Mergeable Dictionaries

TL;DR

This paper introduces a new data structure for mergeable dictionaries that efficiently supports predecessor search, split, and merge operations on interleaved sets in logarithmic time, improving over previous methods.

Contribution

The authors present a novel data structure that supports all mergeable dictionary operations, including interleaved merges, in O(log n) amortized time, surpassing prior solutions.

Findings

Supports interleaved set merges without interval restrictions

Operates in O(log n) amortized time for all operations

Improves over previous structures that had higher complexity or restrictions

Abstract

A data structure is presented for the Mergeable Dictionary abstract data type, which supports the following operations on a collection of disjoint sets of totally ordered data: Predecessor-Search, Split and Merge. While Predecessor-Search and Split work in the normal way, the novel operation is Merge. While in a typical mergeable dictionary (e.g. 2-4 Trees), the Merge operation can only be performed on sets that span disjoint intervals in keyspace, the structure here has no such limitation, and permits the merging of arbitrarily interleaved sets. Tarjan and Brown present a data structure which can handle arbitrary Merge operations in O(log n) amortized time per operation if the set of operations is restricted to exclude the Split operation. In the presence of Split operations, the amortized time complexity of their structure becomes \Omega(n). A data structure which supports both Split and Merge operations in O(log^2 n) amortized time per operation was given by Farach and Thorup. In contrast, our data structure supports all operations, including Split and Merge, in O(log n) amortized time, thus showing that interleaved Merge operations can be supported at no additional cost vis-a-vis disjoint Merge operations.