Helmut Alt

Computing a maximum cardinality matching in a bipartite graph in time O n 1.5 m/ log n

Abstract We show how to compute a maximum cardinality matching in a bipartite graph of n vertices in time O(n1.5 m log n ). For dense graphs this improves on the O( n m) algorithm of Hopcroft and Karp. The speed-up is obtained by an application of the fast adjacency matrix scanning technique of Cheriyan, Hagerup and Mehlhorn.

Approximate motion planning and the complexity of the boundary of the union of simple geometric figures

We study rigid motions of a rectangle amidst polygonal obstacles. The best known algorithms for this problem have a running time of Ω(n2), wheren is the number of obstacle corners. We introduce thetightness of a motion-planning problem as a measure of the difficulty of a planning problem in an intuitive sense and describe an algorithm with a running time ofO((a/b · 1/ɛcrit + 1)n(logn)2), wherea ≥b are the lengths of the sides of a rectangle and ɛcrit is the tightness of the problem. We show further that the complexity (= number of vertices) of the boundary ofn bow ties (see Figure 1) isO(n). Similar results for the union of other simple geometric figures such as triangles and wedges are also presented.

A Method for Obtaining Randomized Algorithms with Small Tail Probabilities

We study strategies for converting randomized algorithms of the Las Vegas type into randomized algorithms with small tail probabilities.

A language over a one symbol alphabet requiring only O (log log n) space

A language over a one symbol alphabe t requiring only 0(log log n) spac e It is well known that the minimal growth function for th e tape complexity of Turing machines is log log n (21. In th e literature, one can find essentially one example of a language requiring only 0(log log n) space, namel y Lo ={ bin(1)

A lower bound for the nondeterministic space complexity of context-free recognition

bin(2) bin(3)-if. .. # bin(n)* ; n E IN } where bin(i) is the binary representation of the integer i. In this note we describe a language over a one symbo l alphabet having space complexity 0(log log n). Let L 1 = { an ; the smallest number q which does not divid e n is a power of two } For every natural number n let q(n) be the smallest numbe r which does not divide n. Lemma : 3c > 0 : q(n) S c log n Proof :. Let n be any natural number and let m = q(n). The n all primes less than m divide n and hence their product P divides them. By the prime number theorem of Gauss there ar e about m/ log m primes less than m and hence P is larger than

Square rooting is as difficult as multiplication

Abstract We prove a log n lower bound on the nondeterministic space complexity of every nonregular deterministic context-free language.

Lower bounds on space complexity for contextfree recognition

It is shown that multiplication of numbers and square rooting have the same complexity, i. e. from a program for multiplication one can construct a program for square rooting with the same asymptotic time complexity (1 step≦1 bit-operation) and vice versa. It follows from the Schönhage-Strassen algorithm that square rooting can be performed in 0 (n logn log logn) bit-operations.ZusammenfassungEs wird gezeigt, daß Multiplikation von Zahlen und Bestimmen der Quadratwurzel von gleicher Komplexität sind, d. h. aus einem Programm zur Multiplikation kann man eines zum Wurzelziehen konstruieren, das größenordnungsmäßig die gleiche Zeitkomplexität hat (1 Schritt ≦ 1 Bit-Operation) und umgekehrt. Mit dem Schönhage-Strassen-Algorithmus erhält man so einen 0 (n logn log logn)-Algorithmus zum Berechnen der Quadratwurzel.

Multiplication is the easiest nontrivial arithmetic function

SummaryUsing methods from linear algebra and crossing-sequence arguments it is shown that logarithmic space is necessary for the recognition of all context-free nonregular subsets of {a1}* ... {an}*, where {a1,...,an} is some alphabet. It then follows that log n is a lower bound on the space complexity for the recognition of any bounded or deterministic non-regular context-free language.

Comparison of arithmetic functions with respect to boolean circuit depth

It is shown that floating point (or integer) multiplication can be reduced to the evalution of a very large class of functions including most of the nontrivial functions used in practice. That means that whenever any such function can be evaluated by boolean circuits of size S(n), then multiplication can be done with circuits of size O(S(n)). as well.

Searching Semisorted Tables

1. IntroducUon This paper investigates reductions between numerical functions with respect to Boolean circuit size or depth as complexity measures. We assume that the input is a binary fixed point representation of the argument, if there are n input bits, the circuit is supposed to compute the n most significant bits of the function value. A function ] is called rsduc#.bls to a function g, (jf ~ g) , iff, whenever g can be evaluated for an n-bit input with complexity C(n), then f can be evaluated with complexity O(U(n)), as well. UsualLy we will require that reductions are unifozlrct in the sense that they can be computed in logarithmic space or at least polynomial time. In this paper Theorem 1 will be uniform even with respect to logarithmic space. Two functions f, g are called e~u~g/e~tt (f ~//) if f ~g andg ~y. Permission to copy without fee all or part of this material is granted provided that the copies arc not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is givcn that copying is by permission of the Association for Computing Machincry. To copy otherwise, or to republish, requires a fcc and/or specific pcrmission. We will investigate reducibilities with respect to Boolean Some reducibilities among arithmetic functions are easy to demonstrate. Let rout. div. and sq denote multiplication , division, and squaring, respectively. Then the reduction of sq to mu/ is trivial, the converse one is possible because sit = ~(=+y)s _=s _ ~) (1) Since addition and subtraction can be done in linear size and logarithmic depth simultaneously (cf. [0]), i.e. optimally with respect to both complexity measures, we can use them for free. So with (1) we have that Trtul ~ sq Additionally, from the equation z2 = 1 Z I i (~) z z+l it follows that and, thus sq ~ d/v The motivation to investigate reducibility and equivalence of functions is, of course, that lower or upper bounds, or both, for one function transfer to the 466