Mergeable Dictionaries

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 Union. While Predecessor-Search and Split work in the normal way, the novel operation is Union. While in a typical mergeable dictionary (e.g. 2-4 Trees), the Union 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. Our data structure supports all operations, including Union, in O(log n) amortized time, thus showing that interleaved Union operations can be supported at no additional cost vis-a-vis disjoint Union operations

Decompressing Lempel-Ziv Compressed Text

We consider the problem of decompressing the Lempel--Ziv 77 representation of a string $S$ of length $n$ using a working space as close as possible to the size $z$ of the input. The folklore solution for the problem runs in $O(n)$ time but requires random access to the whole decompressed text. Another folklore solution is to convert LZ77 into a grammar of size $O(z\log(n/z))$ and then stream $S$ in linear time. In this paper, we show that $O(n)$ time and $O(z)$ working space can be achieved for constant-size alphabets. On general alphabets of size $\sigma$, we describe (i) a trade-off achieving $O(n\log^\delta \sigma)$ time and $O(z\log^{1-\delta}\sigma)$ space for any $0\leq \delta\leq 1$, and (ii) a solution achieving $O(n)$ time and $O(z\log\log (n/z))$ space. The latter solution, in particular, dominates both folklore algorithms for the problem. Our solutions can, more generally, extract any specified subsequence of $S$ with little overheads on top of the linear running time and working space. As an immediate corollary, we show that our techniques yield improved results for pattern matching problems on LZ77-compressed text

A simple quantum algorithm to efficiently prepare sparse states

State preparation is a fundamental routine in quantum computation, for which many algorithms have been proposed. Among them, perhaps the simplest one is the Grover-Rudolph algorithm. In this paper, we analyse the performance of this algorithm when the state to prepare is sparse. We show that the gate complexity is linear in the number of non-zero amplitudes in the state and quadratic in the number of qubits. We then introduce a simple modification of the algorithm, which makes the dependence on the number of qubits also linear. This is competitive with the best known algorithms for sparse state preparatio

A pointer-free data structure for merging heaps and min-max heaps

AbstractIn this paper a data structure for the representation of mergeable heaps and min-max heaps without using pointers is introduced. The supported operations are: Insert, DeleteMax, DeleteMin, FindMax, FindMin, Merge, NewHeap, DeleteHeap. The structure is analyzed in terms of amortized time complexity, resulting in a O(1) amortized time for each operation except for Insert, for which a O(lg n) bound holds

10091 Abstracts Collection -- Data Structures

From February 28th to March 5th 2010, the Dagstuhl Seminar 10091 "Data Structures" was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. It brought together 45 international researchers to discuss recent developments concerning data structures in terms of research, but also in terms of new technologies that impact how data can be stored, updated, and retrieved. During the seminar a fair number of participants presented their current research and open problems where discussed. This document first briefly describes the seminar topics and then gives the abstracts of the presentations given during the seminar