Let’s go back to the original motivation for the Minimum Spanning Tree Problem.

Question

Let’s go back to the original motivation for the Minimum Spanning Tree Problem. We are given a connected, undirected graph G = (V , E) with positive edge lengths {le}, and we want to find a spanning subgraph of it. Now suppose we are willing to settle for a subgraph H = (V , F ) that is “denser” than a tree, and we are interested in guaranteeing that, for each pair of vertices u, v ? V, the length of the shortest u-v path in H is not much longer than the length of the shortest u-v path in G. By the length of a path P here,we mean the sum of leover all edges e in P.

Here’s a variant of Kruskal’s Algorithm designed to produce such a subgraph.

. First we sort all the edges in order of increasing length. (You may assume all edge lengths are distinct.)

. We then construct a subgraph H = (V , F ) by considering each edge in order.

. When we come to edge e = (u,v), we add e to the subgraph H if there is currently no u-v path in H. (This is what Kruskal’s Algorithm would do as well.) On the other hand, if there is a u-v path in H, we let duv denote the length of the shortest such path; again, length is with respect to the values {le}. We add e to H if 3(le) < duv.

In other words, we add an edge even when u and v are already in the same connected component, provided that the addition of the edge reduces their shortest-path distance by a sufficient amount.

Let H = (V , F ) be the subgraph of G returned by the algorithm.

(a) Prove that for every pair of nodes u, v ? V, the length of the shortest u-v path in H is at most three times the length of the shortest u-v path in G.

(b) Despite its ability to approximately preserve shortest-path distances, the subgraph H produced by the algorithm cannot be too dense. Let f(n) denote the maximum number of edges that can possibly be produced as the output of this algorithm, over all n-node input graphs with edge lengths. Prove that

Explanation / Answer

n mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to disconnect the remaining nodes from each other.[1] It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network.

A graph is connected when there is a path between every pair of vertices. In a connected graph, there are no unreachable vertices. A graph that is not connected is disconnected. A graph G is said to be disconnected if there exist two nodes in G such that no path in G has those nodes as endpoints.

A directed graph is called weakly connected if replacing all of its directed edges with undirected edges produces a connected (undirected) graph. It is connected if it contains a directed path from u to v or a directed path from v to u for every pair of vertices u, v. It is strongly connected, diconnected, or simply strong if it contains a directed path from u to v and a directed path from v to u for every pair of verticesu, v. The strong components are the maximal strongly connected subgraphs.

One of the most important facts about connectivity in graphs is Menger's theorem, which characterizes the connectivity and edge-connectivity of a graph in terms of the number of independent paths between vertices.

If u and v are vertices of a graph G, then a collection of paths between u and v is called independent if no two of them share a vertex (other than u and v themselves). Similarly, the collection is edge-independent if no two paths in it share an edge. The number of mutually independent paths between u and v is written as (u, v), and the number of mutually edge-independent paths between u and v is written as (u, v).

Menger's theorem asserts that for distinct vertices u,v, (u, v) equals (u, v), and if u is also not adjacent to v then (u, v) equals (u, v).[2][3] This fact is actually a special case of the max-flow min-cut theorem.

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.