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2003

Volume 32, Issue 6, pp. 1395-1673


Finding a Path of Superlogarithmic Length

Andreas Björklund and Thore Husfeldt

SIAM J. Comput. 32, pp. 1395-1402 (8 pages) | Cited 4 times

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We consider the problem of finding a long, simple path in an undirected graph. We present a polynomial-time algorithm that finds a path of length $\Omega\bigl((\log L/\log\log L)^2\bigr)$, where L denotes the length of the longest simple path in the graph. This establishes the performance ratio O(n(log log n/log n)2) for the longest path problem, where n denotes the number of vertices in the graph.

Better Algorithms for Unfair Metrical Task Systems and Applications

Amos Fiat and Manor Mendel

SIAM J. Comput. 32, pp. 1403-1422 (20 pages) | Cited 2 times

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Unfair metrical task systems are a generalization of online metrical task systems. In this paper we introduce new techniques to combine algorithms for unfair metrical task systems and apply these techniques to obtain improved randomized online algorithms for metrical task systems on arbitrary metric spaces.

Matrix Rounding under the Lp-Discrepancy Measure and Its Application to Digital Halftoning

Tetsuo Asano, Naoki Katoh, Koji Obokata, and Takeshi Tokuyama

SIAM J. Comput. 32, pp. 1423-1435 (13 pages) | Cited 2 times

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We study the problem of rounding a real-valued matrix into an integer-valued matrix to minimize an Lp-discrepancy measure between them. To define the Lp-discrepancy measure, we introduce a family ${\cal F}$ of regions (rigid submatrices) of the matrix and consider a hypergraph defined by the family. The difficulty of the problem depends on the choice of the region family ${\cal F}$. We first investigate the rounding problem by using integer programming problems with convex piecewise-linear objective functions and give some nontrivial upper bounds for the Lp discrepancy. We propose "laminar family" for constructing a practical and well-solvable class of ${\cal F}$. Indeed, we show that the problem is solvable in polynomial time if ${\cal F}$ is the union of two laminar families. Finally, we show that the matrix rounding using L1 discrepancy for the union of two laminar families is suitable for developing a high-quality digital-halftoning software.

Finding Points on Curves over Finite Fields

Joachim von zur Gathen, Igor Shparlinski, and Alistair Sinclair

SIAM J. Comput. 32, pp. 1436-1448 (13 pages)

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We solve two computational problems concerning plane algebraic curves over finite fields: generating a uniformly random point, and finding all points deterministically in amortized polynomial time (over a prime field, for nonexceptional curves).

On Frictional Mechanical Systems and Their Computational Power

John H. Reif and Zheng Sun

SIAM J. Comput. 32, pp. 1449-1474 (26 pages) | Cited 1 time

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In this paper we define a class of mechanical systems consisting of rigid objects (defined by linear or quadratic surface patches) connected by frictional contact linkages between surfaces. (This class of mechanisms is similar to the analytical engine developed by Babbage in the 1800s, except that we assume frictional surfaces instead of toothed gears.) We prove that a universal Turing Machine (TM) can be simulated by a (universal) frictional mechanical system in this class consisting of a constant number of parts. Our universal frictional mechanical system has the property that it can reach a distinguished final configuration through a sequence of legal movements if and only if the universal TM accepts the input string encoded by its initial configuration. There are two implications from this result. First, the robotic mover's problem is undecidable when there are frictional linkages. Second, a mechanical computer can be constructed that has the computational power of any conventional electronic computer and yet has only a constant number of mechanical parts.
Previous constructions for mechanical computing devices (such as Babbage's analytical engine) either provided no general construction for finite state control or the control was provided by electronic devices (as was common in electromechanical computers such as Mark I subsequent to Turing's result). Our result seems to be the first to provide a general proof of the simulation of a universal TM via a purely mechanical mechanism.
In addition, we discuss the universal frictional mechanical system in the context of an error model that allows an error up to $\epsilon$ in each mechanical operation. We first show that, for a universal TM M, a frictional mechanical system in this $\epsilon$-error model can be constructed such that, given any space bound S, the system can simulate the computation of M on any input string $\omega$ if M decides $\omega$ in space bound S, provided that $\epsilon < 2^{-cS}$ for some constant c. We also show that, for any universal TM M and space bound S, there exists a frictional mechanical system in the $\epsilon$-error model with $\epsilon = \Omega(1)$; it has O(S) parts and can simulate M on any input $\omega$ that M decides in space bound S.

Computing Elementary Symmetric Polynomials with a Subpolynomial Numberof Multiplications

Vince Grolmusz

SIAM J. Comput. 32, pp. 1475-1487 (13 pages) | Cited 1 time

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Elementary symmetric polynomials $S_n^k$ are the building blocks of symmetric polynomials. In this work we prove that for constant $k$'s, $S_n^k$ modulo composite numbers m=p1p2 can be computed using only no(1) multiplications if the coefficients of monomials xi1xi2. . . xik are allowed to be 1 either mod p1 or mod p2 but not necessarily both. To the best of our knowledge, no previous result yielded even a sublinear (i.e., $n^{\varepsilon}$, $0 < \varepsilon < 1$) number of multiplications for similar tasks. Moreover, our algorithm fits in the model of the most restrictive depth-3 arithmetic circuits (homogeneous, multilinear, or the graph model). In contrast, by a lower bound of Nisan and Wigderson [Comput. Complexity, 6 (1997), pp. 217--234], any homogeneous depth-3 circuit needs size $\Omega((n/2k)^{k/2})$ for computing $S_n^k$ modulo primes. Moreover, the number of multiplications in our algorithm remains sublinear while k=O(log log n). Our results generalize for other nonprime-power composite moduli as well. The proof uses perfect hashing functions and the famous BBR polynomial of Barrington, Beigel, and Rudich.

Optimal External Memory Interval Management

Lars Arge and Jeffrey Scott Vitter

SIAM J. Comput. 32, pp. 1488-1508 (21 pages) | Cited 6 times

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In this paper we present the external interval tree, an optimal external memory data structure for answering stabbing queries on a set of dynamically maintained intervals. The external interval tree can be used in an optimal solution to the dynamic interval management problem, which is a central problem for object-oriented and temporal databases and for constraint logic programming. Part of the structure uses a weight-balancing technique for efficient worst-case manipulation of balanced trees, which is of independent interest. The external interval tree, as well as our new balancing technique, have recently been used to develop several efficient external data structures.

Covering Rectilinear Polygons with Axis-Parallel Rectangles

V. S. Anil Kumar and H. Ramesh

SIAM J. Comput. 32, pp. 1509-1541 (33 pages) | Cited 3 times

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We give an $O(\sqrt{\log n})$ factor approximation algorithm for covering a rectilinear polygon with holes using axis-parallel rectangles. This is the first polynomial time approximation algorithm for this problem with an $o(\log n)$ approximation factor.

On the Autoreducibility of Random Sequences

Todd Ebert, Wolfgang Merkle, and Heribert Vollmer

SIAM J. Comput. 32, pp. 1542-1569 (28 pages) | Cited 3 times

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A binary sequence $A=A(0)A(1)\ldots$ is called infinitely often (i.o.)~Turing-au\-to\-re\-duc\-ible if $A$~is reducible to itself via an oracle Turing machine that never queries its oracle at the current input, outputs either $A(x)$ or a don't-know symbol on any given input~$x$, and outputs $A(x)$ for infinitely many~$x$. If in addition the oracle Turing machine terminates on all inputs and oracles, $A$~is called i.o.~truth-table-autoreducible. We obtain the somewhat counterintuitive result that every Martin-L\"of random sequence, in fact even every rec-random or p-random sequence, is i.o.~truth-table-autoreducible. Furthermore, we investigate the question of how dense the set of guessed bits can be when i.o.~autoreducing a random sequence. We show that rec-random sequences are never i.o.~truth-table-autoreducible such that the set of guessed bits has positive constant density in the limit and that a similar assertion holds for Martin-L\"of random sequences and i.o.~Turing autoreducibility. On the other hand, we show that for any rational-valued computable function~$r$ that goes nonascendingly to zero, any rec-random sequence is i.o.~truth-table-autoreducible such that on any prefix of length~$m$ at least a fraction of~$r(m)$ of the $m$~bits in the prefix are guessed. We include a self-contained account of the hat problem, a puzzle that has received some attention outside of theoretical computer science. The hat problem asks for guessing bits of a finite sequence, thus illustrating the notion of i.o.~autoreducibility in a finite setting. The solution to the hat problem is then used as a module in the proofs of the positive results on i.o.~autoreducibility.

The Quantum Communication Complexity of Sampling

Andris Ambainis, Leonard J. Schulman, Amnon Ta-Shma, Umesh Vazirani, and Avi Wigderson

SIAM J. Comput. 32, pp. 1570-1585 (16 pages) | Cited 5 times

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Sampling is an important primitive in probabilistic and quantum algorithms. In the spirit of communication complexity, given a function $f: X \times Y \rightarrow \{0,1\}$ and a probability distribution ${\cal D}$ over $X \times Y$, we define the sampling complexity of $(f, {\cal D})$ as the minimum number of bits that Alice and Bob must communicate for Alice to pick $x \in X$ and Bob to pick $y \in Y$ as well as a value $z$ such that the resulting distribution of $(x,y,z)$ is close to the distribution $({\cal D}, f({\cal D}))$.
In this paper we initiate the study of sampling complexity, in both the classical and quantum models. We give several variants of a definition. We completely characterize some of these variants and give upper and lower bounds on others. In particular, this allows us to establish an exponential gap between quantum and classical sampling complexity for the set-disjointness function.

The Expected Number of 3D Visibility Events Is Linear

Olivier Devillers, Vida Dujmovic, Hazel Everett, Xavier Goaoc, Sylvain Lazard, Hyeon-Suk Na, and Sylvain Petitjean

SIAM J. Comput. 32, pp. 1586-1620 (35 pages) | Cited 3 times

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In this paper, we show that, amongst $n$ uniformly distributed unit balls in $\mathbb{R}^3$, the expected number of maximal nonoccluded line segments tangent to four balls is linear. Using our techniques we show a linear bound on the expected size of the visibility complex, a data structure encoding the visibility information of a scene, providing evidence that the storage requirement for this data structure is not necessarily prohibitive. These results significantly improve the best previously known bounds of $O(n^{8/3})$ [F. Durand, G. Drettakis, and C. Puech, {ACM Transactions on Graphics}, 21 (2002), pp. 176-206].
Our results generalize in various directions. We show that the linear bound on the expected number of maximal nonoccluded line segments that are not too close to the boundary of the scene and tangent to four unit balls extends to balls of various but bounded radii, to polyhedra of bounded aspect ratio, and even to nonfat three-dimensional objects such as polygons of bounded aspect ratio. We also prove that our results extend to other distributions such as the Poisson distribution. Finally, we indicate how our probabilistic analysis provides new insight on the expected size of other global visibility data structures, notably the aspect graph.

Pseudotriangulations from Surfaces and a Novel Type of Edge Flip

Oswin Aichholzer, Franz Aurenhammer, Hannes Krasser, and Peter Brass

SIAM J. Comput. 32, pp. 1621-1653 (33 pages) | Cited 4 times

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We prove that planar pseudotriangulations have realizations as polyhedral surfaces in three-space. Two main implications are presented. The spatial embedding leads to a novel flip operation that allows for a drastic reduction of flip distances, especially between (full) triangulations. Moreover, several key results for triangulations, like flipping to optimality, (constrained) Delaunayhood, and a convex polytope representation, are extended to pseudotriangulations in a natural way.

A Subquadratic Sequence Alignment Algorithm for Unrestricted Scoring Matrices

Maxime Crochemore, Gad M. Landau, and Michal Ziv-Ukelson

SIAM J. Comput. 32, pp. 1654-1673 (20 pages) | Cited 13 times

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Given two strings of size $n$ over a constant alphabet, the classical algorithm for computing the similarity between two sequences [D. Sankoff and J. B. Kruskal, eds., {Time Warps, String Edits, and Macromolecules}; Addison-Wesley, Reading, MA, 1983; T. F. Smith and M. S. Waterman, { J.\ Molec.\ Biol., 147 (1981), pp. 195-197] uses a dynamic programming matrix and compares the two strings in O(n2) time. We address the challenge of computing the similarity of two strings in subquadratic time for metrics which use a scoring matrix of unrestricted weights. Our algorithm applies to both {local} and {global} similarity computations. The speed-up is achieved by dividing the dynamic programming matrix into variable sized blocks, as induced by Lempel-Ziv parsing of both strings, and utilizing the inherent periodic nature of both strings. This leads to an $O(n^2 / \log n)$, algorithm for an input of constant alphabet size. For most texts, the time complexity is actually $O(h n^2 / \log n)$, where $h \le 1$ is the entropy of the text. We also present an algorithm for comparing two {run-length} encoded strings of length m and n, compressed into m' and n' runs, respectively, in O(m'n + n'm) complexity. This result extends to all distance or similarity scoring schemes that use an additive gap penalty.
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