At a lecture in a computational biology conference one of us attended a few years ago, a w...

At a lecture in a computational biology conference one of us attended a few years ago, a well-known protein chemist talked about the idea of building a "representative set" for a large collection of protein molecules whose properties we don't understand. The idea would be to intensively study the proteins in the representative set and thereby learn (by inference) about all the proteins in the full collection.

To be useful, the representative set must have two properties.

• It should be relatively small, so that it will not be too expensive to study it.

• Every protein in the full collection should be "similar" to some protein in the representative set. (In this way, it truly provides some information about all the proteins.)

More concretely, there is a large set P of proteins. We define similarity on proteins by a distance function d: Given two proteins p and q, it returns a number d(p, q) > 0. In fact, the function d(:, •) most typically used is the sequence alignment measure, which we looked at when we studied dynamic programming in Chapter 6. We'll assume this is the distance being used here. There is a predefined distance cut-off A that's specified as part of the input to the problem; two proteins p and q are deemed to be "similar" to one another if and only if d(p, q)

We say that a subset of P is a representative set if, for every protein p, there is a protein q in the subset that is similar to it–that is, for which d(p, q) ≤ ∆. Our goal is to find a representative set that is as small as possible.

(a)  Give a polynomial-time algorithm that approximates the minimum representative set to within a factor of 0(log n). Specifically, your algorithm should have the following property: If the minimum possible size of a representative set is s1, your algorithm should return a representative set of size at most O(s2 log n).

(b)   Note the close similarity between this problem and the Center Selection Problem–—a problem for which we considered approximation algorithms in Section Why doesn't the algorithm described there solve the current problem?