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1998

Volume 27, Issue 6, pp. 1515-1811


On Learning Read-k-Satisfy-j DNF

Howard Aizenstein, Avrim Blum, Roni Khardon, Eyal Kushilevitz, Leonard Pitt, and Dan Roth

SIAM J. Comput. 27, pp. 1515-1530 (16 pages) | Cited 3 times

Online Publication Date: July 28, 2006

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We study the learnability of read-k-satisfy-j (RkSj) DNF formulas. These are boolean formulas in disjunctive normal form (DNF), in which the maximum number of occurrences of a variable is bounded by k, and the number of terms satisfied by any assignment is at most j. After motivating the investigation of this class of DNF formulas, we present an algorithm that for any unknown RkSj DNF formula to be learned, with high probability finds a logically equivalent DNF formula using the well-studied protocol of equivalence and membership queries. The algorithm runs in polynomial time for $k\cdot j=O({\log n\over\log\log n})$, where n is the number of input variables.

Log-Space Polynomial End-to-End Communication

Eyal Kushilevitz, Rafail Ostrovsky, and Adi Rosén

SIAM J. Comput. 27, pp. 1531-1549 (19 pages)

Online Publication Date: July 28, 2006

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Communication between processors is the essence of distributed computing: clearly, without communication, distributed computation is impossible. However, as networks become larger and larger, the frequency of link failures increases. The end-to-end communication problem asks how to efficiently carry out fault-free communication between two processors over a network, in spite of such frequent link failures. The sole minimum assumption is that the two processors that are trying to communicate are not permanently disconnected (i.e., the communication should proceed even when there does not (ever) simultaneously exist an operational path between the two processors that are trying to communicate).
We present a protocol to solve the end-to-end problem with logarithmic-space and polynomial communication at the same time. This is an exponential memory improvement to all previous polynomial communication solutions. That is, all previous polynomial communication solutions needed at least linear (in n, the size of the network) amount of memory per link.
Our protocol transfers packets over the network, maintains a simple-to-compute O(log n)-bits potential function at each link in order to perform routing, and uses a novel technique of packet canceling which allows us to keep only one packet per link. The computations of both our potential function and our packet-canceling policy are totally local in nature.

Lower Bounds for Randomized Mutual Exclusion

Eyal Kushilevitz, Yishay Mansour, Michael O. Rabin, and David Zuckerman

SIAM J. Comput. 27, pp. 1550-1563 (14 pages)

Online Publication Date: July 28, 2006

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We establish, for the first time, lower bounds for randomized mutual exclusion algorithms (with a read-modify-write operation). Our main result is that a constant-size shared variable cannot guarantee strong fairness, even if randomization is allowed. In fact, we prove a lower bound of $\Omega (\log\log n)$ bits on the size of the shared variable, which is also tight.
We investigate weaker fairness conditions and derive tight (upper and lower) bounds for them as well. Surprisingly, it turns out that slightly weakening the fairness condition results in an exponential reduction in the size of the required shared variable. Our lower bounds rely on an analysis of Markov chains that may be of interest on its own and may have applications elsewhere.

Adaptive Heuristics for Binary Search Trees and Constant Linkage Cost

Tony W. Lai and Derick Wood

SIAM J. Comput. 27, pp. 1564-1591 (28 pages) | Cited 1 time

Online Publication Date: July 28, 2006

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We present lower and upper bounds on adaptive heuristics for maintaining binary search trees using a constant number of link or pointer changes for each operation (constant linkage cost (CLC)). We show that no adaptive heuristic with an amortized linkage cost of o(log n) can be competitive. In particular, we show that any heuristic that performs f(n)=o(log n) promotions (rotations) amortized over each access has a competitive ratio of at least $\Omega(\log n/f(n))$ against an oblivious adversary, and any heuristic that performs f(n)=o(log n) pointer changes amortized over each access has a competitive ratio of at least $\Omega(\frac{\log n}{f(n)\log(\log n/f(n))})$ against an adaptive online adversary.
In our investigation of upper bounds we present four adaptive heuristics: a randomized, worst-case-CLC heuristic randomized two-promotion (R2P) whose expected search time is within a constant factor of the search time using an optimal tree; that is, it is statically competitive against an oblivious adversary; a randomized, expected-CLC heuristic (locally optimized randomized partial splay (LORPS)) that has O(log n) expected-amortized update time and is statically competitive against an oblivious adversary; a deterministic, amortized-CLC heuristic (locally optimized partial splay (LOPS)) that has O(log n) amortized update time and is statically competitive against an adaptive adversary; a practical, randomized heuristic (randomized partial splay (RPS)) that is not CLC but has performance bounds comparable with those of the splay heuristic of Sleator and Tarjan; it is statically competitive against an adaptive adversary.
The randomized heuristics use only constant extra space, whereas the deterministic heuristic uses O(n) extra space.

Tree Contractions and Evolutionary Trees

Ming-Yang Kao

SIAM J. Comput. 27, pp. 1592-1616 (25 pages) | Cited 4 times

Online Publication Date: July 28, 2006

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An evolutionary tree is a rooted tree where each internal vertex has at least two children and where the leaves are labeled with distinct symbols representing species. Evolutionary trees are useful for modeling the evolutionary history of species. An agreement subtree of two evolutionary trees is an evolutionary tree which is also a topological subtree of the two given trees. We give an algorithm to determine the largest possible number of leaves in any agreement subtree of two trees T1 and T2 with n leaves each. If the maximum degree d of these trees is bounded by a constant, the time complexity is O(n log2n) and is within a log n factor of optimal. For general d, this algorithm runs in O(nd2 log d log2n) time or alternatively in $O(ndsqrt{d}\log^3{n})$ time.

Optimal Prediction for Prefetching in the Worst Case

P. Krishnan and Jeffrey Scott Vitter

SIAM J. Comput. 27, pp. 1617-1636 (20 pages) | Cited 3 times

Online Publication Date: July 28, 2006

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Response time delays caused by I/O are a major problem in many systems and database applications. Prefetching and cache replacement methods are attracting renewed attention because of their success in avoiding costly I/Os. Prefetching can be looked upon as a type of online sequential prediction, where the predictions must be accurate as well as made in a computationally efficient way. Unlike other online problems, prefetching cannot admit a competitive analysis, since the optimal offline prefetcher incurs no cost when it knows the future page requests. Previous analytical work on prefetching [. Vitter Krishnan 1991.] [J. Assoc. Comput. Mach., 143 (1996), pp. 771--793] consisted of modeling the user as a probabilistic Markov source.
In this paper, we look at the much stronger form of worst-case analysis and derive a randomized algorithm for pure prefetching. We compare our algorithm for every page request sequence with the important class of finite state prefetchers, making no assumptions as to how the sequence of page requests is generated. We prove analytically that the fault rate of our online prefetching algorithm converges almost surely for every page request sequence to the fault rate of the optimal finite state prefetcher for the sequence. This analysis model can be looked upon as a generalization of the competitive framework, in that it compares an online algorithm in a worst-case manner over all sequences with a powerful yet nonclairvoyant opponent. We simultaneously achieve the computational goal of implementing our prefetcher in optimal constant expected time per prefetched page using the optimal dynamic discrete random variate generator of [. Matias Matias, Vitter, and Ni [Proc. 4th Annual SIAM/ACM Symposium on Discrete Algorithms, Austin, TX, January 1993].

A Correctness Condition for High-Performance Multiprocessors

Hagit Attiya and Roy Friedman

SIAM J. Comput. 27, pp. 1637-1670 (34 pages) | Cited 2 times

Online Publication Date: July 28, 2006

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Hybrid consistency, a consistency condition for shared memory multiprocessors, attempts to capture the guarantees provided by contemporary high-performance architectures. It combines the expressiveness of strong consistency conditions (e.g., sequential consistency, linearizability) and the efficiency of weak consistency conditions (e.g., pipelined RAM, causal memory). Memory access operations are classified as either strong or weak. A global ordering of strong operations at different processes is guaranteed, but there is very little guarantee on the ordering of weak operations at different processes, except for what is implied by their interleaving with the strong operations. A formal and precise definition of this condition is given and an algorithm for providing hybrid consistency on distributed memory machines is presented. The response time of the algorithm is proved to be within a constant multiplicative factor of the (theoretical) optimal time bounds.

Efficient Algorithms for the Domination Problems on Interval and Circular-Arc Graphs

Maw-Shang Chang

SIAM J. Comput. 27, pp. 1671-1694 (24 pages) | Cited 6 times

Online Publication Date: July 28, 2006

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This paper first presents a unified approach to design efficient algorithms for the weighted domination problem and its three variants, i.e., the weighted independent, connected, and total domination problems, on interval graphs. Given an interval model with endpoints sorted, these algorithms run in time O(n) or O(n log log n) where n is the number of vertices. The results are then extended to solve the same problems on circular-arc graphs in O(n + m) time where m is the number of edges of the input graph.

Computing the Local Consensus of Trees

Sampath Kannan, Tandy Warnow, and Shibu Yooseph

SIAM J. Comput. 27, pp. 1695-1724 (30 pages) | Cited 1 time

Online Publication Date: July 28, 2006

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The inference of consensus from a set of evolutionary trees is a fundamental problem in a number of fields such as biology and historical linguistics, and many models for inferring this consensus have been proposed. In this paper we present a model for deriving what we call a local consensus treeT from a set of trees ${\cal T}$. The model we propose presumes a function f, called a total local consensus function, which determines for every triple A of species, the form that the local consensus tree should take on A. We show that all local consensus trees, when they exist, can be constructed in polynomial time and that many fundamental problems can be solved in linear time. We also consider partial local consensus functions and study optimization problems under this model. We present linear time algorithms for several variations. Finally we point out that the local consensus approach ties together many previous approaches to constructing consensus trees.

Parallel Algorithms with Optimal Speedup for Bounded Treewidth

Hans L. Bodlaender and Torben Hagerup

SIAM J. Comput. 27, pp. 1725-1746 (22 pages) | Cited 4 times

Online Publication Date: July 28, 2006

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We describe the first parallel algorithm with optimal speedup for constructing minimum-width tree decompositions of graphs of bounded treewidth. On n-vertex input graphs, the algorithm works in O((log n)2) time using O(n) operations on the EREW PRAM. We also give faster parallel algorithms with optimal speedup for the problem of deciding whether the treewidth of an input graph is bounded by a given constant and for a variety of problems on graphs of bounded treewidth, including all decision problems expressible in monadic second-order logic. On n-vertex input graphs, the algorithms use O(n) operations together with O(log n log*n) time on the EREW PRAM, or O(log n) time on the CRCW PRAM.

First-Order Queries on Finite Structures Over the Reals

Jan Paredaens, Jan Van den Bussche, and Dirk Van Gucht

SIAM J. Comput. 27, pp. 1747-1763 (17 pages) | Cited 2 times

Online Publication Date: July 28, 2006

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We investigate properties of finite relational structures over the reals expressed by first-order sentences whose predicates are the relations of the structure plus arbitrary polynomial inequalities, and whose quantifiers can range over the whole set of reals. In constraint programming terminology, this corresponds to Boolean real polynomial constraint queries on finite structures. The fact that quantifiers range over all reals seems crucial; however, we observe that each sentence in the first-order theory of the reals can be evaluated by letting each quantifier range over only a finite set of real numbers without changing its truth value. Inspired by this observation, we then show that when all polynomials used are linear, each query can be expressed uniformly on all finite structures by a sentence of which the quantifiers range only over the finite domain of the structure. In other words, linear constraint programming on finite structures can be reduced to ordinary query evaluation as usual in finite model theory and databases. Moreover, if only "generic" queries are taken into consideration, we show that this can be reduced even further by proving that such queries can be expressed by sentences using as polynomial inequalities only those of the simple form x < y.

Spirality and Optimal Orthogonal Drawings

Giuseppe Di Battista, Giuseppe Liotta, and Francesco Vargiu

SIAM J. Comput. 27, pp. 1764-1811 (48 pages) | Cited 8 times

Online Publication Date: July 28, 2006

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We deal with the problem of constructing the orthogonal drawing of a graph with the minimum number of bends along the edges. The problem has been recently shown to be NP-complete in the general case. In this paper we introduce and study the new concept of spirality, which is a measure of how an orthogonal drawing is "rolled up," and develop a theory on the interplay between spirality and number of bends of orthogonal drawings. We exploit this theory to present polynomial time algorithms for two significant classes of graphs: series-parallel graphs and 3-planar graphs. Series-parallel graphs arise in a variety ofproblems such as scheduling, electrical networks, data-flow analysis, database logic programs, and circuit layout. Also, they play a central role in planarity problems. Furthermore, drawings of 3-planar graphs are a classical field of investigation.
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