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2005

Volume 34, Issue 3, pp. 515-773


Top-Down Analysis of Path Compression

Raimund Seidel and Micha Sharir

SIAM J. Comput. 34, pp. 515-525 (11 pages)

Online Publication Date: July 27, 2006

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We present a new analysis of the worst-case cost of path compression, which is an operation that is used in various well-known "union-find" algorithms. In contrast to previous analyses which are essentially based on bottom-up approaches, our method proceeds top-down, yielding recurrence relations from which the various bounds arise naturally. In particular the famous quasi-linear bound involving the inverse Ackermann function can be derived without having to introduce the Ackermann function itself.

Pseudo-Line Arrangements: Duality, Algorithms, and Applications

Pankaj K. Agarwal and Micha Sharir

SIAM J. Comput. 34, pp. 526-552 (27 pages) | Cited 2 times

Online Publication Date: July 27, 2006

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A finite collection of x-monotone unbounded Jordan curves in the plane is called a family of pseudo-lines if every pair of curves intersect in at most one point, and the two curves cross each other there. Let L be such a collection of n pseudo-lines, and let P be a set of m points in $\reals^2$. Extending a result of Goodman [Discrete Math., 32 (1980), pp. 27--35], we define a duality transform that maps L to a set L* of points in $\reals^2$ and P to a set P* of (x-monotone) pseudo-lines in $\reals^2$, so that the incidence and the "above-below" relations between the points and the pseudo-lines are preserved. We present an efficient algorithm for computing the dual arrangement {\eus A}$(P^*)$ under an appropriate model of computation.
We also present a dynamic data structure for reporting, in $O(m^\eps + k)$ time, all k points of P that lie below a query arc, which is either a circular arc or a portion of the graph of a polynomial of fixed degree. This result is needed for computing the dual arrangement for certain classes of pseudo-lines arising in several applications, but is also interesting in its own right. We present a few applications of our dual arrangement algorithm, such as computing incidences between points and pseudo-lines and computing a subset of faces in a pseudo-line arrangement.
Next, we present an efficient algorithm for cutting a set of circles into arcs so that every pair of arcs intersect in at most one point, i.e., the resulting arcs constitute a collection of pseudo-segments. By combining this algorithm with our algorithm for computing the dual arrangement of pseudo-lines, we obtain efficient algorithms for several problems involving arrangements of circles or circular arcs, such as reporting or counting incidences between points and circles and computing a set of marked faces in arrangements of circles.

Layout of Graphs with Bounded Tree-Width

Vida Dujmovic, Pat Morin, and David R. Wood

SIAM J. Comput. 34, pp. 553-579 (27 pages) | Cited 7 times

Online Publication Date: July 27, 2006

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Aqueue layout of a graph consists of a total order of the vertices, and a partition of the edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its queue-number. A three-dimensional (straight-line grid) drawing of a graph represents the vertices by points in $\mathbb{Z}^3$ and the edges by noncrossing line-segments. This paper contributes three main results:
(1) It is proved that the minimum volume of a certain type of three-dimensional drawing of a graph G is closely related to the queue-number of G. In particular, if G is an n-vertex member of a proper minor-closed family of graphs (such as a planar graph), then G has a $\mathcal{O}(1) \times \mathcal{O}(1) \times \mathcal{O}(n)$ drawing if and only if G has a $\mathcal{O}(1)$ queue-number.
(2) It is proved that the queue-number is bounded by the tree-width, thus resolving an open problem due to Ganley and Heath [Discrete Appl. Math., 109 (2001), pp. 215--221] and disproving a conjecture of Pemmaraju [Exploring the Powers of Stacks and Queues via Graph Layouts, Ph. D. thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA, 1992]. This result provides renewed hope for the positive resolution of a number of open problems in the theory of queue layouts.
(3) It is proved that graphs of bounded tree-width have three-dimensional drawings with $\mathcal{O}(n)$ volume. This is the most general family of graphs known to admit three-dimensional drawings with $\mathcal{O}(n)$ volume.
The proofs depend upon our results regarding track layouts and tree-partitions of graphs, which may be of independent interest.

Abstract Combinatorial Programs and Efficient Property Testers

Artur Czumaj and Christian Sohler

SIAM J. Comput. 34, pp. 580-615 (36 pages) | Cited 6 times

Online Publication Date: July 27, 2006

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Property testing is a relaxation of classical decision problems which aims at distinguishing between functions having a predetermined property and functions being far from any function having the property. In this paper we present a novel framework for analyzing property testing algorithms. Our framework is based on a connection of property testing and a new class of problems which we call abstract combinatorial programs. We show that if the problem of testing a property can be reduced to an abstract combinatorial program of small dimension, then the property has an efficient tester.
We apply our framework to a variety of problems. We present efficient property testing algorithms for geometric clustering problems, for the reversaldistance problem, and for graph and hypergraph coloring problems. We also prove that, informally, any hereditary graph property can be efficiently tested if and only if it can be reduced to an abstract combinatorial program of small size.
Our framework allows us to analyze all our testers in a unified way, and the obtained complexity bounds either match or improve the previously known bounds. Furthermore, even if the asymptotic complexity of the testers is not improved, the obtained proofs are significantly simpler than the previous ones. We believe that our framework will help to understand the structure of efficiently testable properties.

Load Balancing in Arbitrary Network Topologies with Stochastic Adversarial Input

Aris Anagnostopoulos, Adam Kirsch, and Eli Upfal

SIAM J. Comput. 34, pp. 616-639 (24 pages) | Cited 1 time

Online Publication Date: July 27, 2006

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We study the long-term (steady state) performance of a simple, randomized, local load balancing technique under a broad range of input conditions. We assume a system of n processors connected by an arbitrary network topology. Jobs are placed in the processors by a deterministic or randomized adversary. The adversary knows the current and past load distribution in the network and can use this information to place the new tasks in the processors. A node can execute one job per step, and can also participate in one load balancing operation in which it can move tasks to a direct neighbor in the network. In the protocol we analyze here, a node equalizes its load with a random neighbor in the graph.
Our analysis of the protocol does not assume any particular input distribution. The input is generated by an arbitrary deterministic or probabilistic adversary subject only to some weak statistical properties. For stability and expected performance of the system we adopt the stochastic adversary model of [Borodin et al., J. ACM, 48 (2001), pp. 13--38]. For high-probability bounds we introduce a more restricted input model, the strongly bounded adversary.
Assuming the stochastic adversarial input model, we show that if the adversary does not trivially overload the network (i.e., there is an integer $w\geq 1$ such that the expected number of new jobs in any interval of length $w$ is bounded by $\lambda nw$ for some $\lambda < 1$), then the system is stable for any connected network topology, regardless of how the adversary allocates the new jobs between the processors.
When the system is stable, the next performance parameter of interest is the waiting time of jobs. We develop expected and high probability bounds on the total load in the system and the waiting time of jobs in terms of the network topology. In particular, in the above stochastic adversary model, if the network is an expander graph, the expected wait of a task is O(w + log n), and in the strongly bounded adversary model the waiting time of a task is O(w + log n) with high probability.
We contrast these results with the work stealing load balancing protocol, where we show that in sparse networks, the load in the system and the waiting time can be exponential in the network size.

A Second-Order Perceptron Algorithm

Nicolò Cesa-Bianchi, Alex Conconi, and Claudio Gentile

SIAM J. Comput. 34, pp. 640-668 (29 pages) | Cited 6 times

Online Publication Date: July 27, 2006

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Kernel-based linear-threshold algorithms, such as support vector machines and Perceptron-like algorithms, are among the best available techniques for solving pattern classification problems. In this paper, we describe an extension of the classical Perceptron algorithm, called second-order Perceptron, and analyze its performance within the mistake bound model of on-line learning. The bound achieved by our algorithm depends on the sensitivity to second-order data information and is the best known mistake bound for (efficient) kernel-based linear-threshold classifiers to date. This mistake bound, which strictly generalizes the well-known Perceptron bound, is expressed in terms of the eigenvalues of the empirical data correlation matrix and depends on a parameter controlling the sensitivity of the algorithm to the distribution of these eigenvalues. Since the optimal setting of this parameter is not known a priori, we also analyze two variants of the second-order Perceptron algorithm: one that adaptively sets the value of the parameter in terms of the number of mistakes made so far, and one that is parameterless, based on pseudoinverses.

Nonmigratory Online Deadline Scheduling on Multiprocessors

Ho-Leung Chan, Tak-Wah Lam, and Kar-Keung To

SIAM J. Comput. 34, pp. 669-682 (14 pages) | Cited 4 times

Online Publication Date: July 27, 2006

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In this paper we consider multiprocessor scheduling with hard deadlines and investigate the cost of eliminating migration in the online setting. Let I be any set of jobs that can be completed by some migratory offline schedule on m processors. We show that I can also be completed by a nonmigratory online schedule using m speed-5.828 processors (i.e., processors 5.828 times faster). This result supplements the previous results that I can also be completed by a nonmigratory offline schedule using 6m unit-speed processors [B. Kalyanasundaram and K. R. Pruhs, J. Algorithms, 38 (2001), pp. 2--24] or a migratory online schedule using m speed-2 processors [C. A. Phillips et al., Algorithmica, 32 (2002), pp. 163--200]. Our result is based on a simple conservative scheduling algorithm called PARK, which commits a processor to a job only when the processor has zero commitment before its deadline. A careful analysis of PARK further shows that the processor speed can be reduced arbitrarily close to 1 by exploiting more processors (say, using 16m speed-1.8 processors). PARK also finds application in overloaded systems; it gives the first online nonmigratory algorithm that can exploit moderately faster processors to match the performance of any migratory offline algorithm.

On Even Triangulations of 2-Connected Embedded Graphs

Huaming Zhang and Xin He

SIAM J. Comput. 34, pp. 683-696 (14 pages)

Online Publication Date: July 27, 2006

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Recently, Hoffmann and Kriegel proved an important combinatorial theorem [SIAM J. Discrete Math., 9 (1996), pp. 210--224]: Every 2-connected bipartite plane multigraph G without 2-cycle faces has a triangulation in which all vertices have even degree (this is called an even triangulation). Combined with the classical Whitney's theorem, this result implies that every such graph has a 3-colorable plane triangulation. Using this theorem, Hoffmann and Kriegel significantly improved the upper bounds of several art gallery and prison guard problems. A complicated O(n2) time algorithm was obtained in [SIAM J. Discrete Math., 9 (1996), pp. 210--224] for constructing an even triangulation of G. Hoffmann and Kriegel conjectured that there is an O(n3/2) time algorithm for solving this problem.
In this paper, we develop a simple proof of the above theorem. Our proof reveals and relies on a natural correspondence between even triangulations of G and certain orientations of G. Based on this new proof, we obtain a very simple O(n) time algorithm for finding an even triangulation of G. We also extend our proof to show the existence of even triangulations for similar graphs on high genus surface.

Fault-Tolerant Scheduling

Bala Kalyanasundaram and Kirk R. Pruhs

SIAM J. Comput. 34, pp. 697-719 (23 pages) | Cited 2 times

Online Publication Date: July 27, 2006

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We study fault-tolerant multiprocessor scheduling under the realistic assumption that the occurrence of faults cannot be predicted. The goal in these problems is to minimize the delay incurred by the jobs. Since this is an online problem we use competitive analysis to evaluate possible algorithms. For the problems of minimizing the makespan and minimizing the average completion time (for static release times), we give nonclairvoyant algorithms (both deterministic and randomized) that have provably asymptotically optimal competitive ratios. The main tool used by these algorithms to combat faults is redundancy. We also show that randomization has the same effect as redundancy.

Classifying the Complexity of Constraints Using Finite Algebras

Andrei Bulatov, Peter Jeavons, and Andrei Krokhin

SIAM J. Comput. 34, pp. 720-742 (23 pages) | Cited 39 times

Online Publication Date: July 27, 2006

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Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NP-complete in general, but certain restrictions on the form of the constraints can ensure tractability. Here we show that any set of relations used to specify the allowed forms of constraints can be associated with a finite universal algebra and we explore how the computational complexity of the corresponding constraint satisfaction problem is connected to the properties of this algebra. Hence, we completely translate the problem of classifying the complexity of restricted constraint satisfaction problems into the language of universal algebra.
We introduce a notion of "tractable algebra," and investigate how the tractability of an algebra relates to the tractability of the smaller algebras which may be derived from it, including its subalgebras and homomorphic images. This allows us to reduce significantly the types of algebras which need to be classified. Using our results we also show that if the decision problem associated with a given collection of constraint types can be solved efficiently, then so can the corresponding search problem. We then classify all finite strictly simple surjective algebras with respect to tractability, obtaining a dichotomy theorem which generalizes Schaefer's dichotomy for the generalized satisfiability problem. Finally, we suggest a possible general algebraic criterion for distinguishing the tractable and intractable cases of the constraint satisfaction problem.

Online Scheduling of Precedence Constrained Tasks

Yumei Huo and Joseph Y. T. Leung

SIAM J. Comput. 34, pp. 743-762 (20 pages)

Online Publication Date: July 27, 2006

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A fundamental problem in scheduling theory is that of scheduling a set of n tasks, with precedence constraints, on $m \ge 1$ identical and parallel processors so as to minimize the makespan (schedule length). In the past, research has focused on the setting whereby all tasks are available for processing at the beginning (i.e., at time t = 0). In this article we consider the situation where tasks, along with their precedence constraints, are released at different times, and the scheduler has to make scheduling decisions without knowledge of future releases. In other words, the scheduler has to schedule tasks in an online fashion. We consider both preemptive and nonpreemptive schedules. We show that optimal online algorithms exist for some cases, while for others it is impossible to have one. Our results give a sharp boundary delineating the possible and the impossible cases.

Primal-Dual Meets Local Search: Approximating MSTs With Nonuniform Degree Bounds

J. Könemann and R. Ravi

SIAM J. Comput. 34, pp. 763-773 (11 pages) | Cited 3 times

Online Publication Date: July 27, 2006

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We present a new bicriteria approximation algorithm for the degree-bounded minimum-cost spanning tree (MST) problem: Given an undirected graph with nonnegative edge weights and a degree bound B, find a spanning tree of maximum node-degree B and minimum total edge-cost. Our algorithm outputs a tree of maximum degree at most a constant times B and total edge-cost at most a constant times that of a minimum-cost degree-B-bounded spanning tree.
While our new algorithm is based on ideas from Lagrangian relaxation, as is our previous work [SIAM J. Comput., 31 (2002), pp. 1783--1793], it does not rely on computing a solution to a linear program. Instead, it uses a repeated application of Kruskal's MST algorithm interleaved with a combinatorial update of approximate Lagrangian node-multipliers maintained by the algorithm. These updates cause subsequent repetitions of the spanning tree algorithm to run for longer and longer times, leading to overall progress and a proof of the performance guarantee. A second useful feature of our algorithm is that it can handle nonuniform degree bounds on the nodes: Given distinct bounds Bv for every node $v \in V$, the output tree has degree at most O(Bv + log|V|) for every $v \in V$. As before, the cost of the tree is at most a constant times that of a minimum-cost tree obeying all degree bounds.
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