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 19.4, Problem 2E
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- Suppose there is undirected graph F with nonnegative edge weights we ≥ 0. You have also calculated the minimum spanning tree of F and also the shortest paths to all nodes from a particular node p ∈ V . Now, suppose that each edge weight is increased by 1, so the new weights are we′ = we + 1. (a) Will there be a change of the minimum spanning tree? Provide an example where it does or prove that it cannot change. (b) (3 points) Will the shortest paths from p change? Provide an example where it does or prove that it cannot change.arrow_forwardGiven an undirected weighted graph G with n nodes and m edges, and we have used Prim’s algorithm to construct a minimum spanning tree T. Suppose the weight of one of the tree edge ((u, v) ∈ T) is changed from w to w′, design an algorithm to verify whether T is still a minimum spanning tree. Your algorithm should run in O(m) time, and explain why your algorithm is correct. You can assume all the weights are distinct. (Hint: When an edge is removed, nodes of T will break into two groups. Which edge should we choose in the cut of these two groups?)arrow_forward* Simulate the node expansion from start state (s) to reach goal state (G) - Using Geedy, A* - Given heuristic function: h(a) = 8.0 h(b) = 9.0 h(c) = 7.0 h(d) = 5.0 h(f) = 4.0 h(h) = 2.0 h(g) = 0 h(j) = 3.0 h(i) = 1.0arrow_forward
- a) Draw the connected subgraph of the given graph above which contains only four nodes ACGB and is also a minimum spanning tree with these four nodes. What is its weighted sum? Draw the adjacency matrix representation of this subgraph (use boolean matrix with only 0 or 1, to show its adjacency in this case).b) Find the shortest path ONLY from source node D to destination node G of the given graph above, using Dijkstra’s algorithm. Show your steps with a table as in our course material, clearly indicating the node being selected for processing in each step.c) Draw ONLY the shortest path obtained above, and indicate the weight in each edge in the diagram. Also determine the weighted sumarrow_forwardLet G = (V, E) be a connected, undirected graph, and let s be a fixed vertex in G. Let TB be the spanning tree of G found by a bread first search starting from s, and similarly TD the spanning tree found by depth first search, also starting at s. (As in problem 1, these trees are just sets of edges; the order in which they were traversed by the respective algorithms is irrelevant.) Using facts, prove that TB = TD if and only if TB = TD = E, i.e., G is itself a tree.arrow_forwardIn a lecture the professor said that for every minimum spanning tree T of G there is an execution of the algorithm of Kruskal which delivers T as a result. ( Input is G). The algorithm he was supposedly talking about is: Kruskal() Precondition. N = (G, cost) is a connected network with n = |V| node and m = |E| ≥ n − 1 edges.All edges of E are uncolored. postcondition: All edges are colored. The green-colored edges together with V form one MST by N. Grand Step 1: Sort the edges of E in increasing weight: e1 , e2, . . . , em Grand step 2: For t = 0.1, . . . , m − 1 execute: Apply Kruskal's coloring rule to the et+1 edge i dont really understand this statement or how it is done. can someone explain me what he meant?arrow_forward
- Given a graph G = (V, E), let us call G an almost-tree if G is connected and G contains at most n + 12 edges, where n = |V |. Each edge of G has an associated cost, and we may assume that all edge costs are distinct. Describe an algorithm that takes as input an almost-tree G and returns a minimum spanning tree of G. Your algorithm should run in O(n) time.arrow_forwardSimulate the node expansion from start state (S) to reach goal state (G) using BFS, DFS, UCS, Greedy, A*, and also given heuristic function that h(A) = 8 , h(B) = 9, h(C) = 7, h(D) = 5, h(F) = 4, h(H) = 2, h(G) = 0, h(J) = 3,h(I) = 1arrow_forwardWe are given a simple connected undirected graph G = (V, E) with edge costs c : E → R+. We would like to find a spanning binary tree T rooted a given node r ∈ T such that T has minimum weight. Consider the following modifiedPrim algorithm that works similar to Prim’s MST algorithm: We maintain a tree T (initially set to be r by itself) and in each iteration of the algorithm, we grow T by attaching a new node T in the cheapest possible way such that we do not violate the binary constraint; if it is not possible to grow the tree, we declare the instance to be infeasible.1: function modifiedPrim(G=(V, E), r)2: T ← {r}3: while |T| < |V| do4: S ← {u ∈ V : u ∈ T and |children(u)| < 2}5: R ← {u ∈ V : u ∈/ T}6: if ∃ (u, v) ∈ E with u ∈ S and v ∈ R then7: let (u, v) be the minimum cost such edge8: Add (u, v) to T9: else10: return infeasible11: return THow would you either prove the correctness of modifiedPrim or provide a counter-example where it fails to return the correct answer.arrow_forward
- Suppose we want to use UCS and the A* algorithm on the graph below to find the shortest path from node S to node G. Each node is labeled by a capital letter and the value of a heuristic function. Each edge is labeled by the cost to traverse that edge. Perform A*, UCS, and BFS on this graph. Indicate the f, g, and h values of each node for the A*. e.g., S = 0 + 6 = 6 (i.e. S = g(S) + h(S) = f(S)). Additionally, show how the priority queue changes with time. Show the order in which the nodes are visited for BFS and UCS. Show the path found by the A*, UCS, and BFS algorithms on the graph above. Make this example inadmissible by changing the heuristic value at one of the nodes. What node do you choose and what heuristic value do you assign? What would be the A* algorithm solution then.arrow_forwardConsider the so-called k-Minimum Spanning Tree (k-MST) problem, which is defined as follows. An instance of the k-MST problem is given by a connected undirected graph G=(V,E) with edge weights w:E→Q and a natural number k>2. The question is to find a tree with exactly k nodes that is a subgraph of G and minimises the weight among all such trees. Informally, k-MST is the variant of the minimum spanning tree problem, where instead of a spanning tree one wants to find a tree with exactly k nodes. What would be the result if we apply Prim's, respectively Kruskal's, algorithm to the problem by stopping both algorithms after k−1 edges have been added? In the following we refer to these versions of Prim's and Kruskal's algorithm as the modified algorithm of Prime or Kruskal, respectively. a) consider the following graph. (image provided- image 1) For which of the following edge weights, assigned to the graph above, does the modified algorithm of Kruskal provide a wrong result assuming…arrow_forwardhow could someone solce this one? The Chinese postman problem: Consider an undirected connected graph and a given starting node. The Chinese postman has to find the shortest route through the graph that starts and ends in the starting node such that all links are passed. The same problem appears for instance for snow cleaning or garbage collection in a city. For a branch-and-bound algorithm, find a possible lower bound function. (Remark: If the problem is to pass through all nodes of the graph, it is called the travelling salesman problem - which needs different solution algorithms, but also of the branch-and-bound type).).arrow_forward
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