In this problem you will design an algorithm that takes as input a directed acyclic graph G = (V,E) andtwo 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 vertexv: pov, poryv, posryv, and psryv. Notice: your algorithm needs only to count the simple paths, not listthem.myWDesign a recursive backtracking (brute-force) algorithm that determines the number ofpaths from s to t.Write down the pseudocode of your algorithm and prove its correctness, i.e., convince us that it worksbeyond any doubt. (Hint: using induction.).

In this question we have to design algorithm that can determine the number of simple paths from a…

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Brad is provided with a graph containing X branches. And it's given that the xth branch has a weight of Px. And for the given condition y-I + Py-Px there exits an edge connecting the branches. You have to develop a Python code that finds the number of connected components in this graph. Example Input: 1 3 152 Result: 1
6. In Java create an algorithm for an undirected graph with n vertices and m edges that...- Takes as a parameter some integer k- Returns the maximum induced subgraph where each vertex of that subgraph has a degree greater than or equal to k (or returns null if no such subgraph exists)- Operates in O(n + m) time
*Discrete Math In the graph above, let ε = {2, 3}, Let G−ε be the graph that is obtained from G by deleting the edge {2,3}. Let G∗ be the graph that is obtain from G − ε by merging 2 and 3 into a single vertex w. (As in the notes, v is adjacent to w in the new if and only if either {2,v} or {3,v is an edge of G.) (a) Draw G − ε and calculate its chromatic polynomial. (b) Give an example of a vertex coloring that is proper for G − ε, but not for G. (c) Explain, in own words, why no coloring can be proper for G but not proper for G − ε. (d) Draw G∗ and calculate its chromatic polynomial. (e) Verify that, for this example,PG(k) = PG−ε(k) − PG∗ (k).
The given inputs consist of two nodes (s, t) and a directed graph G = (V, E). In addition, each edge of the graph is either blue or red. The goal is to find a path from point s to point t such that red edges always follow blue edges. There need not be any red or blue borders on the route, but if there are, the red ones should follow the blue ones. Develop an algorithm that does the task in O(n + m) time and analyze its performance.
If there is an Euler path that starts at AA and ends at some other vertex, give it. Otherwise enter DNE.This should be a list of letters, from A to the other vertex: e.g. ACBDEAD (if that were even possible!). Path: Since this is a planar graph, the number of regions - arcs + nodes equals what number?
(V, E) be a connected, undirected graph. Let A = V, B = V, and f(u) = neighbours of u. Select all that are true. Let G = a) f: AB is not a function Ob) f: A B is a function but we cannot always apply the Pigeonhole Principle with this A, B Odf: A B is a function but we cannot always apply the extended Pigeonhole Principle with this A, B d) none of the above
3. Kleinberg, Jon. Algorithm Design (p. 519, q. 28) Consider this version of the Independent Set Problem. You are given an undirected graph G and an integer k. We will call a set of nodes I "strongly independent" if, for any two nodes v, u € I, the edge (v, u) is not present in G, and neither is there a path of two edges from u to v. That is, there is no node w such that both (v, w) and (u, w) are present. The Strongly Independent Set problem is to decide whether G has a strongly independent set of size at least k. Show that the Strongly Independent Set Problem is NP-Complete.
You are given a weighted, undirected graph G = (V, E) which is guaranteed to be connected. Design an algorithm which runs in O(V E + V 2 log V ) time and determines which of the edges appear in all minimum spanning trees of G. Do not write the code, give steps and methods. Explain the steps of algorithm, and the logic behind these steps in plain English
Alert dont submit AI generated answer. Write a program that reads a weighted graph and an initial vertex.The program must print on the screen the minimum paths obtained by Dijkstra's algorithm. Input: Receives n, m and s; n is the total number of vertices, m the total number of arcs and s is the initial vertex.Next, m lines, each line with a trio of integers, corresponding to the beginning and end of the arc, followed by the weight of the arc.(Vertices are identified from 0 to n-1.) Output: Prints the shortest paths obtained by Dijkstra's algorithm. Exemple: Input: 5 10 00 1 100 4 51 2 11 4 22 3 43 2 63 0 74 1 34 2 94 3 2 Output: [0, 8, 9, 7, 5][-1, 4, 1, 4, 0]
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Suppose you have a graph with 100 nodes and 500 edges and you want to find the shortest path between two nodes using Dijkstra's algorithm. What is the time complexity of this operation?
You are organizing a programming competition, where contestants implement Dijkstra's algorithm. Given adirected graph G = (V, E) with integer-weight edges and a starting vertex s ∈ V , their programs are supposedto output triplets (v, v.d, v.π) for each vertex v ∈ V . Design an O(V +E) time algorithm that takes as inputthe original graph G in both adjacency matrix (G.M) and adjacency list (G.Adj) representations, startingvertex s, and the output of a contestant's program (given as an array A of triplets), and returns whetherA is the correct output for G. Write down the pseudocode for your algorithm, explain why it correctlyveries the output, and analyze your algorithm's running time. You may assume that all edge weights of the input graph provided to the contestantsare nonnegative and A (the output of their programs) is in the valid format, i.e., you don't need to verifythat A is actually an array of triplets, with v and v.π being valid vertices and v.d being an integer.Can you…

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