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1998

Volume 28, Issue 1, pp. 1-381


Optimal Biweighted Binary Trees and the Complexity of Maintaining Partial Sums

Haripriyan Hampapuram and Michael L. Fredman

SIAM J. Comput. 28, pp. 1-9 (9 pages) | Cited 2 times

Online Publication Date: July 28, 2006

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Let A be an array. The partial sum problem concerns the design of a data structure for implementing the following operations. The operation update (j,x) has the effect $A[j] \leftarrow A[j]+x \,$, and the query operation $\ssum(j)$ returns the partial sum $\sum_{i=1}^j \, A[i] \,$. Our interest centers upon the optimal efficiency with which sequences of such operations can be performed, and we derive new upper and lower bounds in the semigroup model of computation. Our analysis relates the optimal complexity of the partial sum problem to optimal binary trees relative to a type of weighting scheme that defines the notion of biweighted binary tree.

Dynamic 2-Connectivity with Backtracking

Johannes A. La Poutré and Jeffery Westbrook

SIAM J. Comput. 28, pp. 10-26 (17 pages)

Online Publication Date: July 28, 2006

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We give algorithms and data structures that maintain the 2-edge and 2-vertex-connected components of a graph under insertions and deletions of edges and vertices, where deletions occur in a backtracking fashion (i.e., deletions undo the insertions in the reverse order). Our algorithms run in $\Theta (\log n)$ worst-case time per operation and use $\Theta (n)$ space, where n is the number of vertices. Using our data structure we can answer queries, which ask whether vertices u and v belong to the same 2-connected component, in $\Theta (\log n)$ worst-case time.

On the Structure of $\cal NP_\Bbb C$

Gregorio Malajovich and Klaus Meer

SIAM J. Comput. 28, pp. 27-35 (9 pages) | Cited 3 times

Online Publication Date: July 28, 2006

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This paper deals with complexity classes ${\cal P}_{\Bbb C}$ and ${\cal NP}_{\Bbb C}$ as they were introduced over the complex numbers by Blum, Shub, and Smale [Bull. Amer. Math. Soc., 21 (1989), p. 1]. Under the assumption ${\cal P}_{\Bbb C} \ne {\cal NP}_{\Bbb C}$ the existence of noncomplete problems in ${\cal NP}_{\Bbb C}$ not belonging to ${\cal P}_{\Bbb C}$ is established.

Weighted NP Optimization Problems: Logical Definability and Approximation Properties

Marius Zimand

SIAM J. Comput. 28, pp. 36-56 (21 pages) | Cited 1 time

Online Publication Date: July 28, 2006

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Extending a well-known property of NP optimization problems in which the value of the optimum is guaranteed to be polynomially bounded in the length of the input, it is observed that, by attaching weights to tuples over the domain of the input, all NP optimization problems admit a logical characterization. It is shown that any NP optimization problem can be stated as a problem in which the constraint conditions can be expressed by a $\Pi_2$ first-order formula. The paper analyzes the weighted analogue of all syntactically defined classes of optimization problems that are known to have good approximation properties in the nonweighted case. Dramatic changes occur when negative weights are allowed.

The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory

Tomás Feder and Moshe Y. Vardi

SIAM J. Comput. 28, pp. 57-104 (48 pages) | Cited 70 times

Online Publication Date: July 28, 2006

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This paper starts with the project of finding a large subclass of NP which exhibits a dichotomy. The approach is to find this subclass via syntactic prescriptions. While the paper does not achieve this goal, it does isolate a class (of problems specified by) "monotone monadic SNP without inequality" which may exhibit this dichotomy. We justify the placing of all these restrictions by showing, essentially using Ladner's theorem, that classes obtained by using only two of the above three restrictions do not show this dichotomy. We then explore the structure of this class. We show that all problems in this class reduce to the seemingly simpler class CSP. We divide CSP into subclasses and try to unify the collection of all known polytime algorithms for CSP problems and extract properties that make CSP problems NP-hard. This is where the second part of the title, "a study through Datalog and group theory," comes in. We present conjectures about this class which would end in showing the dichotomy.

Asymptotically Tight Bounds for Performing BMMC Permutations on Parallel Disk Systems

Thomas H. Cormen, Thomas Sundquist, and Leonard F. Wisniewski

SIAM J. Comput. 28, pp. 105-136 (32 pages) | Cited 3 times

Online Publication Date: July 28, 2006

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This paper presents asymptotically equal lower and upper bounds for the number of parallel I/O operations required to perform bit-matrix-multiply/complement (BMMC) permutations on the Parallel Disk Model proposed by Vitter and Shriver. A BMMC permutation maps a source index to a target index by an affine transformation over GF(2), where the source and target indices are treated as bit vectors. The class of BMMC permutations includes many common permutations, such as matrix transposition (when dimensions are powers of 2), bit-reversal permutations, vector-reversal permutations, hypercube permutations, matrix reblocking, Gray-code permutations, and inverse Gray-code permutations. The upper bound improves upon the asymptotic bound in the previous best known BMMC algorithm and upon the constant factor in the previous best known bit-permute/complement (BPC) permutation algorithm. The algorithm achieving the upper bound uses basic linear-algebra techniques to factor the characteristic matrix for the BMMC permutation into a product of factors, each of which characterizes a permutation that can be performed in one pass over the data.
The factoring uses new subclasses of BMMC permutations: memoryload-dispersal (MLD) permutations and their inverses. These subclasses extend the catalog of one-pass permutations.
Although many BMMC permutations of practical interest fall into subclasses that might be explicitly invoked within the source code, this paper shows how to quickly detect whether a given vector of target addresses specifies a BMMC permutation. Thus, one can determine efficiently at run time whether a permutation to be performed is BMMC and then avoid the general-permutation algorithm and save parallel I/Os by using the BMMC permutation algorithm herein.

L-Printable Sets

Lance Fortnow, Judy Goldsmith, Matthew A. Levy, and Stephen Mahaney

SIAM J. Comput. 28, pp. 137-151 (15 pages)

Online Publication Date: July 28, 2006

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A language is L-printable if there is a logspace algorithm which, on input 1n, prints all members in the language of length n. Following the work of Allender and Rubinstein [SIAM J. Comput., 17 (1988), pp. 1193--1202] on P-printable sets, we present some simple properties of the L-printable sets. This definition of "L-printable" is robust and allows us to give alternate characterizations of the L-printable sets in terms of tally sets and Kolmogorov complexity. In addition, we show that a regular or context-free language is L-printable if and only if it is sparse, and we investigate the relationship between L-printable sets, L-rankable sets (i.e., sets A having a logspace algorithm that, on input x, outputs the number of elements of A that precede x in the standard lexicographic ordering of strings), and the sparse sets in L. We prove that under reasonable complexity-theoretic assumptions, these three classes of sets are all different. We also show that the class of sets of small generalized Kolmogorov space complexity is exactly the class of sets that are L-isomorphic to tally languages.

The Inverse Satisfiability Problem

Dimitris Kavvadias and Martha Sideri

SIAM J. Comput. 28, pp. 152-163 (12 pages) | Cited 3 times

Online Publication Date: July 28, 2006

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We study the complexity of telling whether a set of bit-vectors represents the set of all satisfying truth assignments of a Boolean expression of a certain type. We show that the problem is coNP-complete when the expression is required to be in conjunctive normal form with three literals per clause (3CNF). We also prove a dichotomy theorem analogous to the classical one by Schaefer, stating that, unless P=NP, the problem can be solved in polynomial time if and only if the clauses allowed are all Horn, or all anti-Horn, or all 2CNF, or all equivalent to equations modulo two.

On Syntactic versus Computational Views of Approximability

Sanjeev Khanna, Rajeev Motwani, Madhu Sudan, and Umesh Vazirani

SIAM J. Comput. 28, pp. 164-191 (28 pages) | Cited 18 times

Online Publication Date: July 28, 2006

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We attempt to reconcilethe two distinct views of approximation classes: syntactic and computational. Syntactic classes such as MAX SNP permit structural results and have natural complete problems, while computational classes such as APX allow us to work with classes of problems whose approximability is well understood. Our results provide a syntactic characterization of computational classes and give a computational framework for syntactic classes.
We compare the syntactically defined class MAX SNP with the computationally defined class APX and show that every problem in APX can be "placed" (i.e., has approximation-preserving reduction to a problem) in MAX SNP. Our methods introduce a simple, yet general, technique for creating approximation-preserving reductions which shows that any "well"-approximable problem can be reduced in an approximation-preserving manner to a problem which is hard to approximate to corresponding factors. The reduction then follows easily from the recent nonapproximability results for MAX SNP-hard problems. We demonstrate the generality of this technique by applying it to other classes such as MAX SNP-RMAX(2) and MIN F$^{+}\Pi_2(1)$ which have the clique problem and the set cover problem, respectively, as complete problems.
The syntactic nature of MAX SNP was used by Papadimitriou and Yannakakis [J. Comput. System Sci., 43 (1991), pp. 425--440] to provide approximation algorithms for every problem in the class. We provide an alternate approach to demonstrating this result using the syntactic nature of MAX SNP. We develop a general paradigm, nonoblivious local search, useful for developing simple yet efficient approximation algorithms. We show that such algorithms can find good approximations for all MAX SNP problems, yielding approximation ratios comparable to the best known for a variety of specific MAX SNP-hard problems. Nonoblivious local search provably outperforms standard local search in both the degree of approximation achieved and the efficiency ofthe resulting algorithms.

Maximum k-Chains in Planar Point Sets: Combinatorial Structure and Algorithms

Stefan Felsner and Lorenz Wernisch

SIAM J. Comput. 28, pp. 192-209 (18 pages)

Online Publication Date: July 28, 2006

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A chain of a set P of n points in the plane is a chain of the dominance order on P. A k-chain is a subset C of P that can be covered by k chains. A k-chain C is a maximum k-chain if no other k-chain contains more elements than C. This paper deals with the problem of finding a maximum k-chain of P in the cardinality and in the weighted case.
Using the skeleton S(P) of a point set P introduced by Viennot we describe a fairly simple algorithm that computes maximum k-chains in time O(kn log n) and linear space. The basic idea is that the canonical chain partition of a maximum (k-1)-chain in the skeleton S(P) provides k regions in the plane such that a maximum k-chain for P can be obtained as the union of a maximal chain from each of these regions.
By the symmetry between chains and antichains in the dominance order we may use the algorithm for maximum k-chains to compute maximum k-antichains for planar points in time O(kn log n). However, for large k one can do better. We describe an algorithm computing maximum k-antichains (and, by symmetry, k-chains) in time O((n2k) log n) and linear space. Consequently, a maximum k-chain can be computed in time O(n3/2 log n) for arbitrary k.
The background for the algorithms is a geometric approach to the Greene--Kleitman theory for permutations. We include a skeleton-based exposition of this theory and give some hints on connections with the theory of Young tableaux.
The concept of the skeleton of a planar point set is extended to the case of a weighted point set. This extension allows to compute maximum weighted k-chains with an algorithm that is similar to the algorithm for the cardinality case. The time and space requirements of the algorithm for weighted k-chains are O(2kn log(2kn)) and O(2kn), respectively.

Fast Algorithms for Constructing t-Spanners and Paths with Stretch t

Edith Cohen

SIAM J. Comput. 28, pp. 210-236 (27 pages) | Cited 9 times

Online Publication Date: July 28, 2006

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The distance between two vertices in a weighted graph is the weight of a minimum-weight path between them (where the weight of a path is the sum of the weights of the edges in the path). A path has stretcht if its weight is at most t times the distance between its end points. We present algorithms that compute paths of stretch $2\leq t\leq\log n$ on undirected graphs G=(V,E) with nonnegative weights. The stretch t is of the form $t=\beta(2+\epsilon')$, where $\beta$ is integral and $\epsilon'>0$ is at least as large as some fixed $\epsilon>0$. We present an $\tilde{O}((m+k)n^{(2+\epsilon)/t})$ time randomized algorithm that finds paths between k specified pairs of vertices and an $\tilde{O}((m+ns)n^{2(1+\log_n m+\epsilon)/t})$ deterministic algorithm that finds paths from $s$ specified sources to all other vertices (for any fixed $\epsilon>0$), where n=|V| and m=|E|. This improves significantly over the slower $\tilde{O}(\min\{k,n\}m)$ exact shortest paths algorithms and a previous $\tilde{O}(mn^{64/t}+kn^{32/t})$ time algorithm by Awerbuch {et al.}\ [Proc. 34th IEEE Annual Symposium on Foundations of Computer Science, IEEE, Piscataway, NJ, 1993, pp. 638--647]. A t-spanner of a graph G is a set of weighted edges on the vertices of G such that distances in the spanner are not smaller and within a factor of t from the corresponding distances in G. Previous work was concerned with bounding the size and efficiently constructing t-spanners. We construct t-spanners of size $\tilde{O}(n^{1+(2+\epsilon)/t})$ in $\tilde{O}(mn^{(2+\epsilon)/t})$ expected time (for any fixed $\epsilon>0$), which constitutes a faster construction (by a factor of n3+2/t /m) of sparser spanners than was previously attainable. We also provide efficient parallel constructions. Our algorithms are based on pairwise covers and a novel approach to construct them efficiently.

Smart SMART Bounds for Weighted Response Time Scheduling

Uwe Schwiegelshohn, Walter Ludwig, Joel L. Wolf, John Turek, and Philip S. Yu

SIAM J. Comput. 28, pp. 237-253 (17 pages) | Cited 5 times

Online Publication Date: July 28, 2006

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Consider a system of independent tasks to be scheduled without preemption on a parallel computer. For each task the number of processors required, the execution time, and a weight are known. The problem is to find a schedule with minimum weighted average response time. We present an algorithm called SMART (which stands for scheduling to minimize average response time) for this problem that produces solutions that are within a factor of 8.53 of optimal. To our knowledge this is the first polynomial-time algorithm for the minimum weighted average response time problem that achieves a constant bound. In addition, for the unweighted case (that is, where all the weights are unity) we describe a variant of SMART that produces solutions that are within a factor of 8 of optimal, improving upon the best known bound of 32 for this special case.

New Approximation Guarantees for Minimum-Weight k-Trees and Prize-Collecting Salesmen

Baruch Awerbuch, Yossi Azar, Avrim Blum, and Santosh Vempala

SIAM J. Comput. 28, pp. 254-262 (9 pages) | Cited 15 times

Online Publication Date: July 28, 2006

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We consider a formalization of the following problem. A salesperson must sell some quota of brushes in order to win a trip to Hawaii. This salesperson has a map (a weighted graph) in which each city has an attached demand specifying the number of brushes that can be sold in that city. What is the best route to take to sell the quota while traveling the least distance possible? Notice that unlike the standard traveling salesman problem, not only do we need to figure out the order in which to visit the cities, but we must decide the more fundamental question: which cities do we want to visit?
In this paper we give the first approximation algorithm having a polylogarithmic performance guarantee for this problem, as well as for the slightly more general "prize-collecting traveling salesman problem" (PCTSP) of Balas, and a variation we call the "bank robber problem" (also called the "orienteering problem" by Golden, Levi, and Vohra). We do this by providing an O(log2k) approximation to the somewhat cleaner k-MST problem which is defined as follows. Given an undirected graph on n nodes with nonnegative edge weights and an integer $k \leq n$, find the tree of least weight that spans k vertices. (If desired, one may specify in the problem a "root vertex" that must be in the tree as well.) Our result improves on the previous best bound of $O(\sqrt{k})$ of Ravi et al.

Near-Linear Time Construction of Sparse Neighborhood Covers

Baruch Awerbuch, Bonnie Berger, Lenore Cowen, and David Peleg

SIAM J. Comput. 28, pp. 263-277 (15 pages) | Cited 10 times

Online Publication Date: July 28, 2006

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This paper introduces a near-linear time sequential algorithm for constructing a sparse neighborhood cover. This implies analogous improvements (from quadratic to near-linear time) for any problem whose solution relies on network decompositions, including small edge cuts in planar graphs, approximate shortest paths, and weight- and distance-preserving graph spanners. In particular, an O(log n) approximation to the k-shortest paths problem on an n-vertex, E-edge graph is obtained that runs in $\soh{n + E + k}$ time.

Unoriented $Theta$-Maxima in the Plane: Complexity and Algorithms

David Avis, Bryan Beresford-Smith, Luc Devroye, Hossam Elgindy, Eric Guévremont, Ferran Hurtado, and Binhai Zhu

SIAM J. Comput. 28, pp. 278-296 (19 pages)

Online Publication Date: July 28, 2006

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We introduce the unoriented $\Theta$-maximum as a new criterion for describing the shape of a set of planar points. We present efficient algorithms for computing the unoriented $\Theta$-maximum of a set of planar points. We also propose a simple linear expected time algorithm for computing the unoriented $\Theta$-maximum of a set of planar points when $\Theta=\pi/2$.

A Spectral Algorithm for Seriation and the Consecutive Ones Problem

Jonathan E. Atkins, Erik G. Boman, and Bruce Hendrickson

SIAM J. Comput. 28, pp. 297-310 (14 pages) | Cited 14 times

Online Publication Date: July 28, 2006

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In applications ranging from DNA sequencing through archeological dating to sparse matrix reordering, a recurrent problem is the sequencing of elements in such a way that highly correlated pairs of elements are near each other. That is, given a correlation function f reflecting the desire for each pair of elements to be near each other, find all permutations $\pi$ with the property that if $\pi(i) < \pi(j) < \pi(k)$ then $f(i,j) \ge f(i,k)$ and $f(j,k) \ge f(i,k)$. This seriationproblem is a generalization of the well-studied consecutive ones problem. We present a spectral algorithm for this problem that has a number of interesting features. Whereas most previous applications of spectral techniques provide only bounds or heuristics, our result is an algorithm that correctly solves a nontrivial combinatorial problem. In addition, spectral methods are being successfully applied as heuristics to a variety of sequencing problems, and our result helps explain and justify these applications.

New Collapse Consequences of NP Having Small Circuits

Johannes Köbler and Osamu Watanabe

SIAM J. Comput. 28, pp. 311-324 (14 pages) | Cited 6 times

Online Publication Date: July 28, 2006

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We show that if a self-reducible set has polynomial-size circuits, then it is low for the probabilistic class ZPP (NP). As a consequence we get a deeper collapse of the polynomial-time hierarchy PH to ZPP(NP) under the assumption that NP has polynomial-size circuits. This improves on the well-known result in Karp and Lipton [ Proceedings of the 12th ACM Symposium on Theory of Computing, ACM Press, New York, 1980, pp. 302--309] stating a collapse of PH to its second level $\Sigmap_2$ under the same assumption. Furthermore, we derive new collapse consequences under the assumption that complexity classes like UP, FewP, and C=P have polynomial-size circuits. Finally, we investigate the circuit-size complexity of several language classes. In particular, we show that for every fixed polynomial s, there is a set in ZPP(NP) which does not have O(s(n))-size circuits.

Sublogarithmic Bounds on Space and Reversals

Viliam Geffert, Carlo Mereghetti, and Giovanni Pighizzini

SIAM J. Comput. 28, pp. 325-340 (16 pages) | Cited 1 time

Online Publication Date: July 28, 2006

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The complexity measure under consideration is $\mbox{\rm SPACE}\!\times\!\mbox{\rm REVERSALS}$ for Turing machines that are able to branch both existentially and universally. We show that, for any function $h(n)$ between $\log\log n$ and $\log n$, $\Pi_1 \mbox{\rm SPACE}\!\times\!\mbox{\rm REVERSALS} (h(n))$ is separated {}from $\Sigma_1 \mbox{\rm SPACE}\!\times\!\mbox{\rm REVERSALS} (h(n))$ as well as {}from $\mbox{\sf co}\Sigma_1 \mbox{\rm SPACE}\!\times\!\mbox{\rm REVERSALS} (h(n))$, for middle, accept, and weak modes of this complexity measure. This also separates determinism from the higher levels of the alternating hierarchy. For "well-behaved" functions h(n) between log log n and log n, almost all of the above separations can be obtained by using unary witness languages.
In addition, the construction of separating languages contributes to the research on minimal resource requirements for computational devices capable of recognizing nonregular languages. For any (arbitrarily slow growing) unbounded monotone recursive function f(n), a nonregular unary language is presented that can be accepted by a \middle\ \mbox{$\Pi_1$ alternating} Turing machine in s(n) space and i(n) input head reversals, with $s(n)\cdot i(n)\in{\cal O}(\log\log n\cdot f(n))$. Thus, there is no exponential gap for the optimal lower bound on the product $s(n)\cdot i(n)$ between unary and general nonregular language acceptance---in sharp contrast with the one-way case.

Separator-Based Sparsification II: Edge and Vertex Connectivity

David Eppstein, Zvi Galil, Giuseppe F. Italiano, and Thomas H. Spencer

SIAM J. Comput. 28, pp. 341-381 (41 pages) | Cited 3 times

Online Publication Date: July 28, 2006

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We consider the problem of maintaining a dynamic planar graph subject to edge insertions and edge deletions that preserve planarity but that can change the embedding. We describe algorithms and data structures for maintaining information about 2- and 3-vertex-connectivity, and 3- and 4-edge-connectivity in a planar graph in O(n1/2) amortized time per insertion, deletion, or connectivity query. All of the data structures handle insertions that keep the graph planar without regard to any particular embedding of the graph. Our algorithms are based on a new type of sparsification combined with several properties of separators in planar graphs.
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