob. 7ECh. 24.3 - Prob. 8ECh. 24.3 - Prob. 9ECh. 24.3 - Prob. 10ECh. 24.4 - Prob. 1ECh. 24.4 - Prob. 2ECh. 24.4 - Prob. 3ECh. 24.4 - Prob. 4ECh. 24.4 - Prob. 5ECh. 24.4 - Prob. 6ECh. 24.4 - Prob. 7ECh. 24.4 - Prob. 8ECh. 24.4 - Prob. 9ECh. 24.4 - Prob. 10ECh. 24.4 - Prob. 11ECh. 24.4 - Prob. 12ECh. 24.5 - Prob. 1ECh. 24.5 - Prob. 2ECh. 24.5 - Prob. 3ECh. 24.5 - Prob. 4ECh. 24.5 - Prob. 5ECh. 24.5 - Prob. 6ECh. 24.5 - Prob. 7ECh. 24.5 - Prob. 8ECh. 24 - Prob. 1PCh. 24 - Prob. 2PCh. 24 - Prob. 3PCh. 24 - Prob. 4PCh. 24 - Prob. 5PCh. 24 - Prob. 6P
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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.arrow_forwardGiven a graph that is a tree (connected and acyclic). (1) Pick any vertex v. (II) Compute the shortest path from v to every other vertex. Let w be the vertex with the largest shortest path distance. (III) Compute the shortest path from w to every other vertex. Let x be the vertex with the largest shortest path distance. Consider the path p from w to x. Which of the following are true a. p is the longest path in the graph b. p is the shortest path in the graph c. p can be calculated in time linear in the number of edges/vertices a,c a,b a,b,c b.carrow_forwardGiven a graph that is a tree (connected and acyclic). (I) Pick any vertex v.(II) Compute the shortest path from v to every other vertex. Let w be the vertex with the largest shortest path distance.(III) Compute the shortest path from w to every other vertex. Let x be the vertex with the largest shortest path distance. Consider the path p from w to x. Which of the following are truea. p is the longest path in the graphb. p is the shortest path in the graphc. p can be calculated in time linear in the number of edges/verticesarrow_forward
- We are given a directed graph G = (V, E) with edge weights w(e) for e E E. The edge weights are allowed to be negative. Let -C be the minimum value of the edge weights. Create a new set of edge weights w'(e) = w(e) + C. Thus these new weights are non-negative: w'(e) ≥ 0 for all e € E. Run Dijkstra's algorithm from a specified start vertex s € V on G with these new weights w'. For every graph G, every weights w, and every s E V, the above algorithm is guaranteed to find the shortest paths in G (not the actual lengths but the paths) from s with respect to the original weights w: True O False If you entered true, provide a short explanation. If you entered false provide a counterexample (show the graph and what the algorithm produces vs. the correct solution).arrow_forwardGiven a directed graph with non-negative edge weights, suppose we have computed the shortest paths from a given source to all the other vertices. If we modify the graph in such a way that the weights of all the edges are doubled, then, the shortest paths remain the same and only the total weights of the paths change. O True O Falsearrow_forwardWe are given a directed graph G = (V, E) with edge weights w(e) for e E E. The edge weights are allowed to be negative. Let -C be the minimum value of the edge weights. Create a new set of edge weights w' (e) = w(e) + C. Thus these new weights are non-negative: w' (e) > 0 for all e € E. Run Dijkstra's algorithm from a specified start vertex s E V on G with these new weights w'. For every graph G, every weights w, and every s € V, the above algorithm is guaranteed to find the shortest paths in G (not the actual lengths but the paths) from s with respect to the original weights w: True False If you entered true, provide a short explanation. If you entered false provide a counterexample (show the graph and what the algorithm produces vs. the correct solution).arrow_forward
- Let G be a directed graph with positive and negative weights. You are given that the shortest paths from source node s to every other vertex is at most k edges long. Give a O(k|E|) algorithm that finds all shortest paths from s.arrow_forwardGiven N cities represented as vertices V₁, V2, un on an undirected graph (i.e., each edge can be traversed in both directions). The graph is fully-connected where the edge eij connecting any two vertices vį and vj is the straight-line distance between these two cities. We want to search for the shortest path from v₁ (the source) to VN (the destination). ... Assume that all edges have different values, and €₁,7 has the largest value among the edges. That is, the source and destination have the largest straight-line distance. Compare the lists of explored vertices when we run the uniform-cost search and the A* search for this problem. Hint: The straight-line distance is the shortest path between any two cities. If you do not know how to start, try to run the algorithms by hand on some small cases first; but remember to make sure your graphs satisfy the conditions in the question.arrow_forwardConsider an undirected graph with n nodes and m edges. The goal is to find a path between two specified nodes u and v that maximizes the minimum weight of any edge along the path. Assume that all edge weights are positive and distinct. Design an algorithm to solve this problem with a time complexity of O(m log n).arrow_forward
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