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2006

Volume 35, Issue 6, pp. 1283-1525


An Algorithm for Computing Fundamental Solutions

Klaus Weihrauch and Ning Zhong

SIAM J. Comput. 35, pp. 1283-1294 (12 pages)

Online Publication Date: July 27, 2006

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For a partial differential operator $P=\sum _{|\alpha |\leq m}c_{\alpha }D^{\alpha }$ with constant coefficients, a generalized function $u$ is a fundamental solution if $Pu=\delta$, where $\delta$ is the Dirac distribution. In this article, we provide an algorithm which computes a fundamental solution for every such differential operator $P$ on a Turing machine if the input- and output-data are represented canonically.

Compact Labeling Scheme for Ancestor Queries

Serge Abiteboul, Stephen Alstrup, Haim Kaplan, Tova Milo, and Theis Rauhe

SIAM J. Comput. 35, pp. 1295-1309 (15 pages) | Cited 1 time

Online Publication Date: July 27, 2006

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We consider the following problem. Given a rooted tree $T$, label the nodes of $T$ in the most compact way such that, given the labels of two nodes $u$ and $v$, one can determine in constant time, by looking only at the labels, whether $u$ is ancestor of $v$. The best known labeling scheme is rather straightforward and uses labels of length at most $2\log_2 n$ bits each, where $n$ is the number of nodes in the tree. Our main result in this paper is a labeling scheme with maximum label length $\log_2 n + \Oh(\sqrt{\log n})$. Our motivation for studying this problem is enhancing the performance of web search engines. In the context of this application each indexed document is a tree, and the labels of all trees are maintained in main memory. Therefore even small improvements in the maximum label length are important.

Quantum Query Complexity of Some Graph Problems

Christoph Dürr, Mark Heiligman, Peter HOyer, and Mehdi Mhalla

SIAM J. Comput. 35, pp. 1310-1328 (19 pages) | Cited 1 time

Online Publication Date: July 27, 2006

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Quantum algorithms for graph problems are considered, both in the adjacency matrix model and in an adjacency list-like array model. We give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree, and Single Source Shortest Paths. For example, we show that the query complexity of Minimum Spanning Tree is in $\Theta(n^{3/2})$ in the matrix model and in $\Theta(\sqrt{nm})$ in the array model, while the complexity of Connectivity is also in $\Theta(n^{3/2})$ in the matrix model but in $\Theta(n)$ in the array model. The upper bounds utilize search procedures for finding minima of functions under various conditions.

The Approximability of Three-valued MAX CSP

Peter Jonsson, Mikael Klasson, and Andrei Krokhin

SIAM J. Comput. 35, pp. 1329-1349 (21 pages) | Cited 2 times

Online Publication Date: July 27, 2006

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In the maximum constraint satisfaction problem (MAX CSP), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given domain to the variables so as to maximize the number (or the total weight, for the weighted case) of satisfied constraints. This problem is NP-hard in general, and, therefore, it is natural to study how restricting the allowed types of constraints affects the approximability of the problem. It is known that every Boolean (that is, two-valued) MAX CSP with a finite set of allowed constraint types is either solvable exactly in polynomial time or else APX-complete (and hence can have no polynomial-time approximation scheme unless P=NP). It has been an open problem for several years whether this result can be extended to non-Boolean MAX CSP, which is much more difficult to analyze than the Boolean case. In this paper, we make the first step in this direction by establishing this result for MAX CSP over a three-element domain. Moreover, we present a simple description of all polynomial-time solvable cases of our problem. This description uses the well-known algebraic combinatorial property of supermodularity. We also show that every hard three-valued MAX CSP contains, in a certain specified sense, one of the two basic hard MAX CSPs which are the Maximum k-Colorable Subgraph problems for k=2,3.

Balanced Allocations: The Heavily Loaded Case

Petra Berenbrink, Artur Czumaj, Angelika Steger, and Berthold Vöcking

SIAM J. Comput. 35, pp. 1350-1385 (36 pages)

Online Publication Date: July 27, 2006

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We investigate balls-into-bins processes allocating $m$ balls into $n$ bins based on the multiple-choice paradigm. In the classical single-choice variant each ball is placed into a bin selected uniformly at random. In a multiple-choice process each ball can be placed into one out of $d \ge 2$ randomly selected bins. It is known that in many scenarios having more than one choice for each ball can improve the load balance significantly. Formal analyses of this phenomenon prior to this work considered mostly the lightly loaded case, that is, when $m \approx n$. In this paper we present the first tight analysis in the heavily loaded case, that is, when $m \gg n$ rather than $m \approx n$.
The best previously known results for the multiple-choice processes in the heavily loaded case were obtained using majorization by the single-choice process. This yields an upper bound of the maximum load of bins of $m/n + {\mbox{$\cal O$}}(\sqrt{m \ln n \,/\, n})$ with high probability. We show, however, that the multiple-choice processes are fundamentally different from the single-choice variant in that they have "short memory." The great consequence of this property is that the deviation of the multiple-choice processes from the optimal allocation (that is, the allocation in which each bin has either $\lfloor m/n \rfloor$ or $\lceil m/n \rceil$ balls) does not increase with the number of balls as in the case of the single-choice process. In particular, we investigate the allocation obtained by two different multiple-choice allocation schemes, the greedy scheme due to Azar et al. and the always-go-left scheme due to Vöcking. We show that these schemes result in a maximum load of only $m/n + {\mbox{$\cal O$}}(\ln \ln n)$ with high probability. All our detailed bounds on the maximum load are tight up to an additive constant.
Furthermore, we investigate the two multiple-choice algorithms in a comparative study. We present a majorization result showing that the always-go-left scheme obtains a better load balancing than the greedy scheme for any choice of $n$, $m$, and $d$.

Linearization and Completeness Results for Terminating Transitive Closure Queries on Spatial Databases

Floris Geerts, Bart Kuijpers, and Jan Van den Bussche

SIAM J. Comput. 35, pp. 1386-1439 (54 pages)

Online Publication Date: July 27, 2006

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We study queries to spatial databases, where spatial data are modeled as semi-algebraic sets, using the relational calculus with polynomial inequalities as a basic query language. We work with the extension of the relational calculus with terminating transitive closures. The main result is that this language can express the linearization of semialgebraic databases. We also show that the sublanguage with linear inequalities only can express all computable queries on semilinear databases. As a consequence of these results, we obtain a completeness result for topological queries on semialgebraic databases.

Partial Match Queries in Random k-d Trees

Hua-Huai Chern and Hsien-Kuei Hwang

SIAM J. Comput. 35, pp. 1440-1466 (27 pages)

Online Publication Date: July 27, 2006

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We solve the open problem of characterizing the leading constant in the asymptotic approximation to the expected cost used for random partial match queries in random k-d trees. Our approach is new and of some generality; in particular, it is applicable to many problems involving differential equations (or difference equations) with polynomial coefficients.

Power from Random Strings

Eric Allender, Harry Buhrman, Michal Koucký, Dieter van Melkebeek, and Detlef Ronneburger

SIAM J. Comput. 35, pp. 1467-1493 (27 pages) | Cited 1 time

Online Publication Date: July 27, 2006

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We show that sets consisting of strings of high Kolmogorov complexity provide examples of sets that are complete for several complexity classes under probabilistic and nonuniform reductions. These sets are provably not complete under the usual many-one reductions.
Let ${{R_{\rm C}}}, {{R_{\rm Kt}}}, {{R_{\rm KS}}}, {{R_{\rm KT}}}$ be the sets of strings $x$ having complexity at least $|x|/2$, according to the usual Kolmogorov complexity measure ${\mbox{\rm C}}$, Levin's time-bounded Kolmogorov complexity ${\mbox{\rm Kt}}$ [L. Levin, Inform. and Control, 61 (1984), pp. 15-37], a space-bounded Kolmogorov measure ${\mbox{\rm KS}}$, and a new time-bounded Kolmogorov complexity measure ${\mbox{\rm KT}}$, respectively.
Our main results are as follows:
\begin{remunerate} \item ${{R_{\rm KS}}}$ and ${{R_{\rm Kt}}}$ are complete for ${{\rm{PSPACE}}}$ and {\mbox{\rm EXP}}, respectively, under ${\mbox{\rm P/poly}}$-truth-table reductions. Similar results hold for other classes with ${{\rm{PSPACE}}}$-robust Turing complete sets.
\item ${\mbox{\rm EXP}} = {\mbox{\rm NP}}^{{{R_{\rm Kt}}}}.$
\item ${{\rm{PSPACE}}} = {\mbox{\rm ZPP}}^{{{R_{\rm KS}}}} \subseteq {\mbox{\rm P}}^{{{R_{\rm C}}}}$.
\item The Discrete Log, Factoring, and several lattice problems are solvable in ${\mbox{\rm BPP}}^{{{R_{\rm KT}}}}$. \end{remunerate}
Our hardness result for ${{\rm{PSPACE}}}$ gives rise to fairly natural problems that are complete for ${{\rm{PSPACE}}}$ under ${\mbox{$\leq^{\rm p}_{\rm T}$}}$ reductions, but not under ${\mbox{$\leq^{\rm log}_{\rm m}$}}$ reductions.
Our techniques also allow us to show that all computably enumerable sets are reducible to ${{R_{\rm C}}}$ via ${\mbox{\rm P/poly}}$-truth-table reductions. This provides the first "efficient" reduction of the halting problem to ${{R_{\rm C}}}$.

Fully Dynamic Orthogonal Range Reporting on RAM

Christian Worm Mortensen

SIAM J. Comput. 35, pp. 1494-1525 (32 pages) | Cited 3 times

Online Publication Date: July 27, 2006

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We show that there exists a constant $\omega < 1$ such that the fully dynamic $d$-dimensional orthogonal range reporting problem for any constant $d \ge 2$ can be solved in time $O(\log^{\omega+d-2} n)$ for updates and time $O((\log n / \log\log n)^{d-1} + r)$ for queries. Here $n$ is the number of points stored and $r$ is the number of points reported. The space usage is $O(n \log^{\omega+d-2} n)$. For $d=2$ our results are optimal in terms of time per operation, and this is the main contribution of this paper. Also for $d=2$, we give a new improved fully dynamic structure supporting 3-sided queries. The model of computation is a unit cost RAM@. We order the coordinates of points using list order as defined in the paper.
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