Reachability. You are given a connected undirected graph G = (V, E ) as an adjacency...
Reachability. You are given a connected undirected graph G = (V, E ) as an adjacency list. The graph G might not be connected. You want to fill-in a two-dimensional array R[,] so that R[u,v] is 1 if there is a path from vertex u to vertex v. If no such path exists, then R[u,v] is 0.
From this two-dimensional array, you can determine whether vertex u is reachable from vertex v in O(1) time for any pair of vertices u and v.
Describe an algorithm that fills in R[,] in O(V 2 ) time. Briefly justify the running time of your algorithm.
Homework Answers
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Reachability. You are given a connected
undirected graph G = (V, E ) as an
adjacency...
Suppose you are given an undirected graph G. Find a pair of vertices (u, v) in...
Suppose you are given an undirected graph G. Find a pair of vertices (u, v) in G with the largest number of common adjacent vertices (neighbors). Give pseudocode for this algorithm and show the worst-case running time.
Consider an unweighted, undirected graph G = 〈V, E). The neighbourhood of a node u E V in the gr...
Consider an unweighted, undirected graph G = 〈V, E). The neighbourhood of a node u E V in the graph is the set of all nodes that are adjacent (or directly connected) to v. Subsequently, we can define the neighbourhood degree of the node v as the sum of the degrees of all its neighbours (those nodes that are directly connects to v) (a) Design an algorithm that returns a list containing the neighbourhood degree for each node v V,...
3. Graph Connected Components (25 pts) You are given an undirected, unweighted graph that may be...
3. Graph Connected Components (25 pts) You are given an undirected, unweighted graph that may be disconnected i.e. some vertices may not be reachable from other vertices. Every group of mutually reachable vertices forms an island, called a con- nected component. There is no path between vertices in different connected components. If a graph is not disconnected, then its vertices are in a single connected component, which is the entire graph itself. Implement a method using depth-first search that will...
Consider an unweighted, undirected graph G = 〈V, E). The neighbourhood of a node u E V in the gr...
This question needs to be done using pseudocode (not any
particular programming language). Thanks
Consider an unweighted, undirected graph G = 〈V, E). The neighbourhood of a node u E V in the graph is the set of all nodes that are adjacent (or directly connected) to v. Subsequently, we can define the neighbourhood degree of the node v as the sum of the degrees of all its neighbours (those nodes that are directly connects to v) (a) Design an...
Let G = (V, E, w) be a connected weighted undirected graph. Given a vertex s...
Let G = (V, E, w) be a connected weighted undirected graph. Given a vertex s ∈ V and a shortest path tree Ts with respect to the source s, design a linear time algorithm for checking whether the shortest path tree Ts is correct or not.(C pseudo)
2. Let G = (V, E) be an undirected connected graph with n vertices and with...
2. Let G = (V, E) be an undirected connected graph with n vertices and with an edge-weight function w : E → Z. An edge (u, v) ∈ E is sparkling if it is contained in some minimum spanning tree (MST) of G. The computational problem is to return the set of all sparkling edges in E. Describe an efficient algorithm for this computational problem. You do not need to use pseudocode. What is the asymptotic time complexity of...
Question 1: Given an undirected connected graph so that every edge belongs to at least one...
Question 1: Given an undirected connected graph so that every edge belongs to at least one simple cycle (a cycle is simple if be vertex appears more than once). Show that we can give a direction to every edge so that the graph will be strongly connected. Question 2: Given a graph G(V, E) a set I is an independent set if for every uv el, u #v, uv & E. A Matching is a collection of edges {ei} so...
2. Let G be an undirected graph. For every u,vE V(G), let dc(u,v) be the length of the shoertest ...
2. Let G be an undirected graph. For every u,vE V(G), let dc(u,v) be the length of the shoertest path from u to v. The diameter of G is he maximum distance bet In other words: max (de(u, v) u,vEV(G) the running time of your algorithm
2. Let G be an undirected graph. For every u,vE V(G), let dc(u,v) be the length of the shoertest path from u to v. The diameter of G is he maximum distance bet In...
hello there ,, can anyone give the solution of this Assuming a graph is represented as an adjacency matrix, write the ps...
hello there ,, can anyone give the solution of this Assuming a graph is represented as an adjacency matrix, write the pseudocode for an algorithm that can determine if any path exists between two vertices. The algorithm would accept as input: The nxn adjacency matrix for an undirected, unweighted graph A source vertex A destination vertex Returning as output: A boolean value indicating whether there is a path between the source and destination. You can use anything for variable/function names...
Given a directed graph with positive edge lengths and a specified vertex v in the graph,...
Given a directed graph with positive edge lengths and a
specified vertex v in the graph, the "all-pairs"
v-constrained shortest path problem" is the problem of computing
for each pair of vertices i and j the shortest
path from i to j that goes through the vertex
v. If no such path exists, the answer is
. Describe an algorithm that takes a graph G= (V; E) and vertex
v as input parameters and computes values L(i; j) that
represent...
Suppose you are given an undirected graph G. Find a pair of vertices (u, v) in G with the largest number of common adjacent vertices (neighbors). Give pseudocode for this algorithm and show the worst-case running time.
Consider an unweighted, undirected graph G = 〈V, E). The neighbourhood of a node u E V in the graph is the set of all nodes that are adjacent (or directly connected) to v. Subsequently, we can define the neighbourhood degree of the node v as the sum of the degrees of all its neighbours (those nodes that are directly connects to v) (a) Design an algorithm that returns a list containing the neighbourhood degree for each node v V,...
3. Graph Connected Components (25 pts) You are given an undirected, unweighted graph that may be disconnected i.e. some vertices may not be reachable from other vertices. Every group of mutually reachable vertices forms an island, called a con- nected component. There is no path between vertices in different connected components. If a graph is not disconnected, then its vertices are in a single connected component, which is the entire graph itself. Implement a method using depth-first search that will...
This question needs to be done using pseudocode (not any
particular programming language). Thanks
Consider an unweighted, undirected graph G = 〈V, E). The neighbourhood of a node u E V in the graph is the set of all nodes that are adjacent (or directly connected) to v. Subsequently, we can define the neighbourhood degree of the node v as the sum of the degrees of all its neighbours (those nodes that are directly connects to v) (a) Design an...
2. Let G be an undirected graph. For every u,vE V(G), let dc(u,v) be the length of the shoertest path from u to v. The diameter of G is he maximum distance bet In other words: max (de(u, v) u,vEV(G) the running time of your algorithm
2. Let G be an undirected graph. For every u,vE V(G), let dc(u,v) be the length of the shoertest path from u to v. The diameter of G is he maximum distance bet In...
Given a directed graph with positive edge lengths and a
specified vertex v in the graph, the "all-pairs"
v-constrained shortest path problem" is the problem of computing
for each pair of vertices i and j the shortest
path from i to j that goes through the vertex
v. If no such path exists, the answer is
. Describe an algorithm that takes a graph G= (V; E) and vertex
v as input parameters and computes values L(i; j) that
represent...
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