Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 21.1, Problem 3E
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To find the number of times FIND-SET and UNION functions are called during the execution of CONNECTED-COMPONENETS on an undirected graph
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- Be G=(V, E)a connected graph and u, vEV. The distance Come in u and v, denoted by d(u, v), is the length of the shortest path between u'and v, Meanwhile he width from G, denoted as A(G), is the greatest distance between two of its vertices. a) Show that if A(G) 24 then A(G) <2. b) Show that if G has a cut vertex and A(G) = 2, then Ġhas a vertex with no neighbors.arrow_forwardGiven an undirected graph G, and a path P, we want to verify that P is a cycle that contains all nodes in the graph (each node occurs on P exactly once, no repetition is allowed).In numbered steps, describe a polynomial-time verifier that checks if a given path P has the described properties (meaning that it is a cycle and it contains all nodes in the graph, with each node appearing on the path exactly once).(Hint: you can think of it in terms of this question: what does the verifier need to check to ensure that the path P has the described properties?).arrow_forwardLet G be a graph. We say that a set of vertices C form a vertex cover if every edge of G is incident to at least one vertex in C. We say that a set of vertices I form an independent set if no edge in G connects two vertices from I. For example, if G is the graph above, C = [b, d, e, f, g, h, j] is a vertex cover since each of the 20 edges in the graph has at least one endpoint in C, and I = = [a, c, i, k] is an independent set because none of these edges appear in the graph: ac, ai, ak, ci, ck, ik. 2a In the example above, notice that each vertex belongs to the vertex cover C or the independent set I. Do you think that this is a coincidence? 2b In the above graph, clearly explain why the maximum size of an independent set is 5. In other words, carefully explain why there does not exist an independent set with 6 or more vertices.arrow_forward
- 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…arrow_forwardComputer Sciencearrow_forwardHow do I do this? We say a graph G = (V, E) has a k-coloring for some positive integer k if we can assign k different colors to vertices of G such that for every edge (v, w) ∈ E, the color of v is different to the color w. More formally, G = (V, E) has a k-coloring if there is a function f : V → {1, 2, . . . , k} such that for every (v, w) ∈ E, f(v) 6= f(w).3-Color problem is defined as follows: Given a graph G = (V, E), does it have a 3-coloring?4-Color problem is defined as follows: Given a graph G = (V, E), does it have a 4-coloring?Prove that 3-Color ≤P 4-Color.(hint: add vertex to 3-Color problem instance.)arrow_forward
- I have seen answers on chegg, I want newarrow_forwardGiven an arbitrary collection of k-mers Patterns (where some k-mers may appear multiple times), we define CompositionGraph(Patterns) as a graph with |Patterns| isolated edges. Every edge is labeled by a k-mer from Patterns, and the starting and ending nodes of an edge are labeled by the prefix and suffix of the k-mer labeling that edge. We then define the de Bruijn graph of Patterns, denoted DeBruijn(Patterns), by gluing identically labeled nodes in CompositionGraph(Patterns), which yields the following algorithm. DEBRUIJN(Patterns) represent every k-mer in Patterns as an isolated edge between its prefix and suffix glue all nodes with identical labels, yielding the graph DeBruijn(Patterns) return DeBruijn(Patterns) De Bruijn Graph from k-mers Problem Construct the de Bruijn graph from a collection of k-mers. Given: A collection of k-mers Patterns. Return: The de Bruijn graph DeBruijn(Patterns), in the form of an adjacency list. Sample Dataset GAGG CAGG GGGG…arrow_forwardConsider 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 itarrow_forward
- Suppose we have a graph G = (V, E) with m edges. Prove that there exists a partition of V into three subsets A, B, C such that there are 2m edges between these subsets (i.e. between A and B, between B and C, or between A and C). 3arrow_forwardWrite a program RandomSparseGraph to generate random sparse graphs for a well-chosen set of values of V and E such that you can use it torun meaningful empirical tests on graphs drawn from the Erdös-Renyi modelarrow_forwardIf a graph G = (V, E), |V | > 1 has N strongly connected components, and an edge E(u, v) is removed, what are the upper and lower bounds on the number of strongly connected components in the resulting graph? Give an example of each boundary case.arrow_forward
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