Martti Penttonen

A Reliable Randomized Algorithm for the Closest-Pair Problem

The following two computational problems are studied:Duplicate grouping:Assume thatnitems are given, each of which is labeled by an integer key from the set {0,?,U?1}. Store the items in an array of sizensuch that items with the same key occupy a contiguous segment of the array.Closest pair:Assume that a multiset ofnpoints in thed-dimensional Euclidean space is given, whered?1 is a fixed integer. Each point is represented as ad-tuple of integers in the range {0,?,U?1} (or of arbitrary real numbers). Find a closest pair, i.e., a pair of points whose distance is minimal over all such pairs.In 1976, Rabin described a randomized algorithm for the closest-pair problem that takes linear expected time. As a subroutine, he used a hashing procedure whose implementation was left open. Only years later randomized hashing schemes suitable for filling this gap were developed.In this paper, we return to Rabins classic algorithm to provide a fully detailed description and analysis, thereby also extending and strengthening his result. As a preliminary step, we study randomized algorithms for the duplicate-grouping problem. In the course of solving the duplicate-grouping problem, we describe a new universal class of hash functions of independent interest.It is shown that both of the foregoing problems can be solved by randomized algorithms that useO(n) space and finish inO(n) time with probability tending to 1 asngrows to infinity. The model of computation is a unit-cost RAM capable of generating random numbers and of performing arithmetic operations from the set {+,?,?,div,log2,exp2}, wheredivdenotes integer division andlog2andexp2are the mappings from N to N?{0} withlog2(m)=?log2m? andexp2(m)=2mfor allm?N. If the operationslog2andexp2are not available, the running time of the algorithms increases by an additive term ofO(loglogU). All numbers manipulated by the algorithms consist ofO(logn+logU) bits.The algorithms for both of the problems exceed the time boundO(n) orO(n+loglogU) with probability 2?n?(1). Variants of the algorithms are also given that use onlyO(logn+logU) random bits and have probabilityO(n??) of exceeding the time bounds, where ??1 is a constant that can be chosen arbitrarily.The algorithms for the closest-pair problem also works if the coordinates of the points are arbitrary real numbers, provided that the RAM is able to perform arithmetic operations from {+,?,?,div} on real numbers, whereadivbnow means ?a/b?. In this case, the running time isO(n) withlog2andexp2andO(n+loglog(?max/?max)) without them, where ?maxis the maximum and ?minis the minimum distance between any two distinct input points.

One-sided and two-sided context in formal grammars

A proof for the equivalence of context-sensitive and one-sided contextsensitive languages is given. This yields as a corollary the normal form A → BC, AB → AC, A → a for context-sensitive grammars.

A note on the complexity of the satisfiability of modal Horn clauses

Abstract A bottom-up algorithm is given for testing the satisfiability of sets of propositional modal Horn clauses. For some modal logics, such as S5, the algorithm works in polynomial time, while for other modal logics, like K, Q, T, and S4, the worst case complexity can be considerably higher.

Towards automating of document structure transformations

In this paper we develop a syntax-directed approach to transformation of documents from one structure to another. The aim is to automate a transformation between two grammars that have common parts, although the grammars and names of elements may differ. In an important case, called local transformations, the transformation can be performed by a finite state tree transducer. We propose a system that can generate a transformation semi-automatically if the user defines a matching between the elements containing the text of the document. Multiple possible translations from the target grammar can be restricted using a suitable heuristic function so that the transformation can be completed in a reasonable time period. From the generated transformation specification, it is possible to construct rules for a tree transducer or XSLT script automatically.

Fast simulation of Turing machines by random access machines

We prove that a

Syntax-directed compression of program files

T(n)

Hot-potato routing algorithms for sparse optical torus

time-bounded,

Some decidability problems of sentential formst

S(n)

NP-COMPLETENESS OF THE HAMMING SALESMAN PROBLEM

space-bounded and

Efficient two-level mesh based simulation of PRAMS

U(n)