Approximate Hotlink Assignment

Abstract

Consider a directed rooted tree T = (V,E) of maximal degree d representing a collection V of web pages connected via a set E of links all reachable from a source home page, represented by the root of T. Each leaf web page carries a weight representative of the frequency with which it is visited. By adding hotlinks, shortcuts from a node to one of its descendents, we are interested in minimizing the expected number of steps needed to visit the leaf pages from the home page. We give an O(N ²) time algorithm for assigning hotlinks so that the expected number of steps to reach the leaves from the root of the tree is at most $\tfrac{{H\left( p \right)}} {{log\left( {d + 1} \right) - \left( {d/\left( {d + 1} \right)} \right)log d}} + \tfrac{{d + 1}} {d}$, where H(p) is the entropy of the probability (frequency) distribution p =< p1,p2, , . . . , pN > on the N leaves of the given tree, i.e., pi is the weight on the ith leaf. The best known lower bound for this problem is $\tfrac{{H\left( p \right)}} {{log\left( {d + 1} \right)}}$. Thus our algorithm approximates the optimal hotlink assignment to within a constant for any fixed d.

Research supported in part by NSERC (Natural Sciences and Engineering Research Council) of Canada and MITACS (Mathematics of Information Technology and Complex Systems) grants.