In the Euclidean (metric) Traveling Salesperson Problem (TSP), we started with a DFS traversal of the minimum spanning tree (MST) and then skipped vertices that we had already visited. Why were we able to allow skipping of nodes? O Because DFS run on an MST adds an exponential number of vertices O Ina metric space the triangle inequality states that removing a vertex cannot lengthen the path O In the Euclidean metric, distance in measured in the number of edges, so removing edges makes the path shorter O It feels right

In the Euclidean (metric) Traveling Salesperson Problem (TSP), we started with a DFS traversal of the minimum spanning tree (MST) and then skipped vertices that we had already visited. Why were we able to allow skipping of nodes? O Because DFS run on an MST adds an exponential number of vertices O Ina metric space the triangle inequality states that removing a vertex cannot lengthen the path O In the Euclidean metric, distance in measured in the number of edges, so removing edges makes the path shorter O It feels right

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Design an algorithm to find the shortest (directed) cycle containing a vertex v. What is the space and time complexity of your algorithm? Prove that your algorithm finds the shortest cycle.
Write the PYTHON programming to solve with breadth-first search and build node-arcs with python code for the attached problem:
Given a digraph, find a bitonic shortest path from s to every other vertex (if one exists). A path is bitonic if there is an intermediate vertex v suchthat the edges on the path from s to v are strictly increasing and the edges on the pathfrom v to t are strictly decreasing. The path should be simple (no repeated vertices).
Discrete mathematics. Let G = (V, E) be a simple graph4 with n = |V| vertices, and let A be its adjacency matrix of dimension n × n. We want to count the L-cycles : such a cycle, denoted by C = u0u1 · · · uL with uL = u0 contains L distinct vertices u0, . . . , uL-1 et L edges E(C) = {uiui+1 | 0 ≤ i ≤ L − 1} ⊆ E. Two cycles are distinct if the edge sets are different : C = C' if and only if E(C) = E(C'). We define the matrices D, T, Q, the powers of A by matrix multiplication : D = A · A = A2, T = A · D = A3, Q = A · T = A4. Consider the values on the diagonals. Prove that the number of 3-cycles N3 in the whole graph is N3 = 1/6 ∑ u∈V Tu,u
Please help with algorithm no coding needed
The Algorithm of AlgebraThe adjacency matrix A of a graph G = is used by the algebraic BFS algorithm (V, E).First, we create a vector x with all zeros except for the index of the source vertex s that we wish to use as the starting point for the algorithm; next, we create the matrix A = A+ I; last, AT x chooses all nodes that are at a distance of 1 (level 1) from the source vertex. The vertices with the fewest number of hops are obtained by multiplying the vector x by the matrix A2. As a rule, the product Ak x will produce neighbours that are at most k hops distant, and the multiplication should be done in a boolean manner as in the algorithm shown below. Algorithm for Algebraic BFS1: Input : Adjacency matrix An,n of a graph G = (V, E) connected, unweighted graph G and asource vertex s2: Output : N, visited, levels a matrix that shows level i vertices at its column i, the visitedvertices in sequence and their levels3: x[n] ← 04: x[s] ← 15: A ← A + I6: for i = 1 to n do7: N ← AT · x8:…
1. Input: G = (V, E)2. Output: MIS I of G3. I ← ∅4. V ← V5. while V = ∅6. assign a random number r(v) to each vertex v ∈ V7. for all v ∈ V in parallel8. if r(v) is minimum amongst all neighbors9. I ← I ∪ {v}10. V ← V \ {v ∪ N(v)}This algorithm terminates in O(log n)rounds with good probability. The stepsof the algorithm between lines 7–10 can be performed in parallel which will providea speedup.Make Python Implementation this algorithm in sequential form
Suppose G = (V,E) is a DAG. Design a linear-time algorithm that counts the total number of simple paths (i.e., a path that does not visit any node more than once) between two vertices s and t in G. Note that your algorithm only need to return the number of simple paths and does not need to list the actual paths.
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26. Solve the following a) For the given set U, assume a family of subsets (F) on your own and apply the approximation algorithm to find the set cover. U={2,3,4,6,7,8} b) For the following weighted graph, apply approximation algorithm for solving traveling salesman problem. 3 1 4 2 5 5 27. For the following undirected graph, apply the approximation algorithm to find vertex cover. 2 6 1 3 5 4 7
2. A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. There are 2 very popular algorthms to calcualte MSTs of a graph. These algorithms are Kruskal's algorithm and Prim's algorithm. Figure 2: A weighted graph and its minimum spanning tree (ii) Can a Minimum Spanning Tree finding algorithm like Prim's algorithms be used to find out the Maximum Spanning Tree? If yes, then explain your steps. If not, explain why not with logic.
Design a bottom-up decrease-and-conquer algorithm for the following task. For any even n, mark n cells on an infinite sheet of graph paper so that each marked cell has an odd number of marked neighbors. Two cells are considered neighbors if they are next to each other either horizontally or vertically but not diagonally. The marked cells must form a contiguous region, that is a region in which there is a path between any pair of marked cells that goes through a sequence of marked neighbors. Below are examples of solutions for n=2 and n=4 respectively.