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.

Abstract not found

We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems.

We introduce a novel technique for converting static polynomial space search structures for ordered sets into fullydynamic linear space data structures. Based on this we present optimal bounds for dynamic integer searching, including finger search, and exponentially improved bounds for priority queues.

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

Abstract not found

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.

Abstract. We explore the following problem: given a collection of creases on a piece of paper, each assigned a folding direction of mountain or valley, is there a flat folding by a sequence of simple folds? There are several models of simple folds; the simplest one-layer simple fold rotates a portion of paper about a crease in the paper by ¦ � Æ. We first consider the analogous questions in one dimension lower—bending a segment into a flat object—which lead to interesting problems on strings. We develop efficient algorithms for the recognition of simply foldable 1-D crease patterns, and reconstruction of a sequence of simple folds. Indeed, we prove that a 1-D crease pattern is flat-foldable by any means precisely if it is by a sequence of one-layer simple folds. Next we explore simple foldability in two dimensions, and find a surprising contrast: “map ” folding and variants are polynomial, but slight generalizations are NP-complete. Specifically, we develop a linear-time algorithm for deciding foldability of an orthogonal crease pattern on a rectangular piece of paper, and prove that it is (weakly) NP-complete to decide foldability of (1) an orthogonal crease pattern on a orthogonal piece of paper, (2) a crease pattern of axis-parallel and diagonal (45-degree) creases on a square piece of paper, and (3) crease patterns without a mountain/valley assignment. 1

A new hashing primitive is introduced: dispersing hash functions. A family of hash functions F is dispersing if, for any set S of a certain size and random h ∈ F, the expected value of |S | − |h[S] | is not much larger than the expectancy if h had been chosen at random from the set of all functions. We give tight, up to a logarithmic factor, upper and lower bounds on the size of dispersing families. Such families previously studied, for example universal families, are significantly larger than the smallest dispersing families, making them less suitable for derandomization. We present several applications of dispersing families to derandomization (fast element distinctness, set inclusion, and static dictionary initialization). Also, a tight relationship between dispersing families and extractors, which may be of independent interest, is exhibited. We also investigate the related issue of program size for hash functions which are nearly perfect. In particular, we exhibit a dramatic increase in program size for hash functions more dispersing than a random function. 1

Abstract. We consider the problem of maintaining dynamically a set of points in the plane and supporting range queries of the type [a, b] × (−∞, c]. We assume that the inserted points have their x-coordinates drawn from a class of smooth distributions, whereas the y-coordinates are arbitrarily distributed. The points to be deleted are selected uniformly at random among the inserted points. For the RAM model, we present a linear space data structure that supports queries in O(log log n + t) expected time with high probability and updates in O(log log n) expected amortized time, where n is the number of points stored and t is the size of the output of the query. For the I/O model we support queries in O(log log B n + t/B) expected I/Os with high probability and updates in O(log B log n) expected amortized I/Os using linear space, where B is the disk block size. The data structures are deterministic and the expectation is with respect to the input distribution. 1