This paper presents a simple dictionary structure designed for a hierarchical memory. The proposed data structure is cache oblivious and locality preserving. A cache-oblivious data structure has memory performance optimized for all levels of the memory hierarchy even though it has no memory-hierarchy-speci c parameterization. A localitypreserving dictionary maintains elements of similar key values stored close together for fast access to ranges of data with consecutive keys.

We obtain matching upper and lower bounds for the amount of time to find the predecessor of a given element among the elements of a fixed compactly stored set. Our algorithms are for the unit-cost word RAM with multiplication and are extended to give dynamic algorithms. The lower bounds are proved for a large class of problems, including both static and dynamic predecessor problems, in a much stronger communication game model, but they apply to the cell probe and RAM models.

It is shown that a static dictionary that offers constant-time access to n elements with w-bit keys and occupies O(n) words of memory can be constructed deterministically in O(n log n) time on a unit-cost RAM with word length w and a standard instruction set including multiplication. Whereas a randomized construction working in linear expected time was known, the running time of the best previous deterministic algorithm was Ω(n²). Using a standard dynamization technique, the first deterministic dynamic dictionary with constant lookup time and sublinear update time is derived. The new algorithms are weakly nonuniform; i.e., they require access to a fixed number of precomputed constants dependent on w. The main technical tools employed are unit-cost error-correcting codes, word parallelism, and derandomization using conditional expectations.

Abstract. We propose measures for compressed data structures, in which space usage is measured in a data-aware manner. In particular, we consider the fundamental dictionary problem on set data, where the task is to construct a data structure to represent a set S of n items out of a universe U = {0,..., u − 1} and support various queries on S. We use a well-known data-aware measure for set data called gap to bound the space of our data structures. We describe a novel dictionary structure taking gap+O(n log(u/n) / log n)+O(n log log(u/n)) bits. Under the RAM model, our dictionary supports membership, rank, select, and predecessor queries in nearly optimal time, matching the time bound of Andersson and Thorup’s predecessor structure [AT00], while simultaneously improving upon their space usage. Our dictionary structure uses exactly gap bits in the leading term (i.e., the constant factor is 1) and answers queries in near-optimal time. When seen from the worst case perspective, we present the first O(n log(u/n))-bit dictionary structure which supports these queries in nearoptimal time under RAM model. We also build a dictionary which requires the same space and supports membership, select, and partial rank queries even more quickly in O(log log n) time. To the best of our knowledge, this is the first of a kind result which achieves data-aware space usage and retains near-optimal time. 1

Abstract. We present cache-oblivious data structures based upon exponential structures. These data structures perform well on a hierarchical memory but do not depend on any parameters of the hierarchy, including the block sizes and number of blocks at each level. The problems we consider are searching, partial persistence and planar point location. On a hierarchical memory where data is transferred in blocks of size B, some of the results we achieve are: – We give a linear-space data structure for dynamic searching that supports searches and updates in optimal O(log B N) worst-case I/Os, eliminating amortization from the result of Bender, Demaine, and Farach-Colton (FOCS ’00). We also consider finger searches and updates and batched searches. – We support partially-persistent operations on an ordered set, namely, we allow searches in any previous version of the set and updates to the latest version of the set (an update creates a new version of the set). All operations take an optimal O(log B (m + N)) amortized I/Os, where N is the size of the version being searched/updated, and m is the number of versions. – We solve the planar point location problem in linear space, taking optimal O(log B N) I/Os for point location queries, where N is the number of line segments specifying the partition of the plane. The pre-processing requires O((N/B) log M/B N) I/Os, where M is the size of the ‘inner ’ memory. 1

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We introduce exponential search trees as a novel technique for converting static polynomial space search structures for ordered sets into fully-dynamic linear space data structures. This leads to an optimal bound of O ( � log n / log log n) for searching and updating a dynamic set X of n integer keys in linear space. Searching X for an integer y means finding the maximum key in X which is smaller than or equal to y. This problem is equivalent to the standard text book problem of maintaining an ordered set. The best previous deterministic linear space bound was O(log n / log log n) due to Fredman and Willard from STOC 1990. No better deterministic search bound was known using polynomial space. We also get the following worst-case linear space trade-offs between the number n, the word length W, and the maximal key U < 2W: O(min{log log n + log log U log n / log W, log log n · log log log U}). These trade-offs are, however, not likely to be optimal. Our results are generalized to finger searching and string searching, providing optimal results for both in terms of n.

In a natural variant of the comparison model, we show that there exists a constant ! < 1 such that the fully-dynamic d-dimensional orthogonal range reporting problem for d 2 can be solved in time O(log n) for updates and time O((log n= log log n) + r) for queries. Here n is the number of points stored and r is the number of points reported. The space usage is n). In the standard comparison model the result holds for d 3.

Abstract. In this paper, we present an experimental study of the spacetime tradeoffs for the dictionary problem, where we design a data structure to represent set data, which consist of a subset S of n items out of a universe U = {0, 1,...,u − 1} supporting various queries on S. Our primary goal is to reduce the space required for such a dictionary data structure. Many compression schemes have been developed for dictionaries, which fall generally in the categories of combinatorial encodings and data-aware methods and still support queries efficiently. We show that for many (real-world) datasets, data-aware methods lead to a worthwhile compression over combinatorial methods. Additionally, we design a new data-aware building block structure called BSGAP that presents improvements over other data-aware methods. 1

We address the problem of searching for a two-dimensional pattern in a two-dimensional text (or image), such that the pattern can be found even if it appears rotated and it is brighter or darker than its occurrence. Furthermore, we consider approximate matching under several tolerance models. We obtain algorithms that are almost optimal both in the worst and the average cases simultaneously. The complexities we obtain are very close to the best current results for the case where only rotations, but not lighting invariance, are supported. These are the first results for this problem under a combinatorial approach. 1