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2007

Volume 36, Issue 6, pp. 1513-1806


Zaps and Their Applications

Cynthia Dwork and Moni Naor

SIAM J. Comput. 36, pp. 1513-1543 (31 pages)

Online Publication Date: February 09, 2007

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A zap is a 2‐round, public coin witness‐indistinguishable protocol in which the first round, consisting of a message from the verifier to the prover, can be fixed “once and for all” and applied to any instance. We present a zap for every language in NP, based on the existence of noninteractive zero‐knowledge proofs in the shared random string model. The zap is in the standard model and hence requires no common guaranteed random string. We present several applications for zaps, including 3‐round concurrent zero‐knowledge and 2‐round concurrent deniable authentication, in the timing model of Dwork, Naor, and Sahai [J. ACM, 51 (2004), pp. 851–898], using moderately hard functions. We also characterize the existence of zaps in terms of a primitive called verifiable pseudorandom bit generators.

Complexity of Self‐Assembled Shapes

David Soloveichik and Erik Winfree

SIAM J. Comput. 36, pp. 1544-1569 (26 pages) | Cited 24 times

Online Publication Date: February 09, 2007

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The connection between self‐assembly and computation suggests that a shape can be considered the output of a self‐assembly “program,” a set of tiles that fit together to create a shape. It seems plausible that the size of the smallest self‐assembly program that builds a shape and the shape’s descriptional (Kolmogorov) complexity should be related. We show that when using a notion of a shape that is independent of scale, this is indeed so: in the tile assembly model, the minimal number of distinct tile types necessary to self‐assemble a shape, at some scale, can be bounded both above and below in terms of the shape’s Kolmogorov complexity. As part of the proof, we develop a universal constructor for this model of self‐assembly that can execute an arbitrary Turing machine program specifying how to grow a shape. Our result implies, somewhat counterintuitively, that self‐assembly of a scaled‐up version of a shape often requires fewer tile types. Furthermore, the independence of scale in self‐assembly theory appears to play the same crucial role as the independence of running time in the theory of computability. This leads to an elegant formulation of languages of shapes generated by self‐assembly. Considering functions from bit strings to shapes, we show that the running‐time complexity, with respect to Turing machines, is polynomially equivalent to the scale complexity of the same function implemented via self‐assembly by a finite set of tile types. Our results also hold for shapes defined by Wang tiling—where there is no sense of a self‐assembly process—except that here time complexity must be measured with respect to nondeterministic Turing machines.

First‐Order Languages Expressing Constructible Spatial Database Queries

Bart Kuijpers, Gabriel Kuper, Jan Paredaens, and Luc Vandeurzen

SIAM J. Comput. 36, pp. 1570-1599 (30 pages)

Online Publication Date: February 20, 2007

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The research presented in this paper is situated in the framework of constraint databases introduced by Kanellakis, Kuper, and Revesz in their seminal paper of 1990, specifically, the language with real polynomial constraints (FO+poly). For reasons of efficiency, this model is implemented with only linear polynomial constraints, but this limitation to linear polynomial constraints has severe implications on the expressive power of the query language. In particular, when used for modeling spatial data, important queries that involve Euclidean distance are not expressible. The aim of this paper is to identify a class of two‐dimensional constraint databases and a query language within the constraint model that go beyond the linear model and allow the expression of queries concerning distance. We seek inspiration in the Euclidean constructions, i.e., constructions by ruler and compass. We first present a programming language that captures exactly the first‐order ruler‐and‐compass constructions that are expressible in a first‐order language with real polynomial constraints. If this language is extended with a while operator, we obtain a language that is complete for all ruler‐and‐compass constructions in the plane. We then transform this language in a natural way into a query language on finite point databases, but this language turns out to have the same expressive power as FO+poly and is therefore too powerful for our purposes. We then consider a safe fragment of this language and use this to construct a query language that allows the expression of Euclidean distance without having the full power of FO+poly.

Quickest Flows Over Time

Lisa Fleischer and Martin Skutella

SIAM J. Comput. 36, pp. 1600-1630 (31 pages) | Cited 3 times

Online Publication Date: February 20, 2007

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Flows over time (also called dynamic flows) generalize standard network flows by introducing an element of time. They naturally model problems where travel and transmission are not instantaneous. Traditionally, flows over time are solved in time‐expanded networks that contain one copy of the original network for each discrete time step. While this method makes available the whole algorithmic toolbox developed for static flows, its main and often fatal drawback is the enormous size of the time‐expanded network. We present several approaches for coping with this difficulty. First, inspired by the work of Ford and Fulkerson on maximal $s$‐$t$‐flows over time (or “maximal dynamic $s$‐$t$‐flows”), we show that static length‐bounded flows lead to provably good multicommodity flows over time. Second, we investigate “condensed” time‐expanded networks which rely on a rougher discretization of time. We prove that a solution of arbitrary precision can be computed in polynomial time through an appropriate discretization leading to a condensed time‐expanded network of polynomial size. In particular, our approach yields fully polynomial‐time approximation schemes for the NP‐hard quickest min‐cost and multicommodity flow problems. For single commodity problems, we show that storage of flow at intermediate nodes is unnecessary, and our approximation schemes do not use any.

A Constant‐Factor Approximation Algorithm for Optimal 1.5D Terrain Guarding

Boaz Ben‐Moshe, Matthew J. Katz, and Joseph S. B. Mitchell

SIAM J. Comput. 36, pp. 1631-1647 (17 pages) | Cited 5 times

Online Publication Date: February 26, 2007

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We present the first constant‐factor approximation algorithm for a nontrivial instance of the optimal guarding (coverage) problem in polygons. In particular, we give an $O(1)$‐approximation algorithm for placing the fewest point guards on a 1.5D terrain, so that every point of the terrain is seen by at least one guard. While polylogarithmic‐factor approximations follow from set cover results, our new results exploit the geometric structure of terrains to obtain a substantially improved approximation algorithm.

Finding Paths and Cycles of Superpolylogarithmic Length

Harold N. Gabow

SIAM J. Comput. 36, pp. 1648-1671 (24 pages) | Cited 2 times

Online Publication Date: March 02, 2007

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Let $\ell$ be the number of edges in a longest cycle containing a given vertex $v$ in an undirected graph. We show how to find a cycle through $v$ of length $\exp(\Omega(\sqrt {\log \ell/\log\log \ell}))$ in polynomial time. This implies the same bound for the longest cycle, longest $vw$‐path, and longest path. The previous best bound for longest path is length $\Omega( (\log \ell )^2/\, \log\log \ell)$ due to Björklund and Husfeldt. Our approach, which builds on Björklund and Husfeldt’s, uses cycles to enlarge cycles. This self‐reducibility allows the approximation method to be iterated.

An Optimal Cache‐Oblivious Priority Queue and Its Application to Graph Algorithms

Lars Arge, Michael A. Bender, Erik D. Demaine, Bryan Holland‐Minkley, and J. Ian Munro

SIAM J. Comput. 36, pp. 1672-1695 (24 pages) | Cited 1 time

Online Publication Date: March 19, 2007

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We develop an optimal cache‐oblivious priority queue data structure, supporting insertion, deletion, and delete‐min operations in $O(\frac{1}{B}\log_{M/B}\frac{N}{B})$ amortized memory transfers, where $M$ and $B$ are the memory and block transfer sizes of any two consecutive levels of a multilevel memory hierarchy. In a cache‐oblivious data structure, $M$ and $B$ are not used in the description of the structure. Our structure is as efficient as several previously developed external memory (cache‐aware) priority queue data structures, which all rely crucially on knowledge about $M$ and $B$. Priority queues are a critical component in many of the best known external memory graph algorithms, and using our cache‐oblivious priority queue we develop several cache‐oblivious graph algorithms.
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Online Learning and Resource‐Bounded Dimension: Winnow Yields New Lower Bounds for Hard Sets

John M. Hitchcock

SIAM J. Comput. 36, pp. 1696-1708 (13 pages) | Cited 1 time

Online Publication Date: March 19, 2007

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We establish a relationship between the online mistake‐bound model of learning and resource‐bounded dimension. This connection is combined with the Winnow algorithm to obtain new results about the density of hard sets under adaptive reductions. This improves previous work of Fu [SIAM J. Comput., 24 (1995), pp. 1082–1090] and Lutz and Zhao [SIAM J. Comput., 30 (2000), pp. 1197–1210], and solves one of Lutz and Mayordomo’s “twelve problems in resource‐bounded measure” [Bull. Eur. Assoc. Theor. Comput. Sci. EATSC, 68 (1999), pp. 64–80].

Online Scheduling of Equal‐Length Jobs: Randomization and Restarts Help

Marek Chrobak, Wojciech Jawor, Jiří Sgall, and Tomáš Tichý

SIAM J. Comput. 36, pp. 1709-1728 (20 pages) | Cited 3 times

Online Publication Date: March 19, 2007

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We consider the following scheduling problem. The input is a set of jobs with equal processing times, where each job is specified by its release time and deadline. The goal is to determine a single‐processor nonpreemptive schedule that maximizes the number of completed jobs. In the online version, each job arrives at its release time. We give two online algorithms with competitive ratios below $2$ and show several lower bounds on the competitive ratios. First, we give a barely random $5/3$‐competitive algorithm that uses only one random bit. We also show a lower bound of $3/2$ on the competitive ratio of barely random algorithms that randomly choose one of two deterministic algorithms. If the two algorithms are selected with equal probability, we can further improve the bound to $8/5$. Second, we give a deterministic $3/2$‐competitive algorithm in the model that allows restarts, and we show that in this model the ratio $3/2$ is optimal. For randomized algorithms with restarts we show a lower bound of $6/5$.
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Physical Limits of Heat‐Bath Algorithmic Cooling

Leonard J. Schulman, Tal Mor, and Yossi Weinstein

SIAM J. Comput. 36, pp. 1729-1747 (19 pages) | Cited 5 times

Online Publication Date: March 19, 2007

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Simultaneous near‐certain preparation of qubits (quantum bits) in their ground states is a key hurdle in quantum computing proposals as varied as liquid‐state NMR and ion traps. “Closed‐system” cooling mechanisms are of limited applicability due to the need for a continual supply of ancillas for fault tolerance and to the high initial temperatures of some systems. “Open‐system” mechanisms are therefore required. We describe a new, efficient initialization procedure for such open systems. With this procedure, an $n$‐qubit device that is originally maximally mixed, but is in contact with a heat bath of bias $\varepsilon \gg 2^{-n}$, can be almost perfectly initialized. This performance is optimal due to a newly discovered threshold effect: For bias $\varepsilon \ll 2^{-n}$ no cooling procedure can, even in principle (running indefinitely without any decoherence), significantly initialize even a single qubit.

Whole Genome Duplications and Contracted Breakpoint Graphs

Max A. Alekseyev and Pavel A. Pevzner

SIAM J. Comput. 36, pp. 1748-1763 (16 pages) | Cited 4 times

Online Publication Date: March 19, 2007

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The genome halving problem, motivated by the whole genome duplication events in molecular evolution, was solved by El‐Mabrouk and Sankoff in the pioneering paper [SIAM J. Comput., 32 (2003), pp. 754–792]. The El‐Mabrouk–Sankoff algorithm is rather complex, inspiring a quest for a simpler solution. An alternative approach to the genome halving problem based on the notion of the contracted breakpoint graph was recently proposed in [M. A. Alekseyev and P. A. Pevzner, IEEE/ACM Trans. Comput. Biol. Bioinformatics, 4 (2007), pp. 98–107]. This new technique reveals that while the El‐Mabrouk–Sankoff result is correct in most cases, it does not hold in the case of unichromosomal genomes. This raises a problem of correcting a flaw in the El‐Mabrouk–Sankoff analysis and devising an algorithm that deals adequately with all genomes. In this paper we efficiently classify all genomes into two classes and show that while the El‐Mabrouk–Sankoff theorem holds for the first class, it is incorrect for the second class. The crux of our analysis is a new combinatorial invariant defined on duplicated permutations. Using this invariant we were able to come up with a full proof of the genome halving theorem and a polynomial algorithm for the genome halving problem.

Approximating the Radii of Point Sets

Kasturi Varadarajan, S. Venkatesh, Yinyu Ye, and Jiawei Zhang

SIAM J. Comput. 36, pp. 1764-1776 (13 pages) | Cited 3 times

Online Publication Date: March 19, 2007

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We consider the problem of computing the outer‐radii of point sets. In this problem, we are given integers $n, d$, and $k$, where $k \le d$, and a set $P$ of $n$ points in $\Re^d$. The goal is to compute the outer $k$‐radius of $P$, denoted by ${\cal R}_k(P)$, which is the minimum over all $(d-k)$‐dimensional flats $F$ of $\max_{p \in P} d(p,F)$, where $d(p,F)$ is the Euclidean distance between the point $p$ and flat $F$. Computing the radii of point sets is a fundamental problem in computational convexity with many significant applications. The problem admits a polynomial time algorithm when the dimension $d$ is constant [U. Faigle, W. Kern, and M. Streng, Math. Program., 73 (1996), pp. 1–5]. Here we are interested in the general case in which the dimension $d$ is not fixed and can be as large as $n$, where the problem becomes NP‐hard even for $k=1$. It is known that $R_k(P)$ can be approximated in polynomial time by a factor of $(1 + \varepsilon)$ for any $\varepsilon > 0$ when $d - k$ is a fixed constant [M. Bădoiu, S. Har‐Peled, and P. Indyk, in Proceedings of the ACM Symposium on the Theory of Computing, 2002; S. Har‐Peled and K. Varadarajan, in Proceedings of the ACM Symposium on Computing Geometry, 2002]. A polynomial time algorithm that guarantees a factor of $O(\sqrt{\log n})$ approximation for $R_1(P)$, the width of the point set $P$, is implied by the results of Nemirovski, Roos, and Terlaky [Math. Program., 86 (1999), pp. 463–473] and Nesterov [Handbook of Semidefinite Programming Theory, Algorithms, Kluwer Academic Publishers, Norwell, MA, 2000]. In this paper, we show that $R_k(P)$ can be approximated by a ratio of $O(\sqrt{\log n})$ for any $1 \leq k \leq d$, thus matching the previously best known ratio for approximating the special case $R_1 (P)$, the width of point set $P$. Our algorithm is based on semidefinite programming relaxation with a new mixed deterministic and randomized rounding procedure. We also prove an inapproximability result that gives evidence that our approximation algorithm is doing well for a large range of $k$. We show that there exists a constant $\delta > 0$ such that the following holds for any $0 < \eps < 1$: there is no polynomial time algorithm that approximates $R_k(P)$ within $(\log n)^{\delta}$ for all $k$ such that $k \leq d - d^{\varepsilon}$ unless NP $\subseteq$ DTIME $[2^{(\log m)^{O(1)}}]$. Our inapproximability result for $R_k(P)$ extends a previously known hardness result of Brieden [Discrete Comput. Geom., 28 (2002), pp. 201–209] and is proved by modifying Brieden’s construction using basic ideas from probabilistically checkable proofs (PCP) theory.

Linear Recurrences with Polynomial Coefficients and Application to Integer Factorization and Cartier–Manin Operator

Alin Bostan, Pierrick Gaudry, and Éric Schost

SIAM J. Comput. 36, pp. 1777-1806 (30 pages) | Cited 2 times

Online Publication Date: March 22, 2007

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We study the complexity of computing one or several terms (not necessarily consecutive) in a recurrence with polynomial coefficients. As applications, we improve the best currently known upper bounds for factoring integers deterministically and for computing the Cartier–Manin operator of hyperelliptic curves.
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