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 22.3, Problem 8E
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To give a counter example to the conjecture that if
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Prove that in a breadth-first search on a undirected graph G, every edge iseither a tree edge or a cross edge, where x is neither an ancestor nor descendant of y, in cross edge (x, y).
In an undirected graph G, must every cut edge e be an edge in a depth-first search tree of G, or can e be a back edge? Give a proof or a counterexample. (Any edge whose removal disconnects the graph is called a cut edge.)
COMPLETE-SUBGRAPH problem is defined as follows: Given a graph G = (V, E) and an integer k, output yes if and only if there is a subset of vertices S ⊆ V such that |S| = k, and every pair of vertices in S are adjacent (there is an edge between any pair of vertices).
How do I show that COMPLETE-SUBGRAPH problem is in NP?
How do I show that COMPLETE-SUBGRAPH problem is NP-Complete?
(Hint 1: INDEPENDENT-SET problem is a NP-Complete problem.)
(Hint 2: You can also use other NP-Complete problems to prove NP-Complete of COMPLETE-SUBGRAPH.)
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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- 12 C F 4 1 A 11 10 7 8 3 B 9. D 2]arrow_forwardPart 2: Random GraphsA tournament T is a complete graph whose edges are all oriented. Given a completegraph on n vertices Kn, we can generate a random tournament by orienting each edgewith probability 12 in each direction.Recall that a Hamiltonian path is a path that visits every vertex exactly once. AHamiltonian path in a directed graph is a path that follows the orientations of thedirected edges (arcs) and visits every vertex exactly once. Some directed graphs havemany Hamiltonian paths.In this part, we give a probabilistic proof of the following theorem:Theorem 1. There is a tournament on n vertices with at least n!2n−1 Hamiltonian paths.For the set up, we will consider a complete graph Kn on n vertices and randomlyorient the edges as described above. A permutation i1i2 ...in of 1,2,...,n representsthe path i1 −i2 −···−in in Kn. We can make the path oriented by flipping a coin andorienting each edge left or right: i1 ←i2 →i3 ←···→in.(a) How many permutations of the vertices…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
- A graph is biconnected if every pair of vertices is connectedby two disjoint paths. An articulation point in a connected graph is a vertex that woulddisconnect the graph if it (and its adjacent edges) were removed. Prove that any graphwith no articulation points is biconnected. Hint : Given a pair of vertices s and t and apath connecting them, use the fact that none of the vertices on the path are articulationpoints to construct two disjoint paths connecting s and t.arrow_forwardShow that we can use a depth-first search of an undirected graph G to identify theconnected components of G, and that the depth-first forest contains as many treesas G has connected components. More precisely, show how to modify depth-firstsearch so that it assigns to each vertex an integer label :cc between 1 and k,where k is the number of connected components of G, such that u:cc D :cc ifand only if u and are in the same connected component.arrow_forwardWe 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_forward
- Suppose are you given an undirected graph G = (V, E) along with three distinct designated vertices u, v, and w. Describe and analyze a polynomial time algorithm that determines whether or not there is a simple path from u to w that passes through v. [Hint: By definition, each vertex of G must appear in the path at most once.]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_forwardFor any connected undirected graph G with n nodes and at least n edges, with positive distinct weights on all edges, if e denotes the edge that has the maximum weight in G, then the MST of G does not include e. True Falsearrow_forward
- 5. (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_forwardIs it necessary for every cut edge e in an undirected graph G to be an edge in G's depth-first search tree? Give at least two examples to back up your point. A cut edge is defined as any edge that cuts the graph.)arrow_forwardA Hamiltonian cycle of an undirected graph G = (V, E) is a simple cycle that contains each vertex in Vonly once. A Hamiltonian path of an undirected graph G = (V, E) is a simple path that contains each vertexin V only once. Let Ham-Cycle = {<G, u, v> : there is a Hamiltonian cycle between u and v in graph G} andHam-Path = {< G, u, v>: there is a Hamiltonian path between u and v in graph G}. Give a graph G containing vertex set {fit, sit, cruise, cow, 2023, spinner, filler} and an edge set E such that there is a Hamiltonian path, where "cruise" appears according to its alphabet order in the path;arrow_forward
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