For the weighted shortest path problem, let d, be the cost of reaching the current vertex v, let w be adjacent to v and assume the edge cost is cyw. Suppose that dw Was the costof reaching w prior to examining v. Then under what circumstances is w's distance lowered?A. dw > dyB. dw > dy + CywOC. dw > dy + 1D. dy > dw + Cyw

Answered: For the weighted shortest path problem,…

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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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.
Consider a directed graph G=(V,E) with n vertices, m edges, a starting vertex s∈V, real-valued edge lengths, and no negative cycles. Suppose you know that every shortest path in G from s to another vertex has at most k edges. How quickly can you solve the single-source shortest path problem? (Choose the strongest statement that is guaranteed to be true.) a) O(m+n) b) O(kn) c) O( km) d) O(mn)
Establish the Shortest-Paths Optimality Conditions in the Proposition P. Then, for any v reachable from s, the value of distTo[v] is the length of some path from s to v, with distTo[v] equal to infinity for all v not reachable from s. Let G be an edge-weighted digraph. s is a source vertex in G. distTo[] is a vertex-indexed array of path lengths in G. These numbers represent the lengths of the shortest pathways if and only if each edge e from v to w satisfies the condition that distTo[w] = distTo[v] + e.weight() (i.e., no edge is eligible).
Let A, B, C, D be the vertices of a square with side length 100. If we want to create a minimum-weight spanning tree to connect these four vertices, clearly this spanning tree would have total weight 300 (e.g. we can connect AB, BC, and CD). But what if we are able to add extra vertices inside the square, and use these additional vertices in constructing our spanning tree? Would the minimum-weight spanning tree have total weight less than 300? And if so, where should these additional vertices be placed to minimize the total weight? Let G be a graph with the vertices A, B, C, D, and possibly one or more additional vertices that can be placed anywhere you want on the (two-dimensional) plane containing the four vertices of the square. Determine the smallest total weight for the minimum-weight spanning tree of G. Round your answer to the nearest integer. Note: I encourage you to add n additional points (for n=1, 2, 3) to your graph and see if you can figure out where these point(s) need to…
Let A, B, C, D be the vertices of a square with side length 100. If we want to create a minimum-weight spanning tree to connect these four vertices, clearly this spanning tree would have total weight 300 (e.g. we can connect AB, BC, and CD). But what if we are able to add extra vertices inside the square, and use these additional vertices in constructing our spanning tree? Would the minimum-weight spanning tree have total weight less than 300? And if so, where should these additional vertices be placed to minimize the total weight? Let G be a graph with the vertices A, B, C, D, and possibly one or more additional vertices that can be placed anywhere you want on the (two-dimensional) plane containing the four vertices of the square. Determine the smallest total weight for the minimum-weight spanning tree of G. Round your answer to the nearest integer. Attention: Please don't just copy these two following answers, which are not correct at all. Thank you.…
(Shortest-paths optimality criteria) Demonstrate Proposition P. Let G be an edge-weighted digraph, with s as a source vertex in G and distTo[] a vertex-indexed array of path lengths in G such that the value of distTo[v] is the length of some path from s to v for all v accessible from s, and distTo[v] equal to infinity for all v not reachable from s. These are the lengths of the shortest routes if and only if they meet distTo[w] = distTo[v] + e.weight() for each edge e from v to w (or no edge is eligible).
Suppose that the road network is defined by the undirected graph, where the vertices represent cities and edges represent the road between two cities. The Department of Highways (DOH) decides to install the cameras to detect the bad driver. To reduce the cost, the cameras are strategically installed in the cities that a driver must pass through in order to go from one city to another city. For example, if there are two cities A and B such that the path that goes from A to B and the path that goes from B to A must pass the city C, then C must install the camera. Suppose that there are m cities and n roads. Write an O (m + n) to list all cities where cameras should be installed.
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.
The problem to solve is using a priority queue to compute a minimum spanning tree. Given a fully connected undirected graph where each edge has a weight, find the set of edges with the least total sum of weights. You are a civil engineer and you have been tasked with trying to find out the lowest cost way to build internet access in CIS-Land. There are X (3 ≤ X ≤ 100,000) towns in CIS-Land connected by Y (X ≤ Y ≤ 100,000) roads and you can travel between any two towns in CIS-Land by travelling some sequence of roads. With a limited budget you also know that the cheapest approach to obtain internet access is to install fiber-optic cables along existing roadways. Fortunately, you know the costs of laying fiber-optic cable down along all the roads, and you will be able to cost out how much money CIS-Land will need to spend to successfully complete the internet access project – that is, every town will be connected along some sequence of fiber-optic cables. Good thing, you are also…
With a weighted digraph, determine the monotonic shortest path between s and each of the other nodes. The path is monotonic if the weight of each edge along a route is strictly growing or strictly decreasing. The path ought to be easy to follow.(No vertices are repeated). Hint: To identify the optimal path, first relax the edges in ascending order; then relax the edges in descending order.
Find the monotonic shortest route from s to every other node in a weighted digraph. If the weight of every edge on a route is strictly increasing or strictly declining, the path is monotonic. The route should be straightforward.(no repeated vertices). Hint: First, relax the edges in ascending order to find the best route; then, relax the edges in declining order to find the best path.Find the monotonic shortest route from s to every other node in a weighted digraph. If the weight of every edge on a route is strictly increasing or strictly declining, the path is monotonic. The route should be straightforward.(no repeated vertices). Hint: First, relax the edges in ascending order to find the best route; then, relax the edges in declining order to find the best path.
Suppose that you want to get from vertex s to vertex t in an unweighted graph G = (V, E), but you would like to stop by vertex u if it is possible to do so without increasing the length of your path by more than a factor of α. Describe an efficient algorithm that would determine an optimal s-t path given your preference for stopping at u along the way if doing so is not prohibitively costly. (It should either return the shortest path from s to t or the shortest path from s to t containing u, depending on the situation)

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