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 7E
Program Plan Intro
To calculate the total number of vertices in directed graph having weight function as
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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 be a directed graph where each edge is colored either red, white, or blue. A walk in G is called a patriotic walk if its sequence of edge colors is red, white, blue, red, white, blue, and soon. Formally ,a walk v0 →v1 →...vk is a patriotic walk if for all
0≤i<k, the edge vi →vi+1 is red if i mod3=0, white if i mod3=1,and blue if i mod3=2.
Given a graph G, you wish to find all vertices in G that can be reached from a given vertex v by a patriotic walk. Show that this can be accomplished by efficiently constructing a new graph G′ from G, such that the answer is determined by a single call to DFS in G′. Do not forget to analyze your algorithm.
True or false: let G be an arbitrary connected, undirected graph with a distinct cost c(e) on every edge e. suppose e* is the cheapest edge in G; that is, c(e*) <c(e) for every edge e is not equal to e*. Any minimum spanning tree T of G contains the edge e*
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 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.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_forwardLet G = (V, E) be a graph with vertex-set V = {1, 2, 3, 4, 5} and edge-set E = {(1, 2), (3, 2), (4, 3), (1, 4), (2, 4), (1, 3)}. (a) Draw the graph. Find (b) maximal degree, i.e. ∆(G), (c) minimal degree, i.e. δ(G), (d) the size of biggest clique, i.e. ω(G),(e) the size of biggest independent set, i.e. α(G), ter(f) the minimal number of colours needed to color the graph, i.e. χ(G).arrow_forward
- let us take any standard graph G=(v,e) and let us pretend each edge is the same exact weight. let us think about a minimum spanning tree of the graph G, called T = (V, E' ). under each part a and b illustrate then show that a) s a unique path between u and v in T for all u, v ∈ V . b) tree T is not unique. provide proofarrow_forwardG = (V,E,W) is a weighted connected (undirected) graph where all edges have distinct weights except two edges e and e′ which have the same weight. Suppose there is a Minimum Spanning Tree of G containing both e and e′. Prove that G has a unique Minimum Spanning Tree.arrow_forwardLet G be a connected graph that has exactly 4 vertices of odd degree: v1,v2,v3 and v4. Show that there are paths with no repeated edges from v1 to v2, and from v3 to v4, such that every edge in G is in exactly one of these paths.arrow_forward
- Find the shortest path from S to other nodes, on the given directed acyclic graph.Graph: R → A : 3 S → A : 1 A → C : 6 B → D : 3 C → E : 2R → S : 2 S → B : 2 B → A : 4 C → D : 1 D → E : 1 Answer: Topological Ordering: __________________________ Node Edge Relax? Update Shortest Path from S: Length Path R S A B C D Earrow_forwardConsider a directed graph G=(V,E) with n vertices, m edges, a starting vertex s∈V, real-valued edge lengths, and no negative cycles. Suppose you know that every shortest path in G from s to another vertex has at most k edges. How quickly can you solve the single-source shortest path problem? (Choose the strongest statement that is guaranteed to be true.) a) O(m+n) b) O(kn) c) O( km) d) O(mn)arrow_forwardLet 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 uniquearrow_forward
- Consider 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_forwardLet G= (V,E,w) weighted, directed graph, A ⊂ V and: x,y ∈ V-A. Build an efficient algorithm to find the minimum weight of a track oriented from x to y which: (A) Passes through nodes of A. (B)Does not pass through the nodes of A.arrow_forwardDiscrete mathematics. Let G = (V, E) be a simple graph4 with n = |V| vertices, and let A be its adjacency matrix of dimension n × n. We want to count the L-cycles : such a cycle, denoted by C = u0u1 · · · uL with uL = u0 contains L distinct vertices u0, . . . , uL-1 et L edges E(C) = {uiui+1 | 0 ≤ i ≤ L − 1} ⊆ E. Two cycles are distinct if the edge sets are different : C = C' if and only if E(C) = E(C'). We define the matrices D, T, Q, the powers of A by matrix multiplication : D = A · A = A2, T = A · D = A3, Q = A · T = A4. Consider the values on the diagonals. Prove that the number of 3-cycles N3 in the whole graph is N3 = 1/6 ∑ u∈V Tu,uarrow_forward
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