A lower bound for dynamic fractional cascading

Abstract

We investigate the limits of one of the fundamental ideas in data structures: fractional cascading. This is an important data structure technique to speed up repeated searches for the same key in multiple lists and it has numerous applications. Specifically, the input is a “catalog” graph, Gcat, of constant degree together with a list of values assigned to every vertex of Gcat. The goal is to preprocess the input such that given a connected subgraph G0 of Gcat and a single query value q, one can find the predecessor of q in every list that belongs to G0. The classical result by Chazelle and Guibas shows that in a pointer machine, this can be done in the optimal time of O(log n + |G0|) where n is the total number of values. However, if insertion and deletion of values are allowed, then the query time slows down to O(log n + |G0|log log n). If only insertions (or deletions) are allowed, then once again, an optimal query time can be obtained but by using amortization at update time. We prove a lower bound of Ω(log n√log log n) on the worst-case query time of dynamic fractional cascading, when queries are paths of length O(log n). The lower bound applies both to fully dynamic data structures with amortized polylogarithmic update time and incremental data structures with polylogarithmic worst-case update time. As a side, this also proves that amortization is crucial for obtaining an optimal incremental data structure. This is the first non-trivial pointer machine lower bound for a dynamic data structure that breaks the Ω(log n) barrier. In order to obtain this result, we develop a number of new ideas and techniques that hopefully can be useful to obtain additional dynamic lower bounds in the pointer machine model.