Concept explainers
(a)
To describe a dynamic-
(a)
Explanation of Solution
Given Information:
The shortest closed tour of the graph with length approximately is 24.89. The directed acyclic graph is shown below-
Explanation:
The dynamic-approach used to find the longest length simple path consider the graph G and vertices as V. The longest simple path must go through some edge of weight s or t . The algorithms to compute the longest weight path is
The base condition for the algorithm is
The algorithm to compute longest simple path in a directed acyclic graph is given below-
LONG-PATH(G,u,s,t,len)
If
set
return ( len,u )
else if
Return (len,u).
else
for each adjacent vertex
Check the distance after adding new vertex i.
if
end if.
end for.
end if.
return ( len,u ).
end.
In above algorithm, loop of for is used to determine the longest path and the longest path visit all vertex by checking all adjacent vertex.
The time taken by the algorithm is depends upon the number of vertex visited and the number of edges in the longest simple path. Suppose V represent the number of vertex used in the computing the longest simple path and E represent the number of edges then total running time of the algorithm is equals to
(b)
To describe a dynamic-programming approach for finding longest simple path in directed acyclic graph and also give the running time of the algorithm.
(b)
Explanation of Solution
Given Information:
The shortest closed tour of the graph with length approximately is 25.58. The directed acyclic graph is shown below-
Explanation:
The longest simple path must go through some edge of weight s . The base condition for the algorithm is
The algorithm to compute longest simple path in a directed acyclic graph is given below-
LONG-PATH(G,u,s,t,len)
If
set
return ( len,u )
else if
Return (len,u).
else
for each adjacent vertex
Check the distance after adding new vertex i.
if
end if.
end for.
end if.
return ( len,u ).
end.
The time taken by the algorithm is depends upon the number of vertex visited and the number of edges in the longest simple path. Suppose V represent the number of vertex used in the computing the longest simple path and E represent the number of edges then total running time of the algorithm is equals to
The above algorithm taken consideration of the nodes and generates the output according to the number of nodes in the graph so the longest length is computed by
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Chapter 15 Solutions
Introduction to Algorithms
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- You are given a bipartite graph G=(U,V,E), and an integer K. U and V are the two bipartitions of the graph such that |U| = |V| = N , and E is the edge set. The vertices of U are {1,2,...,N } and that of V are {N+1,N+2,...,2N }. You need to find out whether the total number of different perfect matchings in G is strictly greater than K or not. Recall that a perfect matching is a subset of E such that every vertex of the graph belongs to exactly one edge in the subset. Two perfect matchings are considered to be different even if one edge is different. Write a program in C++ programming language that prints a single line containing “Perfect” if the number of perfect matchings is greater than K, and “Not perfect” in other cases.Sample Input:3 5 21 42 62 53 53 5Output:Not Perfectarrow_forward3) The graph k-coloring problem is stated as follows: Given an undirected graph G= (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in V a color c(v) such that 1arrow_forwardPlease Answer this in Python language: You're given a simple undirected graph G with N vertices and M edges. You have to assign, to each vertex i, a number C; such that 1 ≤ C; ≤ N and Vi‡j, C; ‡ Cj. For any such assignment, we define D; to be the number of neighbours j of i such that C; < C₁. You want to minimise maai[1..N) Di - mini[1..N) Di. Output the minimum possible value of this expression for a valid assignment as described above, and also print the corresponding assignment. Note: The given graph need not be connected. • If there are multiple possible assignments, output anyone. • Since the input is large, prefer using fast input-output methods. Input 1 57 12 13 14 23 24 25 35 Output 2 43251 Qarrow_forwardConsider the Minimum-Weight-Cycle Problem: Input: A directed weighted graph G = :(V, E) (where the weight of edge e is w(e)) and an integer k. Output: TRUE if there is a cycle with total weight at most k and FALSE if there is no cycle with total weight at most k. Remember, a cycle is a list of vertices such that each vertex has an edge to the next and the final vertex has an edge to the first vertex. Each vertex can only occur once in the cycle. A vertex with a self-loop forms a cycle by itself. (a) Assume that all edge weights are positive. Give a polynomial-time algorithm for the Minimum-Weight-Cycle Problem. For full credit, you should: Give a clear description of your algorithm. If you give pseudocode, you should support it with an expla- nation of what the algorithm does. Give the running time of your algorithm in terms of the number of vertices n and the number of edges m. - You do not need to prove the correctness of your algorithm or the correctness of your running time…arrow_forwardSuppose are you given an undirected graph G = (V, E) along with three distinct designated vertices u, v, and w. Describe and analyze a polynomial time algorithm that determines whether or not there is a simple path from u to w that passes through v. [Hint: By definition, each vertex of G must appear in the path at most once.]arrow_forwardFor a directed graph, find if there is a path between two vertices or not. Write the algorithm and C++ code for it. Represent the example diagrammatically. Calculate the time and space complexity for it too.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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