Sequential access in splay trees takes linear time

Sleator and Tarjan have invented a form of self-adjusting binary search tree called thesplay tree. On any sufficiently long access sequence, splay trees are as efficient, to within a constant factor, as both dynamically balanced and static optimum search trees. Sleator and Tarjan have made a much stronger conjecture; namely, that on any sufficiently long access sequence and to within a constant factor, splay trees are as efficient asany form of dynamically updated search tree. Thisdynamic optimality conjecture implies as a special case that accessing the items in a splay tree in sequential order takes linear time, i.e.O(1) time per access. In this paper we prove this special case of the conjecture, generalizing an unpublished result of Wegman. Oursequential access theorem not only supports belief in the dynamic optimality conjecture but provides additional insight into the workings of splay trees. As a corollary of our result, we show that splay trees can be used to simulate output-restricted deques (double-ended queues) in linear time. We pose several open problems related to our result.     

On the power of white pebbles

We construct a family (G _p |p) of directed acyclic graphs such that any black pebble strategy forG _p requiresp ² pebbles whereas 3p+1 pebbles are sufficient when white pebbles are allowed.Supported by the National Science Foundation under contract no. DCR-8407256 and by the office of Naval Research under contract no. N0014-80-0517.     

On rank vs. communication complexity

This paper concerns the open problem of Lovász and Saks regarding the relationship between the communication complexity of a boolean function and the rank of the associated matrix. We first give an example exhibiting the largest gap known. We then prove two related theorems.A preliminary version of this paper appeared in [10].This work was supported by USA-Israel BSF grant 92-00043 and by a Wolfeson research award administered by the Israeli Academy of Sciences.This work was supported by USA-Israel BSF grant 92-00106 and by a Wolfeson research award administered by the Israeli Academy of Sciences.     

The influence of large coalitions

This paper contains two results on influence in collective decision games. The first part deals with general perfect information coin-flipping games as defined in [3].Baton passing (see [8]), ann-player game from this class is shown to have the following property: IfS is a coalition of size at most \(\frac{n}{{3\log n}}\) , then the influence ofS on the game is only \(O\left( {\frac{{\left| S \right|}}{n}} \right)\) . This complements a result from [3] that for everyk there is a coalition of sizek with influence Ω(k/n). Thus the best possible bounds on influences of coalitions of size up to this threshold are known, and there need not be coalitions up to this size whose influence asymptotically exceeds their fraction of the population. This result may be expected to play a role in resolving the most outstanding problem in this area: Does everyn-player perfect information coin flipping game have a coalition ofo(n) players with influence 1?o(1)? (Recently Alon and Naor [1] gave a negative answer to this question.) In a recent paper Kahn, Kalai and Linial [7] showed that for everyn-variable boolean function of expectation bounded away from zero and one, there is a set of \(\frac{{n\omega (n)}}{{\log n}}\) variables whose influence is 1?o(1), wherew(n) is any function tending to infinity withn. They raised the analogous question where 1?o(1) is replaced by any positive constant and speculated that a constant influence may be always achievable by significantly smaller sets of variables. This problem is almost completely solved in the second part of this article where we establish the existence of boolean functions where only sets of at least \(\Omega \left( {\frac{n}{{\log ^2 n}}} \right)\) variables can have influence bounded away from zero.     

Parallel comparison algorithms for approximation problems

Suppose we haven elements from a totally ordered domain, and we are allowed to performp parallel comparisons in each time unit (=round). In this paper we determine, up to a constant factor, the time complexity of several approximation problems in the common parallel comparison tree model of Valiant, for all admissible values ofn, p and , where is an accuracy parameter determining the quality of the required approximation. The problems considered include the approximate maximum problem, approximate sorting and approximate merging. Our results imply as special cases, all the known results about the time complexity for parallel sorting, parallel merging and parallel selection of the maximum (in the comparison model), up to a constant factor. We mention one very special but representative result concerning the approximate maximum problem; suppose we wish to find, among the givenn elements, one which belongs to the biggestn/2, where in each round we are allowed to askn binary comparisons. We show that log^* n+O(1) rounds are both necessary and sufficient in the best algorithm for this problem.Research supported in part by Allon Fellowship, by a Bat Sheva de Rothschild grant and by the Fund for Basic Research administered by the Israel Academy of Sciences.     

Improved processor bounds for combinatorial problems in RNC

Our main result improves the known processor bound by a factor ofn ⁴ (maintaining the expected parallel running time,O(log³ n)) for the following important problem:find a perfect matching in a general or in a bipartite graph with n vertices. A solution to that problem is used in parallel algorithms for several combinatorial problems, in particular for the problems of finding i) a (perfect) matching of maximum weight, ii) a maximum cardinality matching, iii) a matching of maximum vertex weight, iv) a maximums-t flow in a digraph with unit edge capacities. Consequently the known algorithms for those problems are substantially improved.The results of this paper have been presented at the 26-th Annual IEEE Symp. FOCS, Portland, Oregon (October 1985).Partially supported by NSF Grants MCS 8303139 and DCR 8511713.Supporeted by NSF Grants MCS 8203232 and DCR 8507573.     

An analytic approach for the analysis of rotations in fringe-balanced binary search trees

This paper presents an analytic approach to the construction cost of fringe-balanced binary search trees. In [7], Mahmoud used a bottom-up approach and an urn model of Pólya. The present method is top-down and uses differential equations and Hwang's quasi-power theorem to derive the asymptotic normality of the number of rotations needed to construct such afringe balanced search tree. We also obtain the exact expectation and variance with this method. Although Pólya's urn model is no longer needed, we also present an elegant analysis of it based on an operator calculus as in [4].This research was supported by the Austrian Research Society (FWF) under the project number P12599-MAT.     

The complexity of detecting crossingfree configurations in the plane

The computational complexity of the following type of problem is studied. Given a geometric graphG=(P, S) whereP is a set of points in the Euclidean plane andS a set of straight (closed) line segments between pairs of points inP, we want to know whetherG possesses a crossingfree subgraph of a special type. We analyze the problem of detecting crossingfree spanning trees, one factors and two factors in the plane. We also consider special restrictions on the slopes and on the lengths of the edges in the subgraphs.Klaus Jansen acknowledges support by the Deutsche Forschungsgemeinschaft. Gerhard J. Woeginger acknowledges support by the Christian Doppler Laboratorium für Diskrete Optimierung.     

Sorting inc logn parallel steps

We give a sorting network withcn logn comparisons. The algorithm can be performed inc logn parallel steps as well, where in a parallel step we comparen/2 disjoint pairs. In thei-th step of the algorithm we compare the contents of registersR _j(i) , andR _k(i) , wherej(i), k(i) are absolute constants then change their contents or not according to the result of the comparison.     

On Heilbronn’s Problem in Higher Dimension

Heilbronn conjectured that given arbitrary n points in the 2-dimensional unit square [0, 1]², there must be three points which form a triangle of area at most O(1/n²). This conjecture was disproved by a nonconstructive argument of Komlós, Pintz and Szemerédi [10] who showed that for every n there is a configuration of n points in the unit square [0, 1]² where all triangles have area at least (log n/n²). Considering a generalization of this problem to dimensions d3, Barequet [3] showed for every n the existence of n points in the d-dimensional unit cube [0, 1]^d such that the minimum volume of every simplex spanned by any (d+1) of these n points is at least (1/n^d). We improve on this lower bound by a logarithmic factor (log n).     

Independence and port oracles for matroids,with an application to computational learning theory

Given a matroidM with distinguished elemente, aport oracie with respect toe reports whether or not a given subset contains a circuit that containse. The first main result of this paper is an algorithm for computing ane-based ear decomposition (that is, an ear decomposition every circuit of which contains elemente) of a matroid using only a polynomial number of elementary operations and port oracle calls. In the case thatM is binary, the incidence vectors of the circuits in the ear decomposition form a matrix representation forM. Thus, this algorithm solves a problem in computational learning theory; it learns the class ofbinary matroid port (BMP) functions with membership queries in polynomial time. In this context, the algorithm generalizes results of Angluin, Hellerstein, and Karpinski [1], and Raghavan and Schach [17], who showed that certain subclasses of the BMP functions are learnable in polynomial time using membership queries. The second main result of this paper is an algorithm for testing independence of a given input set of the matroidM. This algorithm, which uses the ear decomposition algorithm as a subroutine, uses only a polynomial number of elementary operations and port oracle calls. The algorithm proves a constructive version of an early theorem of Lehman [13], which states that the port of a connected matroid uniquely determines the matroid.Research partially funded by NSF PYI Grant No. DDM-91-96083.Research partially funded by NSF Grant No CCR-92-10957.     

The average-case analysis of some on-line algorithms for bin packing

In this paper we give tighter bounds than were previously known for the performance of the bin packing algorithms Best Fit and First Fit when the inputs are uniformly distributed on [0, 1]. We also give a general lower bound for the performance of any on-line bin packing algorithm on this distribution. We prove these results by analyzing optimal matchings on points randomly distributed in a unit square. We give a new lower bound for the up-right matching problem. A preliminary version of this paper appeared inProc. 25th IEEE Symposium on Foundations of Computer Science, 193–200. This research was done while the author was at the Massachusetts Institute of Technology Dept. of Mathematics and was supported by a NSF Graduate Fellowship and by Air Force grant OSR-82-0326.     

The complexity of computing the tutte polynomial on transversal matroids

The complexity of computing the Tutte polynomialT(M,x,y) is determined for transversal matroidM and algebraic numbersx andy. It is shown that for fixedx andy the problem of computingT(M,x,y) forM a transversal matroid is #P-complete unless the numbersx andy satisfy (x−1)(y−1)=1, in which case it is polynomial-time computable. In particular, the problem of counting bases in a transversal matroid, and of counting various types of “matchable” sets of nodes in a bipartite graph, is #P-complete.     

A fast parallel algorithm to compute the rank of a matrix over an arbitrary field

It is shown that the rank of a matrix over an arbitrary field can be computed inO(log² n) time using a polynomial number of processors. Also appeared in ACM Symposium on Theory of Computing, May 28–30, 1986 Berkeley, California. Research supported by Miller Fellowship, University of California, Berkeley.     

Toward a language theoretic proof of the four color theorem

This paper considers the problem of showing that every pair of binary trees with the same number of leaves parses a common word under a certain simple grammar. We enumerate the common parse words for several infinite families of tree pairs and discuss several ways to reduce the problem of finding a parse word for a pair of trees to that for a smaller pair. The statement that every pair of trees has a common parse word is equivalent to the statement that every planar graph is four-colorable, so the results are a step toward a language theoretic proof of the four color theorem.     

Spectral properties of threshold functions

We examine the spectra of boolean functions obtained as the sign of a real polynomial of degreed. A tight lower bound on various norms of the lowerd levels of the function's Fourier transform is established. The result is applied to derive best possible lower bounds on the influences of variables on linear threshold functions. Some conjectures are posed concerning upper and lower bounds on influences of variables in higher order threshold functions.Supported by an Eshkol fellowship, administered by the National Council for Research and Development—Israel Ministry of Science and Development.     

Dense sets and embedding binary trees into hypercubes

The purpose of this paper is to describe a method for embedding binary trees into hypercubes based on an iterative embedding into their subgraphs induced by dense sets. As a particular application, we present a class of binary trees, defined in terms of size of their subtrees, whose members allow a dilation two embedding into their optimal hypercubes. This provides a partial evidence in favor of a long-standing conjecture of Bhatt and Ipsen which claims that such an embedding exists for an arbitrary binary tree.     

On construction ofk-wise independent random variables

A 0–1probability space is a probability space (, 2,P), where the sample space -{0, 1} ⁿ for somen. A probability space isk-wise independent if, whenY _i is defined to be theith coordinate or the randomn-vector, then any subset ofk of theY _i 's is (mutually) independent, and it is said to be a probability spacefor p ₁,p ₂, ...,p _n ifP[Y _i =1]=p _i .We study constructions ofk-wise independent 0–1 probability spaces in which thep _i 's are arbitrary. It was known that for anyp ₁,p ₂, ...,p _n , ak-wise independent probability space of size always exists. We prove that for somep ₁,p ₂, ...,p _n [0,1],m(n,k) is a lower bound on the size of anyk-wise independent 0–1 probability space. For each fixedk, we prove that everyk-wise independent 0–1 probability space when eachp _i =k/n has size (n _k ). For a very large degree of independence —k=[n], for >1/2- and allp _i =1/2, we prove a lower bound on the size of . We also give explicit constructions ofk-wise independent 0–1 probability spaces.This author was supported in part by NSF grant CCR 9107349.This research was supported in part by the Israel Science Foundation administered by the lsrael Academy of Science and Humanities and by a grant of the Israeli Ministry of Science and Technology.