Give an O(n)-time dynamic programming algorithm for the maximum subarray problem. That is, given an array of real numbers a₁, a2, ..., An your algorithm must compute the maximum sum Σ; ai, where 1 ≤ j≤ k ≤n. Draw the subproblem graph. How many vertices and edges are in the graph? (15 points)
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- 3. We are given a weighted undirected graph G containing exactly 10 cycles. Write an algorithm to compute the MST of G. Your algorithm should have a runtime of O(V + E).Consider an undirected graph with n nodes and m edges. The goal is to find a path between two specified nodes u and v that maximizes the minimum weight of any edge along the path. Assume that all edge weights are positive and distinct. Design an algorithm to solve this problem with a time complexity of O(m log n).Please help me with this practice problem in python : Implement two-level iterative method B = B_{TL} for graph Laplacian matrices. We want the symmetric B. Components: Given a graph, construct its graph Laplacian matrix. Then using Luby's algorithm, construct the P matrix that ensures a prescribed coarsening factor, e.g., 2, 4, or 8 times smaller number of coarse vertices. Since the graph Laplacian matrix is singular (it has the constants in its nullspace), to make it invertible, make its last row and columns zero, but keep the diagonal as it were (nonzero). The resulting modified graph Laplacian matrix A is invertible and s.p.d.. Form the coarse matrix A_c = P^TAP. To implement symmetric two-level cycle use one of the following M and M^T: (i) M is forward Gauss-Seidel, M^T - backward Gauss-Seidel (both corresponding to A) (ii) M = M^T - the ell_1 smoother. Compare the performance (convergence properties in terms of number of iterations) of B w.r.t. just using the smoother M…
- 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…5. Given an undirected graph with n vertices and m edges, find an O(n+m) time algorithm that determines whether it is possible to color all the vertices red and blue such that every edge is between a red vertex and blue vertex. If such a coloring exists, your algorithm should produce one.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).
- Given an undirected graph G = (V, E), a vertex cover is a subset of V so that every edge in E has at least one endpoint in the vertex cover. The problem of finding a minimum vertex cover is to find a vertex cover of the smallest possible size. Formulate this problem as an integer linear programming problem.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 EnglishSuppose 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.]
- The Triangle Vertex Deletion problem is defined as follows: Given: an undirected graph G = (V, E) , with IVI=n, and an integer k>= 0. Is there a set of at most k vertices in G whose deletion results in deleting all triangles in G? (a) Give a simple recursive backtracking algorithm that runs in O(3^k * ( p(n))) where p(n) is a low-degree polynomial corresponding to the time needed to determine whether a certain vertex belongs to a triangle in G. (b) Selecting a vertex that belong to two different triangles can result in a better algorithm. Using this idea, provide an improved algorithm whose running time is O((2.562^n) * p(n)) where 2.652 is the positive root of the equation x^2=x+4You 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…We are given an undirected graph, that is connected. We're also given that each edge is associated with a positive weight. Now, we want to make this graph acyclic by removing some edges. Find this set of edges by designing an algorithm. Note: this edge set should have the smallest total weight out of other potential edge set solutions. The algorithm should run in O((m + n)log n) time.