(a)
To represent the graph’s acyclic topological nature.
(a)
Explanation of Solution
Given Information: A graph
Explanation:
As mentioned, graph does not contained any self-loop therefore; every edge in graph
It means
(b)
To represent the Yen’s improvement in Bellman-Ford
(b)
Explanation of Solution
Given Information: Implementation of Bellman- Ford algorithm and a graph with relaxing edges of
Explanation:
Bellman- Ford algorithm is simpler than Dijkstra’s algorithm and worked very well with distributed systems. In the first step, it calculate the shortest path that is having at-most one edge in bottom-up approach then it will go for at most 2-edges and so-on up to
In Bellman- Ford algorithm and previous part a declare that the graph’s edges
(c)
To calculate the effects of running time complexity for Bellman-Ford algorithm.
(c)
Explanation of Solution
Given Information: Implementation of Bellman- Ford algorithm and a scheme for evaluating the time complexity.
Explanation:
It cannot improve the asymptotic running time of Bellman-Ford algorithm becausehaving a co-efficient of
The final runtime for the algorithm will be
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Chapter 24 Solutions
Introduction to Algorithms
- Given a directed graph with non-negative edge weights, suppose we have computed the shortest paths from a given source to all the other vertices. If we modify the graph in such a way that the weights of all the edges are doubled, then, the shortest paths remain the same and only the total weights of the paths change. O True O Falsearrow_forwardWe are given a directed graph G = (V, E) with edge weights w(e) for e E E. The edge weights are allowed to be negative. Let -C be the minimum value of the edge weights. Create a new set of edge weights w' (e) = w(e) + C. Thus these new weights are non-negative: w' (e) > 0 for all e € E. Run Dijkstra's algorithm from a specified start vertex s E V on G with these new weights w'. For every graph G, every weights w, and every s € V, the above algorithm is guaranteed to find the shortest paths in G (not the actual lengths but the paths) from s with respect to the original weights w: True False If you entered true, provide a short explanation. If you entered false provide a counterexample (show the graph and what the algorithm produces vs. the correct solution).arrow_forwardYou are given a bipartite graph G=(U,V,E), and an integer K. U and V are the two bipartitions of the graph such that |U| = |V| = N , and E is the edge set. The vertices of U are {1,2,...,N } and that of V are {N+1,N+2,...,2N }. You need to find out whether the total number of different perfect matchings in G is strictly greater than K or not. Recall that a perfect matching is a subset of E such that every vertex of the graph belongs to exactly one edge in the subset. Two perfect matchings are considered to be different even if one edge is different. Write a program in C++ programming language that prints a single line containing “Perfect” if the number of perfect matchings is greater than K, and “Not perfect” in other cases.Sample Input:3 5 21 42 62 53 53 5Output:Not Perfectarrow_forward
- Consider the Minimum-Weight-Cycle Problem: Input: A directed weighted graph G = :(V, E) (where the weight of edge e is w(e)) and an integer k. Output: TRUE if there is a cycle with total weight at most k and FALSE if there is no cycle with total weight at most k. Remember, a cycle is a list of vertices such that each vertex has an edge to the next and the final vertex has an edge to the first vertex. Each vertex can only occur once in the cycle. A vertex with a self-loop forms a cycle by itself. (a) Assume that all edge weights are positive. Give a polynomial-time algorithm for the Minimum-Weight-Cycle Problem. For full credit, you should: Give a clear description of your algorithm. If you give pseudocode, you should support it with an expla- nation of what the algorithm does. Give the running time of your algorithm in terms of the number of vertices n and the number of edges m. - You do not need to prove the correctness of your algorithm or the correctness of your running time…arrow_forwardGiven 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.arrow_forwardConsider the Minimum-Weight-Cycle Problem: Input: A directed weighted graph G (V, E) (where the weight of edge e is w(e)) and an integer k. Output: TRUE if there is a cycle with total weight at most k and FALSE if there is no cycle with total weight at most k. Remember, a cycle is a list of vertices such that each vertex has an edge to the next and the final vertex has an edge to the first vertex. Each vertex can only occur once in the cycle. A vertex with a self-loop forms a cycle by itself. (a) Assume that all edge weights are positive. Give a polynomial-time algorithm for the Minimum-Weight-Cycle Problem. For full credit, you should: - Give a clear description of your algorithm. If you give pseudocode, you should support it with an expla- nation of what the algorithm does. Give the running time of your algorithm in terms of the number of vertices n and the number of edges m. You do not need to prove the correctness of your algorithm or the correctness of your running time analysis.arrow_forward
- 4. Run the Bellman-Ford algorithm on the directed graph given blow, using vertex z as the source. In each pass, relax edges in the order (t, x.), (t, y), (t, z), (x, t), (y, x), (y, z), (s, t), (s, y), (z, s), (z, x). Show the d and values after each pass. -2 8 y INarrow_forwardConsider 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)arrow_forwardGiven 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.arrow_forward
- Consider a directed graph G = (V, E), and two distinct vertices u, v V. Recall that a set of U-V paths is non-overlapping if they have no edges in common among them, and a set C of edges disconnects from U if in the graph (V, E-C) there is no path from U to V. Suppose we want to show that for any set of non-overlapping paths P and any disconnecting set C, |P| ≤ |C|. Consider the proof that defines A = P, B = C and f(path q) = qC, and applies the Pigeonhole Principle to obtain the result. True or False: f is a well-defined function (i.e. it satisfies the 3 properties of a well- defined function). True Falsearrow_forwardGiven a graph that is a tree (connected and acyclic). (I) Pick any vertex v.(II) Compute the shortest path from v to every other vertex. Let w be the vertex with the largest shortest path distance.(III) Compute the shortest path from w to every other vertex. Let x be the vertex with the largest shortest path distance. Consider the path p from w to x. Which of the following are truea. p is the longest path in the graphb. p is the shortest path in the graphc. p can be calculated in time linear in the number of edges/verticesarrow_forward2. Let G = (V, E) be a directed weighted graph with the vertices V = {A, B, C, D, E, F) and the edges E= {(A, B, 12), (A, D, 17), (B, C, 8), (B, D, 13), (B, E, 15), (B, F, 13), (C, E, 12), (C, F, 25)}, where the third components is the cost. (a) Write down the adjacency list representation the graph G = (V, E).arrow_forward
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