Prove that The first path from start vertex s to goal vertex r found by monotonic Algorithm A* is optimal.
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Prove that The first path from start vertex s to goal vertex r found by monotonic
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- The first path from start vertex s to goal vertex r found by monotonic Algorithm A* is optimal.The Floyd-Warshall algorithm is a dynamic algorithm for searching the shortest path in a graph. Each vertex pair has its assigned weight. You are asked to draw the initial directed graph and show the tables for each vertex from Mo to Ms by finding all the shortest paths. Below is the algorithm as a guide. Algorithm 1: Pseudocode of Floyd-Warshall Algorithm Data: A directed weighted graph G(V, E) Result: Shortest path between each pair of vertices in G for each de V do | distance|d][d] «= 0; end for each edge (s, p) € E do | distance[s][p] + weight(s, p); end n = cardinality(V); for k = 1 to n do for i = 1 to n do for j = 1 to n do if distancefi][j] > distance/i][k] + distance/k][j] then | distance i]lj] + distancefi|[k] + distance/k|[j]; end end end end Consider the relation R = {(1,4) =4, (2,1)=3, (2,5)=-3, (3,4)=2, (4,2)=1, (4,3)=1, (5,4)=2 } on A = (1,2,3,4,5) solve the Floyd-Warshall Algorithm.In the given graph: Identify the shortest path with the help of Dijikstra algorithm having minimum cost to reach vertex T if S is the source vertex 5 Q 7 P 4 R S 3 X Y 3 5 T U 8. 3 4 W V
- 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).Find the shortest path from any vertex in S to any vertex in T given a digraph with positive edge weights and two distinct subsets of vertices S and T. In the worst-case scenario, your algorithm should run in time proportional to E log V.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)
- 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…The Floyd-Warshall transitive closure algorithm is a dynamic programming algorithm: it makes use of many subproblem solutions to solve the whole problem. In particular, to know if there is a way to get from vertex A to B, what subproblem solutions does it use? OFor every other vertex X, see if there are paths from A to X and from X to B ODo a breadth first search to see if there is a path from A to B Do a depth first search to see if there is a path from A to B OCheck every path from A to B What's the runtime of the Floyd-Warshall transitive closure algorithm on a graph with N vertices, M edges O(N) Check O(N log N) O(M) O(N+M) O(N^2) O(N^3)Suppose 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).
- Create Algorithm for Algorithm A* for an assessment function that is monotonic.start vertex s; target vertex r in: graph G = (V, E); maps as: V VLocal: Heuristic lower limit estimate h(u v), open list S, cost function g(u v),Given an undirected graph G = <V,E>, a vertex cover is a subset of vertices S V such that for each edge (u,v) belongs to E, either u S or v S or both. The Vertex Cover Problem is to find minimum size of the set S. Consider the following algorithm to Vertex Cover Problem: (1) Initialize the result as {} (2) Consider a set of all edges in given graph. Let the set be E’. (3) Do following while E’ is not empty ...a) Pick an arbitrary edge (u,v) from set E’ and add u and v to result ...b) Remove all edges from E which are either incident on u or v. (4) Return result. It claim that this algorithm is exact for undirected connected graphs. Is this claim True or False? Justify the answer.Dijkstra's algorithm for computing lowest cost paths from a single source node is always correct for graphs without negative-cost edges. If a graph has negative-cost edges, the algorithm might or might not give the right answer. For the directed weighted graph below, list all nodes which Dijkstra's algorithm produces the wrong path and write the correct lowest-cost path. -2 7 -1 A E 3 3 1) The correct lowest-cost path from A to E should be 2) The lowest-cost path from A to E that produced from Dijkstra's algorithm B.