(3.) Assume that we have a method for multiplying two 2 x 2 matrices using only 7 scalar mul-tiplications. Use this method to design a divide and conquer algorithm for multiplying twon x n matrices. What is the time complexity of the algorithm? You should start with adescription of your algorithm, then write a recurrence relation which you should solve.4. Let G = (V, E) be a directed graph.Assume that each edge ij e E has a non-negative costc(i, j) associated with it. Design a dynamic programming algorithm for computing the ashortest path between any vertex pair. (FLOYD-WARSHAL ALGORITHM) You shoulddefine all necessary terms and write a recurrence relation. What is the time complexity ofyour algorithm. Finally write a short pseudo-code for your algorithm that uses only onematrix. 1. Prove the following known result on minimum spanning trees.Theorem. Let G = (V, E) be a graph and let w : E → R+ be a weight function. Let A be asubset of edges that is contained in some minimum spanning tree, and let (S, S) be a cut sothat (S, S)N A = 0. Prove that if e is a light edge in (S, S), then e is safe for A.%3D2. Let S =fi and a profit p; associated with it. Design a dynamic programing algorithm for computinga set of compatible activities of maximum profit. You should start with defining some termsand then write a recurrence relation for solving the problem. What is the time complexity ofyour algorithm?{1,2...n} be a set activities, where activity i E S has a start time si, a finish time

[Note: Hello. Since your question has multiple parts, we will solve first question for you. If you…

Assume that we have a method for multiplying two 2 x 2 matrices using only 7 scalar mul- tiplications. Use this method to design a divide and conquer algorithm for multiplying two n x n matrices. What is the time complexity of the algorithm? You should start with a description of your algorithm, then write a recurrence relation which you should solve.

Assume that we have a method for multiplying two 2 x 2 matrices using only 7 scalar mul- tiplications. Use this method to design a divide and conquer algorithm for multiplying two n x n matrices. What is the time complexity of the algorithm? You should start with a description of your algorithm, then write a recurrence relation which you should solve.

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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Let G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Let T be a minimum spanning tree for the graph with the original weight. Suppose we replace eachedge weight ℓ(e) with ℓ(e)^2, then T is still a minimum spanning tree.
let us take any standard graph G=(v,e) and let us pretend each edge is the same exact weight. let us think about a minimum spanning tree of the graph G, called T = (V, E' ). under each part a and b illustrate then show that a) s a unique path between u and v in T for all u, v ∈ V . b) tree T is not unique. provide proof
You are given a graph G = (V, E) with positive edge weights, and a minimum spanning tree T = (V, E') with respect to these weights; you may assume G and T are given as adjacency lists. Now suppose the weight of a particular edge e in E is modified from w(e) to a new value w̃(e). You wish to quickly update the minimum spanning tree T to reflect this change, without recomputing the entire tree from scratch. There are four cases. In each case give a linear-time algorithm for updating the tree. Note, you are given the tree T and the edge e = (y, z) whose weight is changed; you are told its old weight w(e) and its new weight w~(e) (which you type in latex by widetilde{w}(e) surrounded by double dollar signs). In each case specify if the tree might change. And if it might change then give an algorithm to find the weight of the possibly new MST (just return the weight or the MST, whatever's easier). You can use the algorithms DFS, Explore, BFS, Dijkstra's, SCC, Topological Sort as…
Let G = (V, E) be a connected, undirected graph, and let s be a fixed vertex in G. Let TB be the spanning tree of G found by a bread first search starting from s, and similarly TD the spanning tree found by depth first search, also starting at s. (As in problem 1, these trees are just sets of edges; the order in which they were traversed by the respective algorithms is irrelevant.) Using facts, prove that TB = TD if and only if TB = TD = E, i.e., G is itself a tree.
a) Draw the connected subgraph of the given graph above which contains only four nodes ACGB and is also a minimum spanning tree with these four nodes. What is its weighted sum? Draw the adjacency matrix representation of this subgraph (use boolean matrix with only 0 or 1, to show its adjacency in this case).b) Find the shortest path ONLY from source node D to destination node G of the given graph above, using Dijkstra’s algorithm. Show your steps with a table as in our course material, clearly indicating the node being selected for processing in each step.c) Draw ONLY the shortest path obtained above, and indicate the weight in each edge in the diagram. Also determine the weighted sum
. Let G be a weighted, connected, undirected graph, and let V1 and V2 be a partition of the vertices of G into two disjoint nonempty sets. Furthermore, let e be an edge in the minimum spanning tree for G such that e has one endpoint in V1 and the other in V2. Give an example that shows that e is not necessarily the smallest- weight edge that has one endpoint in V1 and the other in V2.
We have undirected graph K, with two distinct vertices z, k. And let O be a minimum spanning tree of K. Prove that the path from z to k along O is a minmax path. Assume that K has distinct edge weights. (Assume for contradiction that the minmax path is not completely on the minimum spanning tree.) Note: A MinMax path in an undirected Graph K is a path between two vertices z, k that minimizes the maximum weight of the edges on the pat h.h.
Consider the so-called k-Minimum Spanning Tree (k-MST) problem, which is defined as follows. An instance of the k-MST problem is given by a connected undirected graph G=(V,E) with edge weights w:E→Q and a natural number k>2. The question is to find a tree with exactly k nodes that is a subgraph of G and minimises the weight among all such trees. Informally, k-MST is the variant of the minimum spanning tree problem, where instead of a spanning tree one wants to find a tree with exactly k nodes. What would be the result if we apply Prim's, respectively Kruskal's, algorithm to the problem by stopping both algorithms after k−1 edges have been added? In the following we refer to these versions of Prim's and Kruskal's algorithm as the modified algorithm of Prime or Kruskal, respectively. a) consider the following graph. (image provided- image 1) For which of the following edge weights, assigned to the graph above, does the modified algorithm of Kruskal provide a wrong result assuming…
Let A, B, C, D be the vertices of a square with side length 100. If we want to create a minimum-weight spanning tree to connect these four vertices, clearly this spanning tree would have total weight 300 (e.g. we can connect AB, BC, and CD). But what if we are able to add extra vertices inside the square, and use these additional vertices in constructing our spanning tree? Would the minimum-weight spanning tree have total weight less than 300? And if so, where should these additional vertices be placed to minimize the total weight? Let G be a graph with the vertices A, B, C, D, and possibly one or more additional vertices that can be placed anywhere you want on the (two-dimensional) plane containing the four vertices of the square. Determine the smallest total weight for the minimum-weight spanning tree of G. Round your answer to the nearest integer. Note: I encourage you to add n additional points (for n=1, 2, 3) to your graph and see if you can figure out where these point(s) need to…
Let A, B, C, D be the vertices of a square with side length 100. If we want to create a minimum-weight spanning tree to connect these four vertices, clearly this spanning tree would have total weight 300 (e.g. we can connect AB, BC, and CD). But what if we are able to add extra vertices inside the square, and use these additional vertices in constructing our spanning tree? Would the minimum-weight spanning tree have total weight less than 300? And if so, where should these additional vertices be placed to minimize the total weight? Let G be a graph with the vertices A, B, C, D, and possibly one or more additional vertices that can be placed anywhere you want on the (two-dimensional) plane containing the four vertices of the square. Determine the smallest total weight for the minimum-weight spanning tree of G. Round your answer to the nearest integer. Attention: Please don't just copy these two following answers, which are not correct at all. Thank you.…
Which is true/false. Let G be a connected undirected weighted graph. Consider the following two statements. S₁: There exists a minimum weight edge in G which is present in every minimum spanning tree of G. S2: If every edge in G has distinct weight, then G has a unique minimum spanning tree.
G = (V,E,W) is a weighted connected (undirected) graph where all edges have distinct weights except two edges e and e′ which have the same weight. Suppose there is a Minimum Spanning Tree of G containing both e and e′. Prove that G has a unique Minimum Spanning Tree.