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 35.2, Problem 4E
Program Plan Intro
To demonstrate that there exist a polynomial time estimation
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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…
Suppose we have the following undirected graph, and we know that the two
bolded edges (B-E and G-E) constitute the global minimum cut of the graph.
1. If we run the Karger’s algorithm for just one time to find the global minimum cut, what is the probability for the algorithm to find the minimum cut correctly? Please show your reasoning process.
2. How many times do we need to run the Karger’s algorithm if we want to guarantee that the probability of success is greater than or equal to
0.95, by “success” we mean that there is at least one time the Karger’s algorithm correctly found the minimum cut. Please show your reasoning process. [You do not have to work out the exact value of a logarithm]
How would you modify the dynamic programming algorithm for the coin collecting problem if some cells on the board are inaccessible for the robot? Apply your algorithm to the board below, where the inaccessible cells are shown by X’s. How many optimal paths are there for this board? You need to provide 1) a modified recurrence relation, 2) a pseudo code description of the algorithm, and 3) a table that stores solutions to the subproblems.
Chapter 35 Solutions
Introduction to Algorithms
Ch. 35.1 - Prob. 1ECh. 35.1 - Prob. 2ECh. 35.1 - Prob. 3ECh. 35.1 - Prob. 4ECh. 35.1 - Prob. 5ECh. 35.2 - Prob. 1ECh. 35.2 - Prob. 2ECh. 35.2 - Prob. 3ECh. 35.2 - Prob. 4ECh. 35.2 - Prob. 5E
Ch. 35.3 - Prob. 1ECh. 35.3 - Prob. 2ECh. 35.3 - Prob. 3ECh. 35.3 - Prob. 4ECh. 35.3 - Prob. 5ECh. 35.4 - Prob. 1ECh. 35.4 - Prob. 2ECh. 35.4 - Prob. 3ECh. 35.4 - Prob. 4ECh. 35.5 - Prob. 1ECh. 35.5 - Prob. 2ECh. 35.5 - Prob. 3ECh. 35.5 - Prob. 4ECh. 35.5 - Prob. 5ECh. 35 - Prob. 1PCh. 35 - Prob. 2PCh. 35 - Prob. 3PCh. 35 - Prob. 4PCh. 35 - Prob. 5PCh. 35 - Prob. 6PCh. 35 - Prob. 7P
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- Demonstrate that the decision problem variant is NP-complete; Exists, given a graph G and a cost objective c, a spanning tree where the maximum payment of any vertex does not exceed c?arrow_forwardPlease answer the following question in full detail. Please be specifix about everything: You have learned before that A∗ using graph search is optimal if h(n) is consistent. Does this optimality still hold if h(n) is admissible but inconsistent? Using the graph in Figure 1, let us now show that A∗ using graph search returns the non-optimal solution path (S,B,G) from start node S to goal node G with an admissible but inconsistent h(n). We assume that h(G) = 0. Give nonnegative integer values for h(A) and h(B) such that A∗ using graph search returns the non-optimal solution path (S,B,G) from S to G with an admissible but inconsistent h(n), and tie-breaking is not needed in A∗.arrow_forwardIt is NP-complete to determine whether an arbitrary graph has chromatic number k, where k >= 3. However, determining whether an arbitrary graph has chromatic number 2 is in P. Given a graph G on n vertices, create an algorithm that will return TRUE if χ(G) = 2 and FALSE if χ(G) ≠ 2. Clearly explain how your algorithm works, why it guarantees the correct output, and determine the running time of your algorithm.arrow_forward
- Consider the G-graph below. The numbers on the Edges are the gain values, not the cost. Now let's consider the Traveling Salesman Problem. The goal is to find the round that gives the maximum winnings (not the minimum cost). a) Apply the branch-and-bound algorithm to find the round with the maximum gain in the graph below. b) Is the branch-and-bound algorithm always guaranteed to find the optimal result? Is it a polynomial time algorithm?arrow_forwardCan we modify Dijkstra’s algorithm to solve the single-source longest path problem by changing minimum to maximum? If so, then explain why this algorithm correct. If not, then provide a counterexample.arrow_forwardConsider a graph G in the following Find a path from a to g in the graph G using each search strategy of depth-first search, breadth-first search, least-cost search, best-first search, and A* search. Is the returned solution path an optimal one?arrow_forward
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