Data Structures with Local Update Operations

Abstract

In this paper we describe dynamic data structures with restrictions on update operations. In the first part of the paper we consider data structures that support operations insert _Δ (x,y) or insert _Δ (x) instead of general insertions, where insert _Δ (x,y) (insert _Δ (x)) inserts a new element x, such that |x − y| ≤ Δ for some element y already stored in the data structure. We present a data structure that supports predecessor queries in a universe of size U in O(loglogU) time, uses O(n) words of space, and supports operations insert _Δ (x,y), and in O(1) amortized time, where $\Delta=2^{2^{O(\sqrt{\log \log U})}}$. We present the dictionary data structure that supports membership queries in O(loglogn) time and insert _Δ (x,y) and delete (x) in O(1) amortized time, where $\Delta=2^{2^{O(\sqrt{\log \log n})}}$ We also present a priority queue that supports , and in O(1) time and in O(loglogn) time, where Δ = log^O(1) U. All above data structures also support incrementation and decrementation of element values by the corresponding parameter Δ.

In the second part of this paper, we consider the data structure for dominance emptiness queries in the case when an update changes the relative order of two points or increments/decrements coordinates of a point by a small parameter. We show that in this case dominance emptiness queries can be answered faster than the lower bound for the fully dynamic data structure.