EBK DATA STRUCTURES AND ALGORITHMS IN C
4th Edition
ISBN: 9781285415017
Author: DROZDEK
Publisher: YUZU
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Write pseudocode to find all pairs shortest paths using the technique used in Bellman-Ford's algorithm so that it will produce the same matrices like Floyd-Warshall algorithm produces. Also provide the algorithm to print the paths for a source vertex and a destination vertex. Describe the properties of the algorithm you provide and the run time for your algorithm in detail.
You are given a connected, undirected graph G. Devise an algorithm that produces a path
that traverses each edge in G exactly once in each direction. A vertex may occur multiple
times on the path. Provide a short justification about why your algorithm is correct, and
analyze its efficiency.
We are given an undirected connected graph G = (V, E) and vertices s and t.Initially, there is a robot at position s and we want to move this robot to position t by moving it along theedges of the graph; at any time step, we can move the robot to one of the neighboring vertices and the robotwill reach that vertex in the next time step.However, we have a problem: at every time step, a subset of vertices of this graph undergo maintenance andif the robot is on one of these vertices at this time step, it will be destroyed (!). Luckily, we are given theschedule of the maintenance for the next T time steps in an array M [1 : T ], where each M [i] is a linked-listof the vertices that undergo maintenance at time step i.Design an algorithm that finds a route for the robot to go from s to t in at most T seconds so that at notime i, the robot is on one of the maintained vertices, or output that this is not possible. The runtime ofyour algorithm should ideally be O((n + m) ·T ) but you will…
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- Draw a simple, connected, weighted graph with 8 vertices and 16 edges, each with unique edge weights. Identify one vertex as a “start” vertex and illustrate a running of Dijkstra’s algorithm on this graph. Problem R-14.23 in the photoarrow_forwardIn this problem you will design an algorithm that takes as input a directed acyclic graph G = (V,E) and two vertices s and t, and returns the number of simple paths from s tot in G. For example, the directed acyclic graph below contains exactly four simple paths from vertex p to vertex v: pov, poryv, posryv, and psryv. Notice: your algorithm needs only to count the simple paths, not list them. m y W Design a recursive backtracking (brute-force) algorithm that determines the number of paths from s to t. Write down the pseudocode of your algorithm and prove its correctness, i.e., convince us that it works beyond any doubt. (Hint: using induction.).arrow_forwardRun Dijkstra's algorithm on the following graph, starting from vertex A. Whenever there are multiple choices of vertex at the same time, choose the one that is alphabetically first. You are expected to show how you initialize the graph, how you picked a vertex and update the d values at the each, and what is final shortest distance of each vertex from A. B 11 A F Earrow_forward
- Bellman-Ford algorithm Draw a graph G with weights of edges ranging from 3 to 9, is it possible to calculate the LONGEST PATH without altering the algorithm at all? Justify your answer by providing solid reasons.arrow_forwardNeed in JAVA. Implement the algorithm(Prim’s algorithm) using an adjacency matrix for weighted graphs based on the graph provided below.arrow_forwardRun the Bellman-Ford Algorithm on the graph below to find the shortest path distances from the vertex S to all other vertices. Make sure that you give the initialization and show all the steps of the algorithm. 10 A 1 2 1 1 -2 -7 3. -1 8. F.arrow_forward
- In the figure below there is a weighted graph, dots represent vertices, links represent edges, and numbers represent edge weights. S 2 1 2 1 2 3 T 1 1 2 4 (a) Find the shortest path from vertex S to vertex T, i.e., the path of minimum weight between S and T. (b) Find the minimum subgraph (set of edges) that connects all vertices in the graph and has the smallest total weight (sum of edge weights). 2. 3.arrow_forwardDescribe an algorithm to insert and delete edges in the adjacency list representation for an undirected graph. Remember that an edge (i, j) appears on the adjacency list for both vertex i and vertex j.arrow_forwardLet V= {cities of Metro Manila} and E = {(x; y) | x and y are adjacent cities in Metro Manila.} (a) Draw the graph G defined by G = (V; E). You may use initials to name a vertex representing a city. (b) Apply the Four-Color Theorem to determine the chromatic number of the vertex coloring for G.arrow_forward
- Write a pseudocode to find all pairs shortest paths using the technique used in Bellman-Ford's algorithm so that it will produce the same matrices like Floyd-Warshall algorithm produces. Also provide the algorithm to print the paths for a source vertex and a destination vertex. For the pseudocode consider the following definition of the graph - Given a weighted directed graph, G = (V, E) with a weight function wthat maps edges to real-valued weights. w(u, v) denotes the weight of an edge (u, v). Assume vertices are labeled using numbers from1 to n if there are n vertices.arrow_forwardFill in the blank Dijkstra's algorithm works because, on every shortest path p from a source vertex u to a target vertex v, there is a (predecessor) vertex w in p immediately before v such that removing v from p yields the shortest path from u to w. In other words, the path through the previous vertex is also the shortest path. Thus, choosing an edge from the previous vertex that brings us to v with the __ cost always yields the shortest path to v.arrow_forwardImplement the dijkstra's algorithm on a directed graph from a given vertex. all edges have non-negative edge weights. Output the edge as they are added to the shortest path trees. Compute and print the weight of the shortest path to every reachable vertex. The source vertex is S. show the following: -program code -Screenshot of the output -Representation of the graph transversalarrow_forward
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