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 wi 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 1.141H(p)+1 if d > 2 and 1.08H(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 2H(p) log(k+1) and H(p) log(k+d)−log d , respectively. Finally, we present an algorithm implementing these methods in O(n log n) time, an improvement over the previous O(n 2) time algorithms.

An Approximation Algorithm for Binary Searching in Trees

“…In fact, the 'average case' version studied here is one more example of problems related to searching and coding whose complexities are unknown [3,8,11,12].…”

“…Employing the entropy hypothesis to the first term of inequality (8) and using the inductive hypothesis we have…”

“…In Step (iv), the modified algorithm finds a permu- An upper bound for this modified version of the algorithm can be derived by the same analysis used previously. In fact, the introduction of the approximate sorting only alters the third term of the upper bound presented in inequality (8), leading to the following bound:…”

An experimental study of recent hotlink assignment algorithms

“…The threshold α is given as the solution of ( α 1−α ) 2(1−α) = α, that is, α ≈ 0.2965. The H/PH-algorithm has been proposed in Douïeb and Langerman [2006], where it is proved to guarantee a path length of at most 1.141H(ω) + 1. Thus, it is asymptotically a (1.141 log( +1))-approximation in terms of the path length, where is the outdegree of the tree (see Section 1).…”

“…In Douïeb and Langerman [2006], the authors propose an implementation of H/PH that runs in worst-case time O(n log n). They observe that it suffices to traverse the root's heavy path when looking for h. As in the worst-case, that path can have a length of O(n), an involved tree representation is employed for finding h in time O(log n).…”