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2005

Volume 34, Issue 6, pp. 1279-1528


SRT Division Algorithms as Dynamical Systems

Mark McCann and Nicholas Pippenger

SIAM J. Comput. 34, pp. 1279-1301 (23 pages)

Online Publication Date: July 27, 2006

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Sweeney--Robertson--Tocher (SRT) division, as it was discovered in the late 1950s, represented an important improvement in the speed of division algorithms for computers at the time. A variant of SRT division is still commonly implemented in computers today. Although some bounds on the performance of the original SRT division method were obtained, a great many questions remained unanswered. In this paper, the original version of SRT division is described as a dynamical system. This enables us to bring modern dynamical systems theory, a relatively new development in mathematics, to bear on an older problem. In doing so, we are able to show that SRT division is ergodic, and is even Bernoulli, for all real divisors and dividends. With the Bernoulli property, we are able to use entropy to prove that the natural extensions of SRT division are isomorphic by way of the Kolmogorov--Ornstein theorem. We demonstrate how our methods and results can be applied to a much larger class of division algorithms.

Polynomial-Time Approximation Schemes for Geometric Intersection Graphs

Thomas Erlebach, Klaus Jansen, and Eike Seidel

SIAM J. Comput. 34, pp. 1302-1323 (22 pages) | Cited 14 times

Online Publication Date: July 27, 2006

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A disk graph is the intersection graph of a set of disks with arbitrary diameters in the plane. For the case that the disk representation is given, we present polynomial-time approximation schemes (PTASs) for the maximum weight independent set problem (selecting disjoint disks of maximum total weight) and for the minimum weight vertex cover problem in disk graphs. These are the first known PTASs for $\mathcal{NP}$-hard optimization problems on disk graphs. They are based on a novel recursive subdivision of the plane that allows applying a shifting strategy on different levels simultaneously, so that a dynamic programming approach becomes feasible. The PTASs for disk graphs represent a common generalization of previous results for planar graphs and unit disk graphs. They can be extended to intersection graphs of other "disk-like" geometric objects (such as squares or regular polygons), also in higher dimensions.

Quantum Algorithms for Element Distinctness

Harry Buhrman, Christoph Dürr, Mark Heiligman, Peter Høyer, Frédéric Magniez, Miklos Santha, and Ronald de Wolf

SIAM J. Comput. 34, pp. 1324-1330 (7 pages) | Cited 8 times

Online Publication Date: July 27, 2006

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We present several applications of quantum amplitude amplification for deciding whether all elements in the image of a given function are distinct, for finding an intersection of two sorted tables, and for finding a triangle in a graph. Our techniques generalize and improve those of Brassard, Hoyer, and Tapp [ACM SIGACT News, 28 (1997), pp. 14--19]. This shows that in the quantum world element distinctness is significantly easier than sorting, in contrast to the classical world.

Output-Sensitive Construction of the Union of Triangles

Esther Ezra and Micha Sharir

SIAM J. Comput. 34, pp. 1331-1351 (21 pages)

Online Publication Date: July 27, 2006

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We present an efficient algorithm for the following problem: Given a collection $T =\{\Delta_1, \ldots, \Delta_n\}$ of $n$ triangles in the plane, such that there exists a subset $S \subset T$ (unknown to us) of $\xi \ll n$ triangles, such that $\bigcup_{\Delta \in S} \Delta = \bigcup_{\Delta \in T} \Delta$, construct efficiently the union of the triangles in $T$. We show that this problem can be solved in randomized expected time $O(n^{4/3}\log{n} + n\xi\log^2{n})$, which is subquadratic for $\xi=o(n/\log^2{n})$. In our solution, we use a variant of the method of Brönnimann and Goodrich [{\it Discrete Comput. Geom.}, 14 (1995), pp. 463--479] for finding a set cover in a set system of finite VC-dimension. We present a detailed implementation of this variant, which makes it run within the asserted time bound. Our approach is fairly general, and we show that it can be extended to compute efficiently the union of simply shaped bodies of constant description complexity in ${\reals}^d$, when the union is determined by a small subset of the bodies.

Extending Downward Collapse from 1-versus-2 Queries to m-versus-m + 1 Queries

Edith Hemaspaandra, Lane A. Hemaspaandra, and Harald Hempel

SIAM J. Comput. 34, pp. 1352-1369 (18 pages) | Cited 1 time

Online Publication Date: July 27, 2006

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The top part of Figure 1.1 shows some classes from the (truth-table) bounded-query and boolean hierarchies. It is well known that if either of these hierarchies collapses at a given level, then all higher levels of that hierarchy collapse to that same level. This is a standard "upward translation of equality" that has been known for over a decade. The issue of whether these hierarchies can translate equality downwards has proven vastly more challenging. In particular, with regard to Figure 1.1, consider the following claim:
\[ \psigkmtt = \psigkmponett \implies \diffmsigk = \codiffmsigk = \bh(\sigmak). (*) \]
Until recently, it was not known whether (*) ever held, except for the degenerate cases m = 0 and k = 0. Then Hemaspaandra, Hemaspaandra, and Hempel [SIAM J. Comput., 28 (1999), pp. 383--393] proved that (*) holds for all m, for k > 2. Buhrman and Fortnow [J. Comput. System Sci., 59 (1999), pp. 182--199] then showed that, when k = 2, (*) holds for the case m = 1. In this paper, we prove that for the case k = 2, (*) holds for all values of m. Since there is an oracle relative to which "for k = 1, (*) holds for all m" fails (see Buhrman and Fortnow), our achievement of the k = 2 case cannot be strengthened to k = 1 by any relativizable proof technique. The new downward translation we obtain also tightens the collapse in the polynomial hierarchy implied by a collapse in the bounded-query hierarchy of the second level of the polynomial hierarchy.

Approximating the Minimum Spanning Tree Weight in Sublinear Time

Bernard Chazelle, Ronitt Rubinfeld, and Luca Trevisan

SIAM J. Comput. 34, pp. 1370-1379 (10 pages) | Cited 5 times

Online Publication Date: July 27, 2006

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We present a probabilistic algorithm that, given a connected graph G (represented by adjacency lists) of average degree d, with edge weights in the set{1,...,w}, and given a parameter $0 < \eps < 1/2$, estimates in time $O( dw \varepsilon^{-2} \log{\frac{dw}\varepsilon})$ the weight of the minimum spanning tree (MST) of G with a relative error of at most $\eps$. Note that the running time does not depend on the number of vertices in G. We also prove a nearly matching lower bound of $\Omega( dw \varepsilon^{-2} )$ on the probe and time complexity of any approximation algorithm for MST weight.
The essential component of our algorithm is a procedure for estimating in time $O(d\eps^{-2}\log \frac{d}\varepsilon)$ the number of connected components of an unweighted graph to within an additive error of $\varepsilon n$. (This becomes $O(\eps^{-2}\log \frac{1}\varepsilon)$ for $d=O(1)$.) The time bound is shown to be tight up to within the $\log \frac{d}\varepsilon$ factor. Our connected-components algorithm picks $O(1/\varepsilon^2)$ vertices in the graph and then grows "local spanning trees" whose sizes are specified by a stochastic process. From the local information collected in this way, the algorithm is able to infer, with high confidence, an estimate of the number of connected components. We then show how estimates on the number of components in various subgraphs of G can be used to estimate the weight of its MST.

Binary Space Partitions of Orthogonal Subdivisions

John Hershberger, Subhash Suri, and Csaba D. Tóth

SIAM J. Comput. 34, pp. 1380-1397 (18 pages) | Cited 2 times

Online Publication Date: July 27, 2006

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We consider the problem of constructing binary space partitions (BSPs) for orthogonal subdivisions (space-filling packings of boxes) in $d$-space. We show that a subdivision with $n$ boxes can be refined into a BSP of size $O(n^{(d+1)/{3}})$ for all $d \geq 3$ and that such a partition can be computed in time ${O(K\log n)}$, where $K$ is the size of the BSP produced. Our upper bound on the BSP size is tight for $3$-dimensional subdivisions; in higher dimensions, this is the first nontrivial result for general full-dimensional boxes. We also present a lower bound construction for a subdivision of $n$ boxes in $d$-space for which every axis-aligned BSP has $\Omega(n^{\beta(d)})$ size, where $\beta(d)$ converges to $(1+\sqrt{5})/2$ as $d \rightarrow \infty$.

A Shortest Path Algorithm for Real-Weighted Undirected Graphs

Seth Pettie and Vijaya Ramachandran

SIAM J. Comput. 34, pp. 1398-1431 (34 pages) | Cited 6 times

Online Publication Date: July 27, 2006

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We present a new scheme for computing shortest paths on real-weighted undirected graphs in the fundamental comparison-addition model. In an efficient preprocessing phase our algorithm creates a linear-size structure that facilitates single-source shortest path computations in O(m log $\alpha$) time, where $\alpha$ = $\alpha$(m,n) is the very slowly growing inverse-Ackermann function, m the number of edges, and n the number of vertices. As special cases our algorithm implies new bounds on both the all-pairs and single-source shortest paths problems. We solve the all-pairs problem in O(mn log $\alpha$(m,n)) time and, if the ratio between the maximum and minimum edge lengths is bounded by n(log n)O(1), we can solve the single-source problem in O(m + n log log n) time. Both these results are theoretical improvements over Dijkstra's algorithm, which was the previous best for real weighted undirected graphs. Our algorithm takes the hierarchy-based approach invented by Thorup.

On the Bounded Sum-of-Digits Discrete Logarithm Problem in Finite Fields

Qi Cheng

SIAM J. Comput. 34, pp. 1432-1442 (11 pages) | Cited 1 time

Online Publication Date: July 27, 2006

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In this paper, we study the bounded sum-of-digits discrete logarithm problem in finite fields. Our results are concerned primarily with fields Fqn, where n|q - 1. The fields are called Kummer extensions of Fq. It is known that we can efficiently construct an element g with order exponential in n. Let $S_q(\bullet)$ be the function from integers to the sum of digits in their q-ary expansions. We first present an algorithm that, given ge (0 $\leq$ e < qn), finds e in random polynomial time, provided that Sq (e) < n. We then show that the problem is solvable in random polynomial time for most of the exponent e with Sq (e) < 1.32 n by exploring an interesting connection between the discrete logarithm problem and the problem of list decoding of Reed--Solomon codes and applying the Guruswami--Sudan algorithm. As far as we are aware, our algorithm is the first one which can solve discrete logarithms of $2^{\log^{1-\epsilon}{q^n}}$ many instances in polynomial time for infinite many constant characteristic fields Fqn. Furthermore, since every finite field has an extension of reasonable degree, which is a Kummer extension, our result revealsan unexpected property of the discrete logarithm problem, namely, the bounded sum-of-digits discrete logarithm problem in any given finite field becomes polynomial-time solvable in certain low degree extensions.
As a side result, we obtain a sharper lower bound on the number of congruent polynomials generated by linear factors than the one based on the Stothers--Mason ABC-theorem. We also prove that, in the field Fqq-1, the bounded sum-of-digits discrete logarithm with respect to g can be computed in random time O(f(w)log4 (qq-1)), where f is a subexponential function and w is the bound on the q-ary sum-of-digits of the exponent; hence the problem is fixed parameter tractable. These results are shown to be generalized to Artin--Schreier extension Fpp, where p is a prime.

Duality Between Prefetching and Queued Writing with Parallel Disks

David A. Hutchinson, Peter Sanders, and Jeffrey Scott Vitter

SIAM J. Comput. 34, pp. 1443-1463 (21 pages) | Cited 1 time

Online Publication Date: July 27, 2006

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Parallel disks promise to be a cost effective means for achieving high bandwidth in applications involving massive data sets, but algorithms for parallel disks can be difficult to devise. To combat this problem, we define a useful and natural duality between writing to parallel disks and the seemingly more difficult problem of prefetching. We first explore this duality for applications involving read-once accesses using parallel disks. We get a simple linear time algorithm for computing optimal prefetch schedules and analyze the efficiency of the resulting schedules for randomly placed data and for arbitrary interleaved accesses to striped sequences. Duality also provides an optimal schedule for prefetching plus caching, where blocks can be accessed multiple times. Another application of this duality gives us the first parallel disk sorting algorithms that are provably optimal up to lower-order terms. One of these algorithms is a simple and practical variant of multiway mergesort, addressing a question that had been open for some time.

Decidable and Undecidable Problems about Quantum Automata

Vincent D. Blondel, Emmanuel Jeandel, Pascal Koiran, and Natacha Portier

SIAM J. Comput. 34, pp. 1464-1473 (10 pages) | Cited 3 times

Online Publication Date: July 27, 2006

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We study the following decision problem: is the language recognized by a quantum finite automaton empty or nonempty? We prove that this problem is decidable or undecidable depending on whether recognition is defined by strict or nonstrict thresholds. This result is in contrast with the corresponding situation for probabilistic finite automata, for which it is known that strict andnonstrict thresholds both lead to undecidable problems.

Novel Transformation Techniques Using Q-Heaps with Applications to Computational Geometry

Qingmin Shi and Joseph JaJa

SIAM J. Comput. 34, pp. 1474-1492 (19 pages)

Online Publication Date: July 27, 2006

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Using the notions of Q-heaps and fusion trees developed by Fredman and Willard, we develop general transformation techniques to reduce a number of computational geometry problems to their special versions in partially ranked spaces. In particular, we develop a fast fractional cascading technique, which uses linear space and enables sublogarithmic iterative search on catalog trees in the case when the degree of each node is bounded by $O(\log^{\epsilon}n)$ for some constant $\epsilon >0$, where $n$ is the total size of allthe lists stored in the tree. We apply the fast fractional cascading technique in combination with the other techniques to derive the first linear-space sublogarithmic time algorithms for two fundamental geometric retrieval problems: orthogonal segment intersection and rectangular point enclosure.

Complexities for Generalized Models of Self-Assembly

Gagan Aggarwal, Qi Cheng, Michael H. Goldwasser, Ming-Yang Kao, Pablo Moisset de Espanes, and Robert T. Schweller

SIAM J. Comput. 34, pp. 1493-1515 (23 pages) | Cited 16 times

Online Publication Date: July 27, 2006

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In this paper, we study the complexity of self-assembly under models that are natural generalizations of the tile self-assembly model. In particular, we extend Rothemund and Winfree's study of the tile complexity of tile self-assembly [Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, Portland, OR, 2000, pp. 459--468]. They provided a lower bound of $\Omega(\frac{\log N}{\log\log N})$ on the tile complexity of assembling an $N\times N$ square for almost all N. Adleman et al. [Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, Heraklion, Greece, 2001, pp. 740--748] gave a construction which achieves this bound. We consider whether the tile complexity for self-assembly can be reduced through several natural generalizations of the model. One of our results is a tile set of size $O(\sqrt{\log N})$ which assembles an $N\times N$ square in a model which allows flexible glue strength between nonequal glues. This result is matched for almost all N by a lower bound dictated by Kolmogorov complexity. For three other generalizations, we show that the $\Omega(\frac{\log N}{\log\log N})$ lower bound applies to $N\times N$ squares. At the same time, we demonstrate that there are some other shapes for which these generalizations allow reduced tile sets. Specifically, for thin rectangles with length N and width k, we provide a tighter lower bound of $\Omega(\frac{N^{1/k}}{k})$ for the standard model, yet we also give a construction which achieves $O(\frac{\log N}{\log\log N})$ complexity in a model in which the temperature of the tile system is adjusted during assembly. We also investigate the problem of verifying whether a given tile system uniquely assembles into a given shape; we show that this problem is NP-hard for three of the generalized models.

Convergence Properties of the Gravitational Algorithm in Asynchronous Robot Systems

Reuven Cohen and David Peleg

SIAM J. Comput. 34, pp. 1516-1528 (13 pages) | Cited 5 times

Online Publication Date: July 27, 2006

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This paper considers the convergence problem in autonomous mobile robot systems. A natural algorithm for the problem requires the robots to move towards their center of gravity. This paper proves the correctness of the gravitational algorithm in the fully asynchronous model. It also analyzes its convergence rate and establishes its convergence in the presence of crash faults.
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