• Let L, R be two sets, with cardinalities |Z| = 1 and |R| = r. What is the maximum number of edges in a bipartite graph G=(L_R_E). • Let L, R be two sets, with cardinalities |Z| = 1 and |R| = r. Give the formula for the number of bipartite graphs G=(L.RE)?
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- Let G be a graph, where each edge has a weight. A spanning tree is a set of edges that connects all the vertices together, so that there exists a path between any pair of vertices in the graph. A minimum-weight spanning tree is a spanning tree whose sum of edge weights is as small as possible. Last week we saw how Kruskal's Algorithm can be applied to any graph to generate a minimum-weight spanning tree. In this question, you will apply Prim's Algorithm on the same graph from the previous quiz. You must start with vertex A. H 4 G D J 9 4 7 10 6 8 В F A 18 E There are nine edges in the spanning tree produced by Prim's Algorithm, including AB, BC, and IJ. Determine the exact order in which these nine edges are added to form the minimum-weight spanning tree. 3.Consider a weighted, directed graph G with n vertices and m edges that have integer weights. A graph walk is a sequence of not-necessarily-distinct vertices v1, v2, ... , Vk such that each pair of consecutive vertices Vi, Vi+1 are connected by an edge. This is similar to a path, except a walk can have repeated vertices and edges. The length of a walk in a weighted graph is the sum of the weights of the edges in the walk. Let s, t be given vertices in the graph, and L be a positive integer. We are interested counting the number of walks from s to t of length exactly L. Assume all the edge weights are positive. Describe an algorithm that computes the number of graph walks from s to t of length exactly L in O((n+ m)L) time. Prove the correctness and analyze the running time. (Hint: Dynamic Programming solution)Given N cities represented as vertices V₁, V2,..., UN on an undirected graph (i.e., each edge can be traversed in both directions). The graph is fully-connected where the edge eij connecting any two vertices v; and vj is the straight-line distance between these two cities. We want to search for the shortest path from v₁ (the source) to VN (the destination). Assume that all edges have different values, and e₁, has the largest value among the edges. That is, the source and destination have the largest straight-line distance. Compare the lists of explored vertices when we run the uniform-cost search and the A* search for this problem. Hint: The straight-line distance is the shortest path between any two cities. If you do not know how to start, try to run the algorithms by hand on some small cases first; but remember to make sure your graphs satisfy the conditions in the question.
- Let DIST (u, v) denote the distance between vertex u and v. It is well known that distances in graphs satisfy the triangle inequality. That is, for any three vertices u, v, w, DIST (u, v) ≤ DIST (u, w) + DIST (w, v). Let D∗ denote the distance between the two farthest nodes in G. Show that for any vertex s D∗ ≤ 2 max DIST (s, v).Given N cities represented as vertices V₁, V2, un on an undirected graph (i.e., each edge can be traversed in both directions). The graph is fully-connected where the edge ei,j connecting any two vertices vį and vj is the straight-line distance between these two cities. We want to search for the shortest path from v₁ (the source) to UN (the destination). Assume that all edges have different values, and e₁,N has the largest value among the edges. That is, the source and destination have the largest straight-line distance. Compare the lists of explored vertices when we run the uniform-cost search and the A* search for this problem. Hint: The straight-line distance is the shortest path between any two cities. If you do not know how to start, try to run the algorithms by hand on some small cases first; but remember to make sure your graphs satisfy the conditions in the question.Consider a graph G, where each of the edges have different weights. Let T1 be the minimum-weight spanning tree produced by Kruskal's Algorithm, and let T2 be the minimum-weight spanning tree produced by Prim's Algorithm. I claim that T1 and T2 must be identical spanning trees - i.e., the exact same set of edges must appear in both trees. Determine whether this claim is TRUE or FALSE. If your answer is TRUE, see if you can figure out why the claim is true. If your answer is FALSE, see if you can come up with a counterexample.
- Let G = (X ∪ Y, E) be a bipartite graph such that the vertices are partitioned into two groups Xand Y , and each edge has one end point in X and one end point in Y .A 2-1 generalized matching is a set of edges S ⊂ E satisfying the following two conditions:1. Every vertex in X belongs to at most two edges in S.2. Every vertex in Y belongs to at most one edge in S.Give an algorithm to find the size (number of edges) of maximum 2-1 generalized matchingLet 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-setBe G=(V, E)a connected graph and u, vEV. The distance Come in u and v, denoted by d(u, v), is the length of the shortest path between u'and v, Meanwhile he width from G, denoted as A(G), is the greatest distance between two of its vertices. a) Show that if A(G) 24 then A(G) <2. b) Show that if G has a cut vertex and A(G) = 2, then Ġhas a vertex with no neighbors.
- Consider the following edge-weighted graph G with 9 vertices and 16 edges: 40 60 90 50 70 30 80 10 130 150 120 100 110 140 Q6.1 Kruskal List the weights of the MST edges in the order that Kruskal's algorithm adds them to the MST. Your answer should be a sequence of 8 integers, with one space between each integer. 20Let G be a graph with n vertices. If the maximum size of an independent set in G is k, clearly explain why the minimum size of a vertex cover in G is n - k.Given N cities represented as vertices V₁, V2,...,UN on an undirected graph (i.e., each edge can be traversed in both directions). The graph is fully-connected where the edge ei, connecting any two vertices vį and vj is the straight-line distance between these two cities. We want to search for the shortest path from v₁ (the source) to UN (the destination). Assume that all edges have different values, and e₁, has the largest value among the edges. That is, the source and destination have the largest straight-line distance. Compare the lists of explored vertices when we run the uniform-cost search and the A* search for this problem. Hint: The straight-line distance is the shortest path between any two cities. If you do not know how to start, try to run the algorithms by hand on some small cases first; but remember to make sure your graphs satisfy the conditions in the question.