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SIAM J. on Computing

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1996

Volume 25, Issue 6, pp. 1123-1358


Kolmogorov Complexity and Instance Complexity of Recursively Enumerable Sets

Martin Kummer

SIAM J. Comput. 25, pp. 1123-1143 (21 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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The way in which way Kolmogorov complexity and instance complexity affect properties of recursively enumerable (r.e.) sets is studied. The well-known $2\log n$ upper bound on the Kolmogorov complexity of initial segments of r.e. sets is shown to be optimal, and the Turing degrees of r.e. sets which attain this bound are characterized. The main part of the paper is concerned with instance complexity, introduced by Ko, Orponen, Schöning, and Watanabe in 1986, as a measure of the complexity of individual instances of a decision problem. They conjectured that for every r.e. nonrecursive set, the instance complexity is infinitely often at least as high as the Kolmogorov complexity. The conjecture is refuted by constructing an r.e. nonrecursive set with instance complexity logarithmic in the Kolmogorov complexity. This bound is optimal up to an additive constant. In the other extreme, the conjecture is established for many classes of complete sets, such as weak-truth-table-complete (wtt-complete) and Q-complete sets. However, there is a Turing-complete set for which it fails.

An $o(n^3 )$-Time Maximum-Flow Algorithm

Joseph Cheriyan, Torben Hagerup, and Kurt Mehlhorn

SIAM J. Comput. 25, pp. 1144-1170 (27 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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We show that a maximum flow in a network with $n$ vertices can be computed deterministically in $O({{n^3 } / {\log n}})$ time on a uniform-cost RAM. For dense graphs, this improves the previous best bound of $O(n^3 )$.
The bottleneck in our algorithm is a combinatorial problem on (unweighted) graphs. The number of operations executed on flow variables is $O(n^{{8 / 3}} (\log n)^{{4 / 3}} )$, in contrast with $\Omega (nm)$ flow operations for all previous algorithms, where $m$ denotes the number of edges in the network. A randomized version of our algorithm executes $O(n^{{3 / 2}} m^{{1 / 2}} \log n + {{n^2 (\log n)^2 } / {\log }}(2 + {{n(\log n)^2 } / m})$ flow operations with high probability.
For the special case in which all capacities are integers bounded by $U$, we show that a maximum flow can be computed deterministically using $O(n^{{3 / 2}} m^{{1 / 2}} + n^2 (\log U)^{{1 / 2}} + \log U$ flow operations and $O(\min \{ {{nm,n^3 } / {\log n\} + n^2 }}(\log U)^{{1 / 2}} + \log U)$ time. We finally argue that several of our results yield parallel algorithms with optimal speedup.

A Deterministic ${\operatorname{Poly}}(\log \log N)$-Time $N$-Processor Algorithm for Linear Programming in Fixed Dimension

Miklos Ajtai and Nimrod Megiddo

SIAM J. Comput. 25, pp. 1171-1195 (25 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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It is shown that for any fixed number of variables, linear-programming problems with $n$ linear inequalities can be solved deterministically by $n$ parallel processors in sublogarithmic time. The parallel time bound (counting only the arithmetic operations) is $O((\log \log n)^d )$, where $d$ is the number of variables. In the one-dimensional case, this bound is optimal. If we take into account the operations needed for processor allocation, the time bound is $O((\log \log n)^{d + c} )$, where $c$ is an absolute constant.

Feasible Time-Optimal Algorithms for Boolean Functions on Exclusive-Write Parallel Random-Access Machines

Martin Dietzfelbinger, Mirosław Kutyłowski, and Rüdiger Reischuk

SIAM J. Comput. 25, pp. 1196-1230 (35 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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It was shown some years ago that the computation time for many important Boolean functions of $n$ arguments on concurrent-read exclusive-write parallel random-access machines (CREW PRAMs) of unlimited size is at least $\varphi (n) \approx 0.72\log _2 n$. On the other hand, it is known that every Boolean function of $n$ arguments can be computed in $\varphi (n) + 1$ steps on a CREW PRAM with $n \cdot 2^{n - 1} $ processors and memory cells. In the case of the OR of $n$ bits, $n$ processors and cells are sufficient. In this paper, it is shown that for many important functions, there are CREW PRAM algorithms that almost meet the lower bound in that they take $\varphi (n) + o(\log n)$ steps but use only a small number of processors and memory cells (in most cases, $n$). In addition, the cells only have to store binary words of bounded length (in most cases, length 1). We call such algorithms “feasible.” The functions concerned include the following: the PARITY function and, more generally, all symmetric functions; a large class of Boolean formulas; some functions over non-Boolean domains $\{ 0, \ldots ,k - 1\} $ for small $k$, in particular, parallel-prefix sums; addition of $n$-bit numbers; and sorting ${n / l}$ binary numbers of length $l$. Further, it is shown that Boolean circuits with fan-in 2, depth $d$, and size $s$ can be evaluated by CREW PRAMs with fewer than $s$ processors in ,$\varphi (2^d ) + o(d) \approx 0.72d + o(d)$ steps. For the exclusive-read exclusive-write (EREW) PRAM model, a feasible algorithm is described that computes PARITY of $n$ bits in $0.86\log _2 n$ steps.

Lower Bounds for Geometrical and Physical Problems

Jürgen Sellen

SIAM J. Comput. 25, pp. 1231-1253 (23 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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Motion planning involving arbitrarily many degrees of freedom is known to be PSPACE-hard. In this paper, we examine the complexity of generalized motion-planning problems for planar mechanisms consisting of independently movable objects.
Our constructions constitute a general framework for reducing problems in information processing to motion planning, leading to easy proofs of known PSPACE-hardness results and to exponential lower bounds for geometrical problems related to motion planning. Particulalrly, we show that the problem of deciding whether a given mechanism $A$ can always avoid a collision with another mechanism $B$ is EXPSPACE-hard.
New lower bounds are also obtained for the problem of planning under given physical side conditions. We consider the case that certain motions require forces, e.g., to subdue friction, and ask for motions that stay under a given energy limit. Within our framework, we show that such shortest-path problems are EXPTIME-hard if we use number representations by mantissa and exponent, and even undecidable if we allow that some motions require no force or an infinite amount. The proof consists of a simulation of Turing machines with infinite tape and shows that the notion of Turing computability can be interpreted in purely geometrical terms. The geometrical model obtained is capable of expressing a variety of physical-planning problems.

Average and Randomized Complexity of Distributed Problems

Nechama Allenberg-Navony, Alon Itai, and Shlomo Moran

SIAM J. Comput. 25, pp. 1254-1267 (14 pages)

Online Publication Date: July 31, 2006

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Yao proved that in the decision-tree model, the average complexity of the best deterministic algorithm is a lower bound on the complexity of randomized algorithms that solve the same problem. Here it is shown that a similar result does not always hold in the common model of distributed computation, the model in which all the processors run the same program (which may depend on the processors’ input).
We therefore construct a new technique that together with Yao’s method enables us to show that in many cases, a similar relationship does hold in the distributed model. This relationship enables us to carry over known lower bounds on the complexity of deterministic computations to the realm of randomized computations, thus obtaining new results.
The new technique can also be used for obtaining results concerning algorithms with bounded error.

Learning Behaviors of Automata from Multiplicity and Equivalence Queries

Francesco Bergadano and Stefano Varricchio

SIAM J. Comput. 25, pp. 1268-1280 (13 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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We consider the problem of identifying the behavior of an unknown automaton with multiplicity in the field $Q$ of rational numbers ($Q$-automaton) from multiplicity and equivalence queries. We provide an algorithm which is polynomial in the size of the $Q$-automaton and in the maximum length of the given counterexamples. As a consequence, we have that $Q$-automata are probably approximately correctly learnable (PAC-learnable) in polynomial time when multiplicity queries are allowed. A corollary of this result is that regular languages are polynomially predictable using membership queries with respect to the representation of unambiguous nondeterministic automata. This is important since there are unambiguous automata such that the equivalent deterministic automaton has an exponentially larger number of states.

Prefix Codes: Equiprobable Words, Unequal Letter Costs

Mordecai J. Golin and Neal Young

SIAM J. Comput. 25, pp. 1281-1292 (12 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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We consider the following variant of Huffman coding in which the costs of the letters, rather than the probabilities of the words, are nonuniform “Given an alphabet of $r$ letters of nonuniform length, find a minimum-average-length prefix free set of $n$ codewords over the alphabet”; equivalently, “Find an optimal $r$-ary search tree with $n$ leaves, where each leaf is accessed with equal probability but the cost to descend from a parent to its $i$th child depends on $i$.” We show new structural properties of such codes, leading to an $O(n\log ^2 r)$ time algorithm for finding them, This new algorithm is simpler and faster than the best previously known $O(nr\min \{ \log n,r\} )$ time algorithm, due to Perl Garey, and Even [J. Assoc. Comput. Mach., 22 (1975), pp. 202–214].

On Unapproximable Versions of $NP$-Complete Problems

David Zuckerman

SIAM J. Comput. 25, pp. 1293-1304 (12 pages) | Cited 5 times

Online Publication Date: July 31, 2006

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We prove that all of Karp’s 21 original $NP$-complete problems have a version that is hard to approximate. These versions are obtained from the original problems by adding essentially the same simple constraint. We further show that these problems are absurdly hard to approximate. In fact, no polynomial-time algorithm can even approximate $\log ^{(k)} $ of the magnitude of these problems to within any constant factor, where $\log ^{(k)} $ denotes the logarithm iterated $k$ times, unless $NP$ is recognized by slightly superpolynomial randomized machines. We use the same technique to improve the constant $\epsilon $ such that MAX CLIQUE is hard to approximate to within a factor of $n^\epsilon $. Finally, we show that it is even harder to approximate two counting problems: counting the number of satisfying assignments to a monotone 2SAT formula and computing the permanent of $ - 1,0,1$ matrices.

A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth

Hans L. Bodlaender

SIAM J. Comput. 25, pp. 1305-1317 (13 pages) | Cited 88 times

Online Publication Date: July 31, 2006

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In this paper, we give for constant $k$ a linear-time algorithm that, given a graph $G = (V,E)$, determines whether the treewidth of $G$ is at most $k$ and, if so, finds a tree-decomposition of $G$ with treewidth at most $k$. A consequence is that every minor-closed class of graphs that does not contain all planar graphs has a linear-time recognition algorithm. Another consequence is that a similar result holds when we look instead for path-decompositions with pathwidth at most some constant $k$.

An Optimal $O(\log \log N)$-Time Parallel Algorithm for Detecting all Squares in a String

Alberto Apostolico and Dany Breslauer

SIAM J. Comput. 25, pp. 1318-1331 (14 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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An optimal $O(\log \log n)$-time concurrent-read concurrent-write parallel algorithm for detecting all squares in a string is presented. A tight lower bound shows that over general alphabets, this is the fastest possible optimal algorithm. When $p$ processors are available, the bounds become $\Theta \lceil {({{n\log n)} p}\rceil + \log \log _{\lceil {1 + {p / n}} \rceil } 2p} )$. The algorithm uses an optimal parallel string-matching algorithm together with periodicity properties to locate the squares within the input string.

The Wakeup Problem

Michael J. Fischer, Shlomo Moran, Steven Rudich, and Gadi Taubenfeld

SIAM J. Comput. 25, pp. 1332-1357 (26 pages) | Cited 4 times

Online Publication Date: July 31, 2006

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We study a new problem—the wakeup problem—that seems to be fundamental in distributed computing. We present efficient solutions to the problem and show how these solutions can be used to solve the consensus problem, the leader-election problem, and other related problems. The main question we try to answer is “How much memory is needed to solve the wakeup problem?” We assume a model that captures important properties of real systems that have been largely ignored by previous work on cooperative problems.

Erratum: Fast Parallel Computation of the Polynomial Remainder Sequence via Bezout and Hankel Matrices

Dario Bini and Luca Gemignani

SIAM J. Comput. 25, pp. 1358-1358 (1 page)

Online Publication Date: July 31, 2006

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