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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Describe the values of k is
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Given 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?)
We know that when we have a graph with negative edge costs, Dijkstra’s algorithm is not guaranteed to work.
(a) Does Dijkstra’s algorithm ever work when some of the edge costs are negative? Explain why or why not.
(b) Find an algorithm that will always find a shortest path between two nodes, under the assumption that at most one edge in the input has a negative weight. Your algorithm should run in time O(m log n), where m is the number of edges and n is the number of nodes. That is, the runnning time should be at most a constant factor slower than Dijkstra’s algorithm. To be clear, your algorithm takes as input
(i) a directed graph, G, given in adjacency list form. (ii) a weight function f, which, given two adjacent nodes, v,w, returns the weight of the edge between them. For non-adjacent nodes v,w, you may assume f(v,w) returns +1. (iii) a pair of nodes, s, t. If the input contains a negative cycle, you should find one and output it. Otherwise, if the graph contains at least one…
We 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.
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- Let 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_forwarda) Why can Dijkstra's algorithm not work properly on graphs with negative weighted edges? Explain with example. Implement the greedy approach for coin change algorithm and show your step by step approach to give change of 139 in coin change with {1, 3, 5, 25, 45, 60} unit values. c) Write the procedure for calculating GCD and LCM of the following numbers. Also write the answers at last. 2, 4, 5, 20arrow_forward
- If n points are connected to fom a closed polygon as shown below, the area of the polygon can be compuled as n-2 Area = (%)E (*»1 + x ) (y»1 - y ) =0 Notice that although the ilustrated polygon has only 6 distinct comers, n for his polygon is 7 because the algorithmexpects that the last point (x.ya) will be repeat of the initial point, (Ko.yo). Define a structure for a point. Each point contains x coordinate and y coordinate. The represe ntation of a Polygon must be an array of structures in your program. Write a C program that takes the number of actual points (n-1) from the user. After that, user enters x and y coordinates of each point. (The last point will be repeat of the initial point). Writo a compute Are a function which returns the area of the Polygon. Print he area of the Polygon in main. Display the area with wo digts after the decimal point. Note: The absolute value can be computed with fabs function. Example: double x.50: fabs(x) is 5.0 double x 0.0: fabs(x) is 0.0 double…arrow_forwardThe following solution designed from a problem-solving strategy has been proposed for finding a minimum spanning tree (MST) in a connected weighted graph G: Randomly divide the vertices in the graph into two subsets to form two connected weighted subgraphs with equal number of vertices or differing by at most Each subgraph contains all the edges whose vertices both belong to the subgraph’s vertex set. Find a MST for each subgraph using Kruskal’s Connect the two MSTs by choosing an edge with minimum wight amongst those edges connecting Is the final minimum spanning tree found a MST for G? Justify your answer.arrow_forwardLet G = (V, E) be a DAG, where every edge e = ij and every vertex x have positive weighs w(i,j) and w(x), respectively, associated with them. Design an algorithm for computing a maximum weight path. What is the time complexity of your algorithm? (You must start with the correct definitions, and then write a recurrence relation.)arrow_forward
- the shortest path from S to C indeed has distance 7, and this path is S -> E -> D -> A -> C.arrow_forwardLet 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_forwardSuppose you are given a connected undirected weighted graph G with a particular vertex s designated as the source. It is also given to you that weight of every edge in this graph is equal to 1 or 2. You need to find the shortest path from source s to every other vertex in the graph. This could be done using Dijkstra’s algorithm but you are told that you must solve this problem using a breadth-first search strategy. Design a linear time algorithm (Θ(|V | + |E|)) that will solve your problem. Show that running time of your modifications is O(|V | + |E|). Detailed pseudocode is required. Hint: You may modify the input graph (as long as you still get the correct shortest path distances).arrow_forward
- The book demonstrated that a poisoned reverse will prevent the count-to-infinity problem caused when there is a loop involving three directly connected nodes. However, other loops are possible. Will the poisoned reverse solve the general case count-to-infinity problem encountered by Bellman-Ford? -Yes, the poisoned reverse will prevent a node from offering a path that includes preceding nodes in the loop. -It will not, preceeding nodes may still be used in the computation of the distance vector offered by a given node.arrow_forwardYou are given a graph G = (V, E) with positive edge weights, and a minimum spanning tree T = (V, E') with respect to these weights; you may assume G and T are given as adjacency lists. Now suppose the weight of a particular edge e in E is modified from w(e) to a new value w̃(e). You wish to quickly update the minimum spanning tree T to reflect this change, without recomputing the entire tree from scratch. There are four cases. In each case give a linear-time algorithm for updating the tree. Note, you are given the tree T and the edge e = (y, z) whose weight is changed; you are told its old weight w(e) and its new weight w~(e) (which you type in latex by widetilde{w}(e) surrounded by double dollar signs). In each case specify if the tree might change. And if it might change then give an algorithm to find the weight of the possibly new MST (just return the weight or the MST, whatever's easier). You can use the algorithms DFS, Explore, BFS, Dijkstra's, SCC, Topological Sort as…arrow_forwardConsider the problem of finding the length of a "longest" path in a weighted, not necessarily connected, dag. We assume that all weights are positive, and that a "longest" path is a path whose edge weights add up to the maximal possible value. For example, for the following graph, the longest path is of length 15: 9. 9. (h) 2 3 Use a dynamic programming approach to the problem of finding longest path in a weighted dag.arrow_forward
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