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2000

Volume 29, Issue 6, pp. 1761-2097


Improved Data Structures for Fully Dynamic Biconnectivity

Monika R. Henzinger

SIAM J. Comput. 29, pp. 1761-1815 (55 pages)

Online Publication Date: July 27, 2006

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We present fully dynamic algorithms for maintaining the biconnected components in general and plane graphs.
A fully dynamic algorithm maintains a graph during a sequence of insertions and deletions of edges or isolated vertices. Let m be the number of edges and n be the number of vertices in a graph. The time per operation of the best deterministic algorithms is $O(\sqrt n)$ in general graphs and O(log n) in plane graphs for fully dynamic connectivity and O(min m2/3,n}) in general graphs and $O(\sqrt n)$ in plane graphs for fully dynamic biconnectivity. We improve the later running times to $O(\sqrt {m\log n})$ in general graphs and O(log 2n) in plane graphs. Our algorithm for general graphscan also find the biconnected components of all vertices in time O(n).

Optimal Combinatorial Functions Comparing Multiprocess Allocation Performance in Multiprocessor Systems

HÅkan Lennerstad and Lars Lundberg

SIAM J. Comput. 29, pp. 1816-1838 (23 pages) | Cited 1 time

Online Publication Date: July 27, 2006

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For the execution of an arbitrary parallel program P, consisting of a set of processes with any executable interprocess dependency structure, we consider two alternative multiprocessors.
The first multiprocessor has q processors and allocates parallel programs dynamically; i.e., processes may be reallocated from one processor to another. The second employs cluster allocation with k clusters and u processors in each cluster: here processes may be reallocatedwithin a cluster only. Let Td(P,q) and Tc(P,k,u) be execution times for the parallel program P with optimal allocations. We derive a formula for the program independent performance function $$ G(k,u,q)=\sup_{\mbox{{\footnotesize all parallel programs $P$}}} \:\frac{T_c(P,k,u)}{T_d(P,q)}. $$
Hence, with optimal allocations, the execution of P can never take more than a factor G(k,u,q) longer time with the second multiprocessor than with the first, and there exist programs showing that the bound is sharp.
The supremum is taken over all parallel programs consisting of any number of processes. Overhead for synchronization and reallocation is neglected only.
We further present a tight bound which exploits a priori knowledge of the class of parallel programs intended for the multiprocessors, thus resulting in a sharper bound. The function g(n,k,u,q) is the above maximum taken over all parallel programs consisting of n processes.
The functions G and g can be used in various ways to obtain tight performance bounds, aiding in multiprocessor architecture decisions.

The CREW PRAM Complexity of Modular Inversion

Joachim von zur Gathen and Igor E. Shparlinski

SIAM J. Comput. 29, pp. 1839-1857 (19 pages)

Online Publication Date: July 27, 2006

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One of the long-standing open questions in the theory of parallel computation is the parallel complexity of the integer gcd and related problems, such as modular inversion. We present a lower bound $\Omega (\log n)$ for the parallel time on a concurrent-read exclusive-write parallel random access machine (CREW PRAM) computing the inverse modulo certain n-bit integers, including all such primes. For infinitely many moduli, our lower bound matches asymptotically the known upper bound. We obtain a similar lower bound for computing a specified bit in a large power of an integer. Our main tools are certain estimates for exponential sums in finite fields.

Dynamic Maintenance of Maxima of 2-d Point Sets

Sanjiv Kapoor

SIAM J. Comput. 29, pp. 1858-1877 (20 pages) | Cited 5 times

Online Publication Date: July 27, 2006

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This paper describes an efficient scheme for the dynamic maintenance of the set of maxima of a 2-d set of points. Using the fact that the maxima can be stored in a staircase structure, we use a technique in which we maintain approximations to the staircase structure. We first describe how to maintain the maxima in O(log n) time per insertion and deletion when there are n insertions and deletions. O(log n) is charged per change for reporting changes to the staircase structure which stores the maxima. O(n) space is used. We also show another scheme which requires a total of O(n log n + r) time when r maximal points are listed. We finally consider extensions to higher dimensions.

The Complexity of the A B C Problem

Jin-yi Cai, Richard J. Lipton, and Yechezkel Zalcstein

SIAM J. Comput. 29, pp. 1878-1888 (11 pages)

Online Publication Date: July 27, 2006

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We present a deterministic polynomial-time algorithm for the A B C problem, which is the membership problem for 2-generated commutative linear semigroups over an algebraic number field. We also obtain a polynomial-time algorithm for the (easier) membership problem for 2-generated abelian linear groups. Furthermore, we provide a polynomial-sized encoding for the set of all solutions.

The Load and Availability of Byzantine Quorum Systems

Dahlia Malkhi, Michael K. Reiter, and Avishai Wool

SIAM J. Comput. 29, pp. 1889-1906 (18 pages) | Cited 3 times

Online Publication Date: July 27, 2006

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Replicated services accessed via quorums enable each access to be performed at only a subset (quorum) of the servers and achieve consistency across accesses by requiring any two quorums to intersect. Recently, b-masking quorum systems, whose intersections contain at least 2b+1 servers, have been proposed to construct replicated services tolerant of b-arbitrary (Byzantine) server failures. In this paper we consider a hybrid fault model allowing benign failures in addition to the Byzantine ones. We present four novel constructions for b-masking quorum systems in this model, each of which has optimal load (the probability of access of the busiest server) or optimal availability (probability of some quorum surviving failures). To show optimality we also prove lower bounds on the load and availability of any b-masking quorum system in this model.

An Online Algorithm for Improving Performance in Navigation

Avrim Blum and Prasad Chalasani

SIAM J. Comput. 29, pp. 1907-1938 (32 pages)

Online Publication Date: July 27, 2006

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We consider the following scenario. A point robot is placed at some start location s in a 2-dimensional scene containing oriented rectangular obstacles. The robot must repeatedly travel back and forth between s and a second location t in the scene. The robot knows the coordinates of s and t but initially knows nothing about the positions or sizes of the obstacles. It can only determine the obstacles' locations by bumping into them. We would like an intelligent strategy for the robot so that its trips between s and t both are relatively fast initially and improve as more trips are taken and more information is gathered.
In this paper we describe an algorithm for this problem with the following guarantee: in the first $k \leq n$ trips, the average distance per trip is at most $O(\sqrt{n/k})$ times the length of the shortest s-t path in the scene, where n is the Euclidean distance between s and t. We also show a matching lower bound for deterministic strategies. These results generalize known bounds on the one-trip problem. Our algorithm is based on a novel method for making an optimal trade-off between search effort and the goodness of the path found. We improve this algorithm to a "smooth" variant having the property that for every $i \leq n,$ the robot's ith trip length is $O(\sqrt{n/i})$ times the shortest s-t path length.
A key idea of this paper is a method for analyzing obstacle scenes using a tree structure that can be defined based on the positions of the obstacles.

On Interpolation and Automatization for Frege Systems

Maria Luisa Bonet, Toniann Pitassi, and Ran Raz

SIAM J. Comput. 29, pp. 1939-1967 (29 pages) | Cited 7 times

Online Publication Date: July 27, 2006

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The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the proof system, sometimes assuming a (usually modest) complexity-theoretic assumption. In this paper, we show that this method cannot be used to obtain lower bounds for Frege systems, or even for TC0-Frege systems. More specifically, we show that unless factoring (of Blum integers) is feasible, neither Frege nor TC0-Frege has the feasible interpolation property. In order to carry out our argument, we show how to carry out proofs of many elementary axioms/theorems of arithmetic in polynomial-sized TC0-Frege.
As a corollary, we obtain that TC0-Frege, as well as any proof system that polynomially simulates it, is not automatizable (under the assumption that factoring of Blum integers is hard). We also show under the same hardness assumption that the k-provability problem for Frege systems is hard.

Space-Time Tradeoffs for Emptiness Queries

Jeff Erickson

SIAM J. Comput. 29, pp. 1968-1996 (29 pages) | Cited 2 times

Online Publication Date: July 27, 2006

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We develop the first nontrivial lower bounds on the complexity of online hyperplane and halfspace emptiness queries. Our lower bounds apply to a general class of geometric range query data structures called partition graphs. Informally, a partition graph is a directed acyclic graph that describes a recursive decomposition of space. We show that any partition graph that supports hyperplane emptiness queries implicitly defines a halfspace range query data structure in the Fredman/Yao semigroup arithmetic model, with the same asymptotic space and time bounds. Thus, results of Brönnimann, Chazelle, and Pach imply that any partition graph of size s that supports hyperplane emptiness queries in time t satisfies the inequality $st^d = \Omega((n/\log n)^{d - (d-1)/(d+1)})$. Using different techniques, we improve previous lower bounds for Hopcroft's problem---Given a set of points and hyperplanes, does any hyperplane contain a point?---in dimensions four and higher. Using this offline result, we show that for online hyperplane emptiness queries, $\Omega(n^d/{\mbox{ polylog }} n)$ space is required to achieve polylogarithmic query time, and $\Omega(n^{(d-1)/d}/{\mbox{ polylog }} n)$ query time is required if only O(n polylog n) space is available. These two lower bounds are optimal up to polylogarithmic factors. For two-dimensional queries, we obtain an optimal continuous tradeoff $st^2=\Omega(n^2)$ between these two extremes. Finally, using a lifting argument, we show that the same lower bounds hold for both offline and online halfspace emptiness queries in ${\mathbb{R}}^{d(d+3)/2}$.

Parallel Sorting with Limited Bandwidth

Micah Adler, John W. Byers, and Richard M. Karp

SIAM J. Comput. 29, pp. 1997-2015 (19 pages) | Cited 1 time

Online Publication Date: July 27, 2006

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We study the problem of sorting on a parallel computer with limited communication bandwidth. By using the PRAM(m) model, where p processors communicate through a globally shared memory which can service m requests per unit time, we focus on the trade-off between the amount of local computation and the amount of interprocessor communication required for parallel sorting algorithms. Our main result is a lower bound of $\Omega(\frac{n \log m}{m \log n})$ on the time required to sort n numbers on the exclusive-read and queued-read variants of the PRAM(m). We also show that Leighton's Columnsort can be used to give an asymptotically matching upper bound in the case where m grows as a fractional power of n. The bounds are of a surprising form in that they have little dependence on the parameter p. This implies that attempting to distribute the workload across more processors while holding the problem size and the size of the shared memory fixed will not improve the optimal running time of sorting in this model. We also show that both the lower and the upper bounds can be adapted to bridging models that address the issue of limited communication bandwidth: the LogP model and the bulk-synchronous parallel (BSP) model. The lower bounds provide further convincing evidence that efficient parallel algorithms for sorting rely strongly on high communication bandwidth.

Constructing Planar Cuttings in Theory and Practice

Sariel Har-Peled

SIAM J. Comput. 29, pp. 2016-2039 (24 pages) | Cited 3 times

Online Publication Date: July 27, 2006

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We present several variants of a new randomized incremental algorithm for computing a cutting in an arrangement of n lines in the plane. The algorithms produce cuttings whose expected size is O(r2), and the expected running time of the algorithms is O(nr). Both bounds are asymptotically optimal for nondegenerate arrangements. The algorithms are also simple to implement, and we present empirical results showing that they perform well in practice. We also present another efficient algorithm (with slightly worse time bound) that generates small cuttings whose size is guaranteed to be close to the best known upper bound of J. Matou{s}ek [Discrete Comput. Geom., 20 (1998), pp. 427--448].

On Quiescent Reliable Communication

Marcos Kawazoe Aguilera., Wei Chen, and Sam Toueg

SIAM J. Comput. 29, pp. 2040-2073 (34 pages) | Cited 6 times

Online Publication Date: July 27, 2006

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We study the problem of achieving reliable communication with quiescent algorithms (i.e., algorithms that eventually stop sending messages) in asynchronous systems with process crashes and lossy links. We first show that it is impossible to solve this problem in asynchronous systems (with no failure detectors). We then show that, among failure detectors that output lists of suspects, the weakest one that can be used to solve this problem is $\diamond \cal P,$ a failure detector that cannot be implemented. To overcome this difficulty, we introduce an implementable failure detector called Heartbeat and show that it can be used to achieve quiescent reliable communication. Heartbeat is novel: in contrast to typical failure detectors, it does not output lists of suspects and it is implementable without timeouts. With Heartbeat, many existing algorithms that tolerate only process crashes can be transformed into quiescent algorithms that tolerate both process crashes and message losses. This can be applied to consensus, atomic broadcast, k-set agreement, atomic commitment, etc.

Gadgets, Approximation, and Linear Programming

Luca Trevisan, Gregory B. Sorkin, Madhu Sudan, and David P. Williamson

SIAM J. Comput. 29, pp. 2074-2097 (24 pages) | Cited 12 times

Online Publication Date: July 27, 2006

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We present a linear programming-based method for finding "gadgets," i.e., combinatorial structures reducing constraints of one optimization problem to constraints of another. A key step in this method is a simple observation which limits the search space to a finite one. Using this new method we present a number of new, computer-constructed gadgets for several different reductions. This method also answers a question posed by Bellare, Goldreich, and Sudan [SIAM J. Comput., 27 (1998), pp. 804--915] of how to prove the optimality of gadgets: linear programming duality gives such proofs.
The new gadgets, when combined with recent results of Hå stad [ Proceedings of the 29th ACM Symposium on Theory of Computing, 1997, pp. 1--10], improve the known inapproximability results for MAX CUT and MAX DICUT, showing that approximating these problems to within factors of $16/17 + \epsilon$ and $12/13+ \epsilon,$ respectively, is NP-hard for every $\epsilon > 0$. Prior to this work, the best-known inapproximability thresholds for both problems were 71/72 (M. Bellare, O. Goldreich, and M. Sudan [ SIAM J. Comput., 27 (1998), pp. 804--915]). Without using the gadgets from this paper, the best possible hardness that would follow from Bellare, Goldreich, and Sudan and Hå{s}tad is $18/19$. We also use the gadgets to obtain an improved approximation algorithm for MAX3 SAT which guarantees an approximation ratio of .801. This improves upon the previous best bound (implicit from M. X. Goemans and D. P. Williamson [J. ACM, 42 (1995), pp. 1115--1145]; U. Feige and M. X. Goemans [Proceedings of the Third Israel Symposium on Theory of Computing and Systems, 1995, pp. 182--189]) of .7704.
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