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 24.5, Problem 6E
Program Plan Intro
To prove that for every vertex
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Let G be a graph and R be a subset of its vertices. A subset of vertices DR is called R-dominating target if every connected subgraph of G containing Da dominates R. In addition, if each vertex of DR has some R verter in its closed neighborhood, then we call it an essential-R-dominating-target.
Let G = (V, E) denote an weighted undirected graph, in which every edge has unit weight, and let T = (V, E') denote the minimum spanning tree of G. Prove formally that for all u, v ∈ V , the path between u and v in tree T is unique
Let e be a maximum-weight edge on some cycle of graph G=(V, E).
Prove or disprove that after deleting edge a from graph, there is a minimum spanning tree of G=(V, E-{e}) that is also a minimum spanning tree of G.
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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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- Let G = (V, E) be an undirected graph with at least two distinct vertices a, b ∈ V . Prove that we can assign a direction to each edge e ∈ E as to form a directed acyclic graph G′ where a is a source and b is a sink.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_forwardShow that if all edges of a graph G have pairwise distinct weights, then thereis exactly one MST for G.arrow_forward
- Hall's theorem Let d be a positive integer. We say that a graph is d-regular if every node has degree exactly d. Show that every d-regular bipartite graph G = (L ∪ R, E) with bipartition classes L and R has |L| = |R|. Show that every d-regular bipartite graph has a perfect matching by (directly) arguing that a minimum cut of the corresponding flow network has capacity |L|arrow_forwardLet S be the skeleton of any graph. Prove that if we add any edge to S with an unchanged set of vertices, then a cycle is created.arrow_forwardProve 1 For a graph G = (V, E), a forest F is any set of edges of G that doesnot contain any cycles. M = (E, F) where F = {F ⊆ E : F is a forest of G} is amatroid.arrow_forward
- Let G be a connected graph, and let T1, T2 be two spanning trees. Prove thatT1 can be transformed to T2 by a sequence of intermediate trees, each obtainedby deleting an edge from the previous tree and adding another.arrow_forwardConsider a graph with nodes and directed edges and let an edge from node a to node b berepresented by a fact edge (a,b). Define a binary predicate path that is true for nodes c and dif, and only if, there is a path from c to d in the graph.arrow_forward. Let G be a weighted, connected, undirected graph, and let V1 and V2 be a partition of the vertices of G into two disjoint nonempty sets. Furthermore, let e be an edge in the minimum spanning tree for G such that e has one endpoint in V1 and the other in V2. Give an example that shows that e is not necessarily the smallest- weight edge that has one endpoint in V1 and the other in V2.arrow_forward
- Show that a bottleneck SPT of a graph is identical to an MST of an undirected graph. It provides the path between each pair of vertices v and w whose longest edge is as short as feasible.arrow_forwardProve Proposition : For any vertex v reachable from s, BFS computes a shortest path from s to v (no path from s to v has fewer edges).arrow_forwardSuppose G = (V, E) is an undirected connected weighted graph such that all its edge weightsare distinct. Prove that that the minimum spanning tree of G is unique.arrow_forward
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