Dynamic fractional cascading

Abstract

The problem of searching for a key in many ordered lists arises frequently in computational geometry. Chazelle and Guibas recently introduced fractional cascading as a general technique for solving this type of problem. In this paper we show that fractional cascading also supports insertions into and deletions from the lists efficiently. More specifically, we show that a search for a key in n lists takes time O(log N+ n log log N) and an insertion or deletion takes time O(log log N). Here N is the total size of all lists. If only insertions or deletions have to be supported the O(log log N) factor reduces to O(1). As an application we show that queries, insertions, and deletions into segment trees or range trees can be supported in time O(log n log log n), when n is the number of segments (points).

Marked Ancestor Problems (Preliminary Version)

“…The upper bound for this problem is slightly larger, amortised O(log n log log n) using fractional cascading [31], yet for dominance queries it is O(log n/ log log n) using [29,40], matching our lower bound.…”

Sequential dependency computation via geometric data structures

“…The augmented segment tree of Mehlhorn and Näher [5] guarantees the existence of a O(N log N log log N )-time algorithm [4]. In fact, a data structure giving a running time of O(N log N log log N ) is likely to be implicit in Gabow, Bentley, and Tarjan [2]; however their result as stated (Theorem 3.3 and the discussion above it) is for the case when all key(i) values are known in advance.…”

An Optimal Algorithm for Computing Angle-Constrained Spanners

“…Mehlhorn and Näher [18] have shown that both of these operations can be done in O(log d−1 n) time. Moreover, such a range tree can be built in O(n log d−1 n) time.…”

“…First, the amount of space used by the algorithm is O((1/ d−1 )n log d−1 n). Second, the algorithms in [18] do not work in the algebraic computation-tree model. Thus, even though the running time is O(n log n) in the case when d = 2, the Ω(n log n) lower bound of [8] on the time to compute any spanner does not apply.…”