18-434 Problem Set 1 1 18.434: Seminar in Theoretical Computer Science Spring 2015, Problem Set 1 This problem set has a total of 7 problems. You can do any 4 of them. Solutions can be either presented to the instructor or handed in. 1 Converting Walks to Paths Given a graph G with n vertices, m edges, and a walk of length k from vertex s to vertex t. Give an O(k) time algorithm that produces a path from s to t. (recall that a walk is a sequence of vertices v0 = s, v1 . . . vk = t such that vi vi+1 is an edge, and a path is a walk with no repeating vertices.) 2 Minimum Diameter Spanning Tree on a Plane Consider the minimum diameter spanning tree problem on the plane: all edges are present, and distance between two vertices equals to the Euclidean distance between them. Prove that there is a minimum diameter spanning tree whose underlying combinatorial tree has (unweighted) diameter at most 3. 3 Minimum Spanning Trees and Minimum Cost Arborescences In class we saw that the minimum cost arborescence can be different than the minimum spanning tree. Prove or disprove: given an undirected graph G, we can always oriented the edges and pick a root suitably so the minimum spanning tree is a minimum cost arborescence. 4 Low Stretch Spanning Trees What is a good low-stretch spanning tree for: 1. A complete graph on n vertices? 2. A 3-D cube with side lengths n1/3 ? 3. A cycle on n vertices? 5 Alternate Definition of LSST An alternate definition of low-stretch spanning tree is to measure the maximum stretch instead of the average stretch. Give a class of unweighted graphs where any spanning tree must have an edge with stretch Ω(n). 6 Broadcast Domination Revisited Consider the following variant of broadcast domination: the cost of broadcasting a distance i from node u is given by a function c(u, i). Also, the broadcast doesn’t need to be efficient: a node can 18-434 Problem Set 1 2 be covered by multiple other nodes’ broadcasts. Given a polynomial time algorithm for computing a plan of minimum total cost. 7 Ultrametrics A formal definition of ultra-metric is a function d(u, v) that satisfies d(x, y) ≤ max{d(x, z), d(y, z)}. Let c(u, v) denote the size of the minimum cut separating u and v, aka: min S⊂V,u∈S,v ∈S / |∂(S)|, where ∂(S) denotes the set of edges leaving S. Prove c(u, v) is an ultra-metric.