Multiple choice tries and distributed hash tables

Abstract

Tries were introduced in 1960 by Fredkin as an efficient method for searching and sorting digital data. Recent years have seen a resurgence of interest in tries that find applications in dynamic hashing, conflict resolution algorithms, leader election algorithms, IP addresses lookup, Lempel-Ziv compression schemes, and distributed hash tables. In some of these applications, most notably in distributed hash tables one needs to design a well balanced trie, that is, a trie with the height as close as possible to its fillup level. In this paper we consider tries built from n strings such that each string can be chosen from a pool of k strings, each of them generated by a discrete i.i.d. source. Three cases are considered: k = 2, k is large but fixed, and k ∼ c log n. The goal in each case is to obtain tries as balanced as possible. Various parameters such as height and fill-up level are analyzed. It is shown that for two-choice tries a 50% reduction in height is achieved when compared to ordinary tries. In a greedy on-line construction when the string that minimizes the depth of insertion for every pair is inserted, the height is only reduced by 25%. In order to further reduce the height by another 25%, we design a more refined on-line algorithm. The total computation time of the algorithm is O(n log n). Furthermore, when we choose the best among k ≥ 2 strings, then for large but fixed k the height is asymptotically equal to the typical depth in a trie, a result that cannot be improved. Finally, we show that further improvement can be achieved if the number of choices for each string is proportional to log n. In this case for unbiased memoryless sources highly balanced trees can be constructed by a simple greedy algorithm for which the difference between the height and the fill-up level is bounded by a constant with high probability. This, in turn, has implications for distributed hash tables, leading to a randomized ID management algorithm in peer-topeer networks such that, with high probability, the ratio between the maximum and the minimum load of a processor is O(1).

Information Transfer Ion Wireless Networks

“…Please note that [2], [20], [21], [24], [25], and [27] were published shortly after the commencement of the project before its number was changed, therefore acknowledge the funding under the original grant number FA8655-07-1-3071. Finally, [19], [22], and [26] are recent enhanced versions of publications using some results and approaches developed within the former Grant FA8655-04-1-3074, and therefore acknowledge the funding under that grant number.…”

A generalization of multiple choice balls-into-bins

“…Azar et al [1] showed that the maximum load is reduced to ln ln n/ ln d + O(1) in the d-choice process where each ball is placed into the least loaded among d bins chosen i.u.r. Since the multiple choice paradigm was introduced in 1990's, a number of variants of the load balancing process have been studied [8,5,7,2,4].…”