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 25.3, Problem 5E
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To show that if weighted directed graph G contains 0-weight cycle then for every edge ( u, v ) in the graph G , w’ ( u, v ) = 0 while applying Johnson’s
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Suppose we have a graph G = (V, E) with m edges. Prove that there exists a partition of V into
three subsets A, B, C such that there are 2m edges between these subsets (i.e. between A and B,
between B and C, or between A and C).
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Let 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-set
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.
Chapter 25 Solutions
Introduction to Algorithms
Ch. 25.1 - Prob. 1ECh. 25.1 - Prob. 2ECh. 25.1 - Prob. 3ECh. 25.1 - Prob. 4ECh. 25.1 - Prob. 5ECh. 25.1 - Prob. 6ECh. 25.1 - Prob. 7ECh. 25.1 - Prob. 8ECh. 25.1 - Prob. 9ECh. 25.1 - Prob. 10E
Ch. 25.2 - Prob. 1ECh. 25.2 - Prob. 2ECh. 25.2 - Prob. 3ECh. 25.2 - Prob. 4ECh. 25.2 - Prob. 5ECh. 25.2 - Prob. 6ECh. 25.2 - Prob. 7ECh. 25.2 - Prob. 8ECh. 25.2 - Prob. 9ECh. 25.3 - Prob. 1ECh. 25.3 - Prob. 2ECh. 25.3 - Prob. 3ECh. 25.3 - Prob. 4ECh. 25.3 - Prob. 5ECh. 25.3 - Prob. 6ECh. 25 - Prob. 1PCh. 25 - Prob. 2P
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- 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_forwardWe are given a graph G = (V, E); G could be a directed graph or undirected graph. Let M bethe adjacency matrix of G. Let n be the number of vertices so that the matrix M is n ×n matrix. For anymatrix A, let us denote the element of i-th row and j-th column of the matrix A by A[i, j].1. Consider the square of the adjacency matrix M . For all i and j, show that M 2[i, j] is the number ofdifferent paths of length 2 from the i-th vertex to the j-th vertex. It should be explained or proved asclearly as possible.2. For any positive integer k, show that M k[i, j] is the number of different paths of length k from the i-th vertex to the j-th vertex. You may use induction on k to prove it.3. Assume that we are given a positive integer k. Design an algorithm to find the number of different paths of length k from the i-th vertex to j-th vertex for all pairs of (i, j). The time complexity of your algorithm should be O(n3 log k). You can get partial credits if you design an algorithm of O(n3k).arrow_forwardShow that if all edges of a graph G have pairwise distinct weights, then thereis exactly one MST for G.arrow_forward
- 1.6 If G is a simple graph of with number of vertices n > 0, show that deg(v) id(v) = od(v) = q. VEV VEVarrow_forwardConsider an undirected graph G with 100 nodes. The maximum number of edges to be included in G so that the graph is not connected isarrow_forwardConsider a graph G that has k vertices and k −2 connected components,for k ≥ 4. What is the maximum possible number of edges in G? Proveyour answer.arrow_forward
- If a graph has a collection of subsets of the edge set E, with the edges of at most one cycle: Show that if |X| and |Y| are independent sets, show that |X| < |Y| implies that there exists {m} E Y\X such that X U {m} is independent.arrow_forward1. 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_forwardLet G = (V, E) be weighted, directed graph with a source node s where each (u, v, w) ∈E is an edge between vertices u and v with weight w(u, v). Suppose that for any(u, v) ∈ E, the weight is strictly non-negative such that w(u, v) ≥ 0. Let s be somesource vertex. Prove it is possible to design an algorithm obtain δ(s, v) ≤ v.d.Hint: This is a formal proof rather than writing the code to achieve this. Provide thehigh-level construction for this proof using inductive reasoning.arrow_forward
- We are given an undirected connected graph G = (V, E) and vertices s and t.Initially, there is a robot at position s and we want to move this robot to position t by moving it along theedges of the graph; at any time step, we can move the robot to one of the neighboring vertices and the robotwill reach that vertex in the next time step.However, we have a problem: at every time step, a subset of vertices of this graph undergo maintenance andif the robot is on one of these vertices at this time step, it will be destroyed (!). Luckily, we are given theschedule of the maintenance for the next T time steps in an array M [1 : T ], where each M [i] is a linked-listof the vertices that undergo maintenance at time step i.Design an algorithm that finds a route for the robot to go from s to t in at most T seconds so that at notime i, the robot is on one of the maintained vertices, or output that this is not possible. The runtime ofyour algorithm should ideally be O((n + m) ·T ) but you will…arrow_forwardAdvanced Physics Chegg experts gave the wrong answer the last time I asked this, so I am asking it again. Please only answer if you know how to solve the problem! Consider a directed graph G = (V, E) having a source vertex s and sink vertex t. Suppose that it has positive integer edge capacities c_e for all edges in the graph. Also suppose that is has a flow f = {f(e)} for all edges in the graph. We consider an edge to be saturated if f(e) = c_e. Suppose that f is a maximum s-t flow. Let S represent the set of all saturated edges. Consider the minimum total capacity of any given s-t cut. Will it be equal to the total capacity of S? If true, please provide a proof. Otherwise, if it is false, give a counterexample.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_forward
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