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 16.4, Problem 3E
Program Plan Intro
To show that if ( S , I ) is matriod then ( S , I’ ) is also matriod.
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Prove that equivalences of sets are transitive, for example if X ~ Y and Y ~ Z then X ~ Z.
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Chapter 16. Introduction to algorithms, Cormen
Let P be a simple polygon, let W be a finite minimal witness set for P, and let be an element of W. If sees past any reflex vertex of P, then must lie on an edge of P such that lies on.give Proof
Chapter 16 Solutions
Introduction to Algorithms
Ch. 16.1 - Prob. 1ECh. 16.1 - Prob. 2ECh. 16.1 - Prob. 3ECh. 16.1 - Prob. 4ECh. 16.1 - Prob. 5ECh. 16.2 - Prob. 1ECh. 16.2 - Prob. 2ECh. 16.2 - Prob. 3ECh. 16.2 - Prob. 4ECh. 16.2 - Prob. 5E
Ch. 16.2 - Prob. 6ECh. 16.2 - Prob. 7ECh. 16.3 - Prob. 1ECh. 16.3 - Prob. 2ECh. 16.3 - Prob. 3ECh. 16.3 - Prob. 4ECh. 16.3 - Prob. 5ECh. 16.3 - Prob. 6ECh. 16.3 - Prob. 7ECh. 16.3 - Prob. 8ECh. 16.3 - Prob. 9ECh. 16.4 - Prob. 1ECh. 16.4 - Prob. 2ECh. 16.4 - Prob. 3ECh. 16.4 - Prob. 4ECh. 16.4 - Prob. 5ECh. 16.5 - Prob. 1ECh. 16.5 - Prob. 2ECh. 16 - Prob. 1PCh. 16 - Prob. 2PCh. 16 - Prob. 3PCh. 16 - Prob. 4PCh. 16 - Prob. 5P
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- Given a graph G = (V;E) an almost independent set I V is a generalization of an independent set whereeach vertex u 2 I is connected to at most one other vertex in I but no more. In other words for all u; v 2 Iif (u; v) 2 E then (u; z) =2 E for all z 2 I (equvalently (v; z) =2 E). Note that every independent set is alsoalmost independent. Prove that the problem of nding whether there exists an almost independent set ofsize k for some k, is NP-complete.arrow_forwardnot handwritten Find with proof every graph G for which ex(n, G) is defined for all positive integers n andthere exists a constant c such that ex(n, G) = ex(m, G) for all integers m, n > c.arrow_forwardFor the following simple graphs G_1=(V_1,E_1)G1=(V1,E1) and G_2=(V_2,E_2)G2=(V2,E2) (described by their vertex and edge sets) decide whether they are isomorphic. If they are then prove it, and if they are not give a convincing argument that explains why not: V_1=\{ a,b,c,d,e,f,g,h\}V1={a,b,c,d,e,f,g,h}, E_1=\{ ac,ag,ah,bc,bd,bh,ce,de,df,fg,fh\}E1={ac,ag,ah,bc,bd,bh,ce,de,df,fg,fh}, V_2=\{ 1,2,3,4,5,6,7,8\}V2={1,2,3,4,5,6,7,8}, E_2=\{ 12,15,16,25,27,34,36,37,38,48,67\}E2={12,15,16,25,27,34,36,37,38,48,67}arrow_forward
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- Write A greedy algorithm to find the maximal independent sets for matroids step by step.arrow_forwardRecall the Clique problem: given a graph G and a value k, check whether G has a set S of k vertices that's a clique. A clique is a subset of vertices S such that for all u, v € S, uv is an edge of G. The goal of this problem is to establish the NP-hardness of Clique by reducing VertexCover, which is itself an NP-hard problem, to Clique. Recall that a vertex cover is a set of vertices S such that every edge uv has at least one endpoint (u or v) in S, and the VertexCover problem is given a graph H and a value 1, check whether H has a vertex cover of size at most 1. Note that all these problems are already phrased as decision problems, and you only need to show the NP-Hardness of Clique. In other words, we will only solve the reduction part in this problem, and you DO NOT need to show that Clique is in NP. Q4.1 Let S be a subset of vertices in G, and let C be the complement graph of G (where uv is an edge in C if and only if uv is not an edge in G). Prove that for any subset of vertices…arrow_forward5. (This question goes slightly beyond what was covered in the lectures, but you can solve it by combining algorithms that we have described.) A directed graph is said to be strongly connected if every vertex is reachable from every other vertex; i.e., for every pair of vertices u, v, there is a directed path from u to v and a directed path from v to u. A strong component of a graph is then a maximal subgraph that is strongly connected. That is all vertices in a strong component can reach each other, and any other vertex in the directed graph either cannot reach the strong component or cannot be reached from the component. (Note that we are considering directed graphs, so for a pair of vertices u and v there could be a path from u to v, but no path path from v back to u; in that case, u and v are not in the same strong component, even though they are connected by a path in one direction.) Given a vertex v in a directed graph D, design an algorithm for com- puting the strong connected…arrow_forward
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