Problem Set 1

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.