Leonidas J. Guibas

Primitives for the manipulation of general subdivisions and the computation of Voronoi

The following problem is discussed: given n points in the plane (the sites) and an arbitrary query point q, find the site that is closest to q. This problem can be solved by constructing the Voronoi diagram of the griven sites and then locating the query point inone of its regions. Two algorithms are given, one that constructs the Voronoi diagram in O(n log n) time, and another that inserts a new sit on O(n) time. Both are based on the use of the Voronoi dual, or Delaunay triangulation, and are simple enough to be of practical value. the simplicity of both algorithms can be attributed to the separation of the geometrical and topological aspects of the problem and to the use of two simple but powerful primitives, a geometric predicate and an operator for manipulating the topology of the diagram. The topology is represented by a new data structure for generalized diagrams, that is, embeddings of graphs in two-dimensional manifolds. This structure represents simultaneously an embedding, its dual, and its mirror image. Furthermore, just two operators are sufficients for building and modifying arbitrary diagrams.

String overlaps, pattern matching, and nontransitive games

Abstract This paper studies several topics concerning the way strings can overlap. The key notion of the correlation of two strings is introduced, which is a representation of how the second string can overlap into the first. This notion is then used to state and prove a formula for the generating function that enumerates the q -ary strings of length n which contain none of a given finite set of patterns. Various generalizations of this basic result are also discussed. This formula is next used to study a wide variety of seemingly unrelated problems. The first application is to the nontransitive dominance relations arising out of a probabilistic coin-tossing game. Another application shows that no algorithm can check for the presence of a given pattern in a text without examining essentially all characters of the text in the worst case. Finally, a class of polynomials arising in connection with the main result are shown to be irreducible.

Periods in strings

Abstract In this paper we explore the notion of periods of a string. A period can be thought of as a shift that causes the string to match over itself. We obtain two sets of necessary and sufficient conditions for a set of integers to be the set of periods of some string (what we call the correlation of the string). We show that the number of distinct correlations of length n is independent of the alphabet size and is of order nlogn. By using generating function methods we enumerate the strings having a given correlation, and investigate certain related questions.

The power of geometric duality

This paper uses a new formulation of the notion of duality that allows the unified treatment of a number of geometric problems. In particular, we are able to apply our approach to solve two long-standing problems of computational geometry: one is to obtain a quadratic algorithm for computing the minimum-area triangle with vertices chosen among n points in the plane; the other is to produce an optimal algorithm for the half-plane range query problem. This problem is to preprocess n points in the plane, so that given a test half-plane, one can efficiently determine all points lying in the half-plane. We describe an optimal O(k + log n) time algorithm for answering such queries, where k is the number of points to be reported. The algorithm requires O(n) space and O(n log n) preprocessing time. Both of these results represent significant improvements over the best methods previously known. In addition, we give a number of new combinatorial results related to the computation of line arrangements.

Visibility and intersectin problems in plane geometry

We develop new data structures for solving various visibility and intersection problems about a simple polygon <italic>P</italic> on <italic>n</italic> vertices. Among our results are a simple <italic>&Ogr;</italic>(<italic>n</italic> log <italic>n</italic>) algorithm for computing the illuminated subpolygon of <italic>P</italic> from a luminous side, and an <italic>&Ogr;</italic>(log <italic>n</italic>) algorithm for determining which side of <italic>P</italic> is first hit by a bullet fired from a point in a certain direction. The latter method requires preprocessing on <italic>P</italic> which takes time <italic>&Ogr;</italic>(<italic>n</italic> log <italic>n</italic>) and space <italic>&Ogr;</italic>(<italic>n</italic>). Our main new tool in attacking these problems is geometric duality on the two-sided plane.

Long repetitive patterns in random sequences

SummaryAppearances of long repetitive sequences such as 00...0 or 1010...101 in random sequences are studied. The expected length of the longest repetitive run of any specified type in a random binary sequence of length n is shown to tend to the binary logarithm of n plus a periodic function of log n. Necessary and sufficient conditions are derived to ensure that with probability 1 an infinite random sequence should contain repetitive runs of specified lengths in given initial segments. Finally, the number of long repetitive runs of a specified kind that occur in a random sequence is studied. These results are derived from simple expressions for the generating functions for the probabilities of occurrences of various repetitive runs. These generating functions are rational, and lead to sharp asymptotic estimates for the probabilities.

Finding extremal polygons

Given n points in the plane, we present algorithms for finding maximum perimeter or area convex k-gons with vertices k of the given n points. Our algorithms work in linear space and time

MAXIMAL PREFIX-SYNCHRONIZED CODES.

O(kn\lg n + n\lg ^2 n)

On translating a set of rectangles

. For the special case

The analysis of double hashing

k = 3