Consider the graph G with n vertices. We define a Covid-Friends set as a set of vertices none of which is connected to another vertex in the set (they are social distancing!). In other words, any subgraph of G with Covid-Friends vertices has no edge. Clearly, an empty set or a set with a single vertex is a Covid-Friend vertex. However, given a graph, we want to find a such set with maximum number of vertices. (a) works - it may not even give the optimal solution!). Justify why your algorithm makes sense (in terms of the goal: maximizing the Covid-Friend set solution). Design (explain in words) a greedy algorithm for this problem (no need to prove it

Consider the graph G with n vertices. We define a Covid-Friends set as a set of vertices none of which is connected to another vertex in the set (they are social distancing!). In other words, any subgraph of G with Covid-Friends vertices has no edge. Clearly, an empty set or a set with a single vertex is a Covid-Friend vertex. However, given a graph, we want to find a such set with maximum number of vertices. (a) works - it may not even give the optimal solution!). Justify why your algorithm makes sense (in terms of the goal: maximizing the Covid-Friend set solution). Design (explain in words) a greedy algorithm for this problem (no need to prove it

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 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.
Let G = (V, E) be a directed graph, and let wv be the weight of vertex v for every v ∈ V . We say that a directed edgee = (u, v) is d-covered by a multi-set (a set that can contain elements more than one time) of vertices S if either u isin S at least once, or v is in S at least twice. The weight of a multi-set of vertices S is the sum of the weights of thevertices (where vertices that appear more than once, appear in the sum more than once).1. Write an IP that finds the multi-set S that d-cover all edges, and minimizes the weight.2. Write an LP that relaxes the IP.3. Describe a rounding scheme that guarantees a 2-approximation to the best multi-set
Consider a set of real number pairs that represent a collection of V intervals on the real line. An interval graph with one vertex for each interval is defined by such a collection, with edges connecting vertices in the event that the associated intervals cross (have any points in common). Create a programme that creates V random intervals, each of length d, in the unit interval and then creates the interval graph that goes with it.Use a BST, as a tip.
Consider an undirected graph G = (V;E). An independent set is a subset I V such that for any vertices i; j 2 I, there is no edge between i and j in E. A set i is a maximal independent set if no additional vertices of V can be added to I without violating its independence. Note, however, that a maximal independent sent is not necessarily the largest independent set in G. Let (G) denote the size of the largest maximal independent set in G. 1) What is (G) if G is a complete graph on n vertices? What if G is a cycle on n vertices?
Let G = (V, E) be a graph with vertex-set V = {1, 2, 3, 4, 5} and edge-set E = {(1, 2), (3, 2), (4, 3), (1, 4), (2, 4), (1, 3)}. (a) Draw the graph. Find (b) maximal degree, i.e. ∆(G), (c) minimal degree, i.e. δ(G), (d) the size of biggest clique, i.e. ω(G),(e) the size of biggest independent set, i.e. α(G), ter(f) the minimal number of colours needed to color the graph, i.e. χ(G).
We have a connected graph G=(V,E), and a specific vertex u∈V. Suppose we compute a depth-first search tree rooted at u, and obtain a tree T that includes all nodes of G. Suppose we then compute a breadth-first search tree rooted at u, and obtain the same tree T. Prove that G=T. (In other words, if T is both a depth-first search tree and a breadth-first search tree rooted at u, then G cannot contain any edges that do not belong to T.)
5. (This question goes slightly beyond what was covered in the lectures, but you can solve it by combining algorithms that we have described.) A directed graph is said to be strongly connected if every vertex is reachable from every other vertex; i.e., for every pair of vertices u, v, there is a directed path from u to v and a directed path from v to u. A strong component of a graph is then a maximal subgraph that is strongly connected. That is all vertices in a strong component can reach each other, and any other vertex in the directed graph either cannot reach the strong component or cannot be reached from the component. (Note that we are considering directed graphs, so for a pair of vertices u and v there could be a path from u to v, but no path path from v back to u; in that case, u and v are not in the same strong component, even though they are connected by a path in one direction.) Given a vertex v in a directed graph D, design an algorithm for com- puting the strong connected…
Throughout, a graph is given as input as an adjacency list. That is, G is a dictionary where the keysare vertices, and for a vertex v,G[v] = [u such that there is an edge going from v to u].In the case that G is undirected, for every edge u − v, v is in G[u] and u is in G[v]. 3. Write the full pseudocode for the following problem.Input: A directed graph G, and an ordering on the vertices given in a list A.Output: Is A a topological order? In other words, is there an i, j such that i < j and there is an edge fromA[j] to A[i]?
Consider an undirected graph G = (V;E). An independent set is a subset I V such that for any vertices i; j 2 I, there is no edge between i and j in E. A set i is a maximal independent set if no additional vertices of V can be added to I without violating its independence. Note, however, that a maximal independent sent is not necessarily the largest independent set in G. Let (G) denote the size of the largest maximal independent set in G. One way of trying to avoid this dependence on ordering is the use of randomized algorithms. Essentially, by processing the vertices in a random order, you can potentially avoid (with high probability) any particularly bad orderings. So consider the following randomized algorithm for constructing independent sets: @ First, starting with an empty set I, add each vertex of G to I independently with probability p. @ Next, for any edges with both vertices in I, delete one of the two vertices from I (at random). @ Note - in this second step,…
Given a graph G = (V, E), let us call G an almost-tree if G is connected and G contains at most n + 12 edges, where n = |V |. Each edge of G has an associated cost, and we may assume that all edge costs are distinct. Describe an algorithm that takes as input an almost-tree G and returns a minimum spanning tree of G. Your algorithm should run in O(n) time.
A directed graph G= (V,E) consists of a set of vertices V, and a set of edges E such that each element e in E is an ordered pair (u,v), denoting an edge directed from u to v. In a directed graph, a directed cycle of length three is a triple of vertices (x,y,z) such that each of (x,y) (y,z) and (z,x) is an edge in E. Write a Mapreduce algorithm whose input is a directed graph presented as a list of edges (on a file in HDFS), and whose output is the list of all directed cycles of length three in G. Write the pseudocode for the mappers/reducers methods. Also, assuming that there are M mappers, R reducers, m edges and n vertices -- analyze the (upper-bound of the) communication cost(s).
Given a graph G = (V;E) an almost independent set I V is a generalization of an independent set whereeach vertex u 2 I is connected to at most one other vertex in I but no more. In other words for all u; v 2 Iif (u; v) 2 E then (u; z) =2 E for all z 2 I (equvalently (v; z) =2 E). Note that every independent set is alsoalmost independent. Prove that the problem of nding whether there exists an almost independent set ofsize k for some k, is NP-complete.