Michael S. Paterson

A faster algorithm computing string edit distances

The edit distance between two character strings can be defined as the minimum cost of a sequence of editing operations which transforms one string into the other. The operations we admit are deleting, inserting and replacing one symbol at a time, with possibly different costs for each of these operations. The problem of finding the longest common subsequence of two strings is a special case of the problem of computing edit distances. We describe an algorithm for computing the edit distance between two strings of length n and m, n ⪖ m, which requires O(n · max(1, mlog n)) steps whenever the costs of edit operations are integral multiples of a single positive real number and the alphabet for the strings is finite. These conditions are necessary for the algorithm to achieve the time bound.

On the Number of Nonscalar Multiplications Necessary to Evaluate Polynomials

We present algorithms which use only

Efficient binary space partitions for hidden-surface removal and solid modeling

O(\sqrt n )

Finding the Median

nonscalar multiplications (i.e. multiplications involving “x” on both sides) to evaluate polynomials of degree n, and proofs that at least

Improved Sorting Networks with O(log n) Depth

\sqrt ...

A deterministic subexponential algorithm for solving parity games

We consider schemes for recursively dividing a set of geometric objects by hyperplanes until all objects are separated. Such abinary space partition, or BSP, is naturally considered as a binary tree where each internal node corresponds to a division. The goal is to choose the hyperplanes properly so that the size of the BSP, i.e., the number of resulting fragments of the objects, is minimized. For the two-dimensional case, we construct BSPs of sizeO(n logn) forn edges, while in three dimensions, we obtain BSPs of sizeO(n2) forn planar facets and prove a matching lower bound of Θ(n2). Two applications of efficient BSPs are given. The first is anO(n2)-sized data structure for implementing a hidden-surface removal scheme of Fuchset al. [6]. The second application is in solid modeling: given a polyhedron described by itsn faces, we show how to generate anO(n2)-sized CSG (constructive-solid-geometry) formula whose literals correspond to half-spaces supporting the faces of the polyhedron. The best previous results for both of these problems wereO(n3).

Deterministic one-counter automata

An algorithm is described which determines the median of n elements using in the worst case a number of comparisons asymptotic to 3n.

Optimal binary space partitions for orthogonal objects

A rectangleA and a setS ofn points inA are given. We present a new simple algorithm for the so-called largest empty rectangle problem, i.e., the problem of finding a maximum area rectangle contained inA and not containing any point ofS in its interior. The computational complexity of the presented algorithm isO(n logn + s), where s is the number of possible restricted rectangles considered. Moreover, the expected performance isO(n · logn).

On counting homomorphisms to directed acyclic graphs

The existence of polynomial time algorithms for the solution of parity games is a major open problem. The fastest known algorithms for the problem are randomized algorithms that run in subexponential time. These algorithms are all ultimately based on the randomized subexponential simplex algorithms of Kalai and of Matousek, Sharir and Welzl. Randomness seems to play an essential role in these algorithms. We use a completely different, and elementary, approach to obtain a deterministic subexponential algorithm for the solution of parity games. Our deterministic algorithm is almost as fast as the randomized algorithms mentioned above.

Better approximation guarantees for job-shop scheduling

The equivalence problem for deterministic one-counter automata is shown to bedecidable. A corollary for schema theory is that equivalence is decidable for Ianov schemas with an auxiliary counter.