A rout map connecting cities and the available roads with their distances are given in the weighted graph. B 2 3 2 a) Use Dijkstra's algorithm to find the shortest paths from city A to other cities. A 6 D b) What are the shortest paths and their lengths from A to the following cities. 4 K E i. F ii. K iii. L 7, 2.

A rout map connecting cities and the available roads with their distances are given in the weighted graph. B 2 3 2 a) Use Dijkstra's algorithm to find the shortest paths from city A to other cities. A 6 D b) What are the shortest paths and their lengths from A to the following cities. 4 K E i. F ii. K iii. L 7, 2.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Explain what adjustments if any need to be made in Dijkstra’s algorithm and/or in an underlying graph to solve the following problems. a. Solve the single-source shortest-paths problem for directed weighted graphs. b. Find a shortest path between two given vertices of a weighted graph or digraph. (This variation is called the single-pair shortest-path problem.) c. Find the shortest paths to a given vertex from each other vertex of a weighted graph or digraph. (This variation is called the single-destination shortest-paths problem.) d. Solve the single-source shortest-paths problem in a graph with nonnegative numbers assigned to its vertices (and the length of a path defined as the sum of the vertex numbers on the path).
26. Solve the following a) For the given set U, assume a family of subsets (F) on your own and apply the approximation algorithm to find the set cover. U={2,3,4,6,7,8} b) For the following weighted graph, apply approximation algorithm for solving traveling salesman problem. 3 1 4 2 5 5 27. For the following undirected graph, apply the approximation algorithm to find vertex cover. 2 6 1 3 5 4 7
Consider 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?
Algorithm : (1. Single-destination Shortest Path, 2. Bellman-Ford, 3. Negative-Weight Cycles), Dynamic Programming Define and prove a recurrence for the following problem: Given a directed graph G = (V, E) with edge weight function w : E → R and a source vertex s ∈ V , find a shortest path from s to v for every vertex v ∈ V .
1. Let G = (V, E) be a DAG with weights associated with verticesand edges. Design a Dynamic Programming Algorithm for solvingthe longest path problem.
ou are given a directed graph G = (V, E) and two vertices s and t. Moreover, each edge of this graph is colored either blue or red. Your goal is to find whether there is at least one path from s to t such that all red edges in this path appear after all blue edges (the path may not contain any blue edges or any red edges, but if it has both types of edges, all red edges should appear after all blue edges). Design and analyze an algorithm for solving this problem in O(n + m) time.
Algorithm: Network Flow(Maximu Flow, Ford-Fulkerson) and Application of Flow (Minimum Cuts, Bipartite Matching) Consider a flow network and an arbitrary s, t-cut (S, T). We know that by definition s must always be on the S "side" of a cut and t is always going to be on the T "side" of the cut. Obviously, this is true for any cut. Now, consider minimum cuts. This is obviously still true for s and t, but what about other vertices in the flow network? Are there vertices that will always be on one side or the other in every minimum cut? Let's define these notions more concretely. • We say a vertex v is source-docked if v ∈ S for all minimum cuts (S, T). • We say a vertex v is sink-docked if v ∈ T for all minimum cuts (S, T). • We say a vertex v is undocked if v is neither source-docked nor sink-docked. That is, there exist minimum cuts (S, T) and (S 0 , T0 ) such that v ∈ S and v ∈ T' Give an algorithm that takes as input a flow network G and assigns each vertex to one of the three…
Use Dijkstra to find all shortest path from H to all other nodes in following graph. Please show written work with answer plss help!
In the Given graph G=(V, E), find Shortest paths among all pairs of vertices by using Floyed Warshall Algorithm. V={a, b, c, d, e} and E={(a, b, 3), (a, c, 8), (a, e, 4), (b, d, 1), (b, e, 7), (b, c, 4), (d, c, -5), (d, b, 6), (d, a, 2), (e, d, 6)}. Note: Numeric value is the weight of the corresponding edge.
Problem 1 A jogger wants to follow the least undesirable cycle of roads starting at her home. Each road has an "index of undesirability" and can be traversed in either direction; the jogger must follow a nonempty cycle of roads and no road can be used twice. Formulated as a graph problem, the jogger has an undirected weighted graph G = (V, E), and must determine the nonempty cycle of minimum weight starting (and ending) at vertex s. 1. Show how to use multiple applications of Dijkstra's shortest path algorithm to obtain the optimum jogger's route in time O(|V | 2 log |V | + |E||V |). Be precise: each time you want to use Dijkstra's explain which graph is the input of the algorithm, 2. Let T be the shortest path tree constructed by Dijkstra's shortest path algorithm for starting vertex s in G. Prove that some optimum jogger's route has all but one of its edges in T , and furthermore, that s is the lowest common ancestor in T of the end points of that edge. 3. Use the result in…
Assume that we are given an undirected graph G=(V,E). Consider that Dijkstra's algorithm found a shortest path in G, called SP, between two nodes A and X of V. Is it true or false that if we reverse the nodes on SP, we get a shortest path from X to A? Prove or disprove.
please answer both of the questions. 7. The Bellman-Ford algorithm for single-source shortest paths on a graph G(V,E) as discussed in class has a running time of O|V |3, where |V | is the number of vertices in the given graph. However, when the graph is sparse (i.e., |E| << |V |2), then this running time can be improved to O(|V ||E|). Describe how how this can be done.. 8. Let G(V,E) be an undirected graph such that each vertex has an even degree. Design an O(|V |+ |E|) time algorithm to direct the edges of G such that, for each vertex, the outdegree is equal to the indegree. Please give proper explanation and typed answer only.