Near-Entropy Hotlink Assignments

Abstract

Consider a rooted tree T of arbitrary maximum degree d representing a collection of n web pages connected via a set of links, all reachable from a source home page represented by the root of T. Each web page i carries a weight w _i representative of the frequency with which it is visited. By adding hotlinks — shortcuts from a node to one of its descendents — we wish to minimize the expected number of steps l needed to visit pages from the home page, expressed as a function of the entropy H(p) of the access probabilities p. This paper introduces several new strategies for effectively assigning hotlinks in a tree. For assigning exactly one hotlink per node, our method guarantees an upper bound on l of 1.141 H(p)+1 if d>2 and 1.08 H(p)+ 2/3 if d=2. We also present the first efficient general methods for assigning at most k hotlinks per node in trees of arbitrary maximum degree, achieving bounds on l of at most $\frac{2H(p)}{\log (k+1)}$ and $\frac{H(p)}{\log (k+d) -- \log d}$, respectively. Finally, we present an algorithm implementing these methods in O(n logn) time, an improvement over the previous O(n ²) time algorithms.