Homework Answers
3)
take graph in adjacency matrix: A and compute A3...then we get all distinct paths of length 3 between every pair of nodes in the graph...
explanation: if we compute A3 then each value in A3[i][j] represents the number of disting walks between the nodes i to j in the graph
the time complexity: is O(n^3),, where n ithe number of nodes///
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Triangle is a complete graph on 3 vertices (see below) You are given a graph G, and you need to calculate the number of triangles contained in G. Develop an efficient (better than cubic) algorithm to...
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.
You are given a weighted graph G, two designated vertices s and t. Your goal is...
You are given a weighted graph G, two designated vertices s and t. Your goal is to find a path from s to t in which the minimum edge weight is maximized i.e. if there are two paths with weights 10→1→5 and 2→7→3 then the second path is considered better since the minimum weight (2) is greater than the minimum weight of the first (1). Describe an efficient algorithm to solve this problem and show its complexity.
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...
Suppose you are given a connected graph G, with edge costs that you may assume are...
Suppose you are given a connected graph G, with edge costs that you may assume are all distinct. G has n vertices and m edges. A particular edge e of G is specified. Give an algorithm with running time O(m + n) to decide whether e is contained in a minimum spanning tree of G.
(a) Given a graph G = (V, E) and a number k (1 ≤ k ≤...
(a) Given a graph G = (V, E) and a number k (1 ≤ k ≤ n), the CLIQUE problem asks us whether there is a set of k vertices in G that are all connected to one another. That is, each vertex in the ”clique” is connected to the other k − 1 vertices in the clique; this set of vertices is referred to as a ”k-clique.” Show that this problem is in class NP (verifiable in polynomial time)...
Subject: Algorithm need this urgent please thank you. 4. Give pseudocode for an algorithm that will solve the following problem. Given an array A[1..n) that contains every number between 1 and n +...
Subject: Algorithm
need this urgent please thank you.
4. Give pseudocode for an algorithm that will solve the following problem. Given an array A[1..n) that contains every number between 1 and n +1 in order, except that one of the numbers is missing. Find the miss sorted ing mber. Your algorithm should run in time (log n). (Hint: Modify Binary search). A pseudocode means an algorithm with if statements and loops, etc. Don't just write a paragraph. Also, if your...
MST For an undirected graph G = (V, E) with weights w(e) > 0 for each edge e ∈ E, you are given a MST T. Unfortunately one of the edges e* = (u, z) which is in the MST T is deleted from the graph G...
MST For an undirected graph G = (V, E) with weights w(e) > 0 for each edge e ∈ E, you are given a MST T. Unfortunately one of the edges e* = (u, z) which is in the MST T is deleted from the graph G (no other edges change). Give an algorithm to build a MST for the new graph. Your algorithm should start from T. Note: G is connected, and G − e* is also connected. Explain...
10) Shortest Paths (10 marks) Some pseudocode for the shortest path problem is given below. When DIJKSTRA (G, w,s) is called, G is a given graph, w contains the weights for edges in G, and s is a sta...
10) Shortest Paths (10 marks) Some pseudocode for the shortest path problem is given below. When DIJKSTRA (G, w,s) is called, G is a given graph, w contains the weights for edges in G, and s is a starting vertex DIJKSTRA (G, w, s) INITIALIZE-SINGLE-SOURCE(G, s) 1: RELAX (u, v, w) 1: if dlv] > dlu (u, v) then 2d[v] <- d[u] +w(u, v) 3 4: end if 4: while Q φ do 5: uExTRACT-MIN Q) for each vertex v...
Answer B java Problem 2 For each problem given below, do the following: 1. Create an...
Answer B
java
Problem 2 For each problem given below, do the following: 1. Create an algorithm in pseudocode to solve the problem. 2.Identify the factors that would influence the running time of your algorithm. For example, if your algorithm is to search an array the factor that influences the running time is the array size. Assign names (such as n) to each factor. 3. Determine the number of operations in each step of the pseudocode. To do that, identify...
Definition: Given a Graph \(\mathrm{G}=(\mathrm{V}, \mathrm{E})\), define the complement graph of \(\mathrm{G}, \overline{\boldsymbol{G}}\), to be \(\bar{G}=(\mathrm{V},...
Definition: Given a Graph \(\mathrm{G}=(\mathrm{V}, \mathrm{E})\), define the complement graph of \(\mathrm{G}, \overline{\boldsymbol{G}}\), to be \(\bar{G}=(\mathrm{V}, E)\) where \(E\) is the complement set of edges. That is \((\mathrm{v}, \mathrm{w})\) is in \(E\) if and only if \((\mathrm{v}, \mathrm{w}) \notin \mathrm{E}\) Theorem: Given \(\mathrm{G}\), the complement graph of \(\mathrm{G}, \bar{G}\) can be constructed in polynomial time. Proof: To construct \(G\), construct a copy of \(\mathrm{V}\) (linear time) and then construct \(E\) by a) constructing all possible edges of between vertices 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.
Subject: Algorithm
need this urgent please thank you.
4. Give pseudocode for an algorithm that will solve the following problem. Given an array A[1..n) that contains every number between 1 and n +1 in order, except that one of the numbers is missing. Find the miss sorted ing mber. Your algorithm should run in time (log n). (Hint: Modify Binary search). A pseudocode means an algorithm with if statements and loops, etc. Don't just write a paragraph. Also, if your...
10) Shortest Paths (10 marks) Some pseudocode for the shortest path problem is given below. When DIJKSTRA (G, w,s) is called, G is a given graph, w contains the weights for edges in G, and s is a starting vertex DIJKSTRA (G, w, s) INITIALIZE-SINGLE-SOURCE(G, s) 1: RELAX (u, v, w) 1: if dlv] > dlu (u, v) then 2d[v] <- d[u] +w(u, v) 3 4: end if 4: while Q φ do 5: uExTRACT-MIN Q) for each vertex v...
Answer B
java
Problem 2 For each problem given below, do the following: 1. Create an algorithm in pseudocode to solve the problem. 2.Identify the factors that would influence the running time of your algorithm. For example, if your algorithm is to search an array the factor that influences the running time is the array size. Assign names (such as n) to each factor. 3. Determine the number of operations in each step of the pseudocode. To do that, identify...
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