COS 423 Theory of Algorithms Spring 2005
Assignment 2
Answer problems 1โ4. This assignment is due Wednesday, February 16 at the beginning of lecture. Collab-
oration is allowed (according to the rules specified in the handout). If you work with a group, be sure to
clearly acknowledge the other members of your study group on the first page.
Read 4.1โ4.7 in Kleinberg-Tardos. Review Chapter 3 as needed.
1. Problem 4.7.
2. Problem 4.13.
3. An overlay network is a network architecture that runs on top of another network called the underlay
network. Overlay networks are widely used to deploy bandwidth-intensive content. For example, peer-
to-peer networks and PlanetLab are two examples of overlay networks that run on top of the Internet.
We represent an overlay network as a directed graph G=(V,E), where Vis the set of overlay nodes
and Eis the set of logical links (or directed edges) between overlay nodes. Each logical link corresponds
to a physical path in the underlay network. A rel ay pa th is a directed path between two overlay nodes
in the overlay network. Suppose that you have estimated the throughput of each logical link (e.g., by
measuring packet-loss probabilities and packet delays). Let tij >0 denote the throughput of logical
link (i, j). You want to broadcast data from a distinguished overlay node sto every other overlay node
in the network. For each overlay node i, find the bandwidth of the widest relay path (the path whose
smallest bandwidth link is as large as possible) from sto i. Your algorithm should run in O(mlog n)
time where n=|V|is the number of overlay nodes and m=|E|is the number of logical links.
4. Problem 4.18.
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