10918731
Randomness-Optimal Unique Element Isolation with Applications to Perfect Matching and Related Problems
In this paper, we precisely characterize the randomness complexity of the unique element isolation problem, a crucial step in the $RNC$ algorithm for perfect matching due to Mulmuley, Vazirani, and Vazirani [Combinatorica, 7 (1987), pp. 105--113] and in several other applications. Given a set $S$ and an unknown family ${\cal F} \subseteq 2^S$ with $|{\cal F}| \leq Z$, we present a scheme to assign polynomially bounded weights to the elements of $S$, using only $O(\log Z + \log |S|)$ random bits, such that the minimum weight set in $\cal F$ is unique with high probability. This generalizes the solution of Mulmuley, Vazirani, and Vazirani who use $O(S\log S)$ bits, independent of $Z$. We also provide a matching lower bound for the randomness complexity of this problem. The new weight assignment scheme yields a randomness-efficient $RNC^2$ algorithm for perfect matching which uses $O(\log Z + \log n)$ random bits where $Z$ is any given upper bound on the number of perfect matchings in the input graph. This generalizes the result of Grigoriev and Karpinski [Proc. IEEE Symposium on Foundations of Computer Science, 1987, pp. 166--172], who present an $NC^3$ algorithm when $Z$ is polynomial and also improves the running time in this case. The worst-case randomness complexity of our algorithm is $O(n \log(m/n))$ random bits improving on the previous bound of $O(m \log n)$. Our scheme also gives randomness-efficient solutions for several problems where unique element isolation is used, such as $RNC$ algorithms for variants of matching and basic problems on linear matroids. We obtain a randomness-efficient random reduction from $SAT$ to $USAT$, the language of uniquely satisfiable formulas which can be derandomized in the case of languages in $FewP$ to yield new proofs of the results $FewP\subseteq \oplus P$ and $FewP\subseteq C_{\!\!=}P$.
8959226
Low-Discrepancy Sets For High-Dimensional Rectangles: A Survey
A sub-area of discrepancy theory that has received much attention in computer science recently, is that of explicit constructions of low-discrepancy point sets for various types of rectangle families in high dimension. This research has led to interesting applications in error-control coding, distributed protocols, Web document filtering, derandomization, and other areas. We give a short survey of this area here.
23028632
Mechanism design for intellectual property rights protection
Estimates for the annual size of software piracy are in the region of $15 billion. The problem is also manifest in other forms of intellectual property. The explosive growth of the Internet and other communications networks has accentuated this problem. The new infrastructure has facilitated piracy by effectively reducing the cost of copying to negligible levels. (Recently, the FBI launched an operation called “Cyber Strike,” with the objective of curbing illegal copying and distribution of software over computer networks.) A naive suggestion has been that intellectual property rights (IPR) violation is actually beneficial from the social point of view, since it decreases the monopoly power of the provider. However, the flip side of this argument is that the IPR violations decrease the incentive for innovation because they lead to lower profits. In this paper, several mechanisms for IPR protection are designed. The mechanisms involve two tools: the level of enforcement (i.e., the probability that an IPR violator will be apprehended) and the penalty to be levied on an apprehended IPR violator. These tools are analyzed by considering the tradeoff between social welfare and the provider’s profit. The paper considers a provider who wants to maximize his profit and a policy maker who wants to maximize “net” social welfare (defined as welfare less any costs of enforcement), subject to a target profit for the provider. The consumers choose from among three options: buy, copy, or stay out of the market. A game-theoretic framework is adopted, where the policy maker, the provider, and the consumers get to move in that order. Among the results, the research shows that
8868988
Contention resolution with constant expected delay
We study contention resolution in a multiple-access channel such as the Ethernet channel. In the model that we consider, n users generate messages for the channel according to a probability distribution. Raghavan and Upfal have given a protocol in which the expected delay (time to get serviced) of every message is O(log n) when messages are generated according to a Bernoulli distribution with generation rate up to about 1/10. Our main results are the following protocols: (a) one in which the expected average message delay is O(1) when messages are generated according to a Bernoulli distribution with a generation rate smaller than 1/e, and (b) one in which the expected delay of any message is O(1) for an analogous model in which users are synchronized (i.e., they agree about the time), there are potentially an infinite number of users, and messages are generated according to a Poisson distribution with generation rate up to 1/e. (Each message constitutes a new user.)
To achieve (a), we first show how to simulate (b) using n synchronized users, and then show how to build the synchronization into the protocol.
12370019
New algorithmic aspects of the Local Lemma with applications to routing and partitioning
The Lovasz local lemma (LLL)is a powerful tool that is increasingly playing a valuable role in computer science. The original lemma was nonconstructive; a breakthrough of Beck and its generalizations (due to Alon and Molloy and Reed)have led to constructive versions. However, these methods do not capture some classes of applications of the LLL. We make progress on this by providing algorithmic approaches to two families of applications of the LLL. The first provides constructive versions of certain applications of an extension of the LLL (modeling, e.g., hypergraph-partitioning and low-congestion routing problems); the second provides new algorithmic results on constructing disjoint paths in graphs. Our results can also be seen as constructive upper bounds on the integrality gap of certain packing problems. One common theme of our work is a "gradual rounding" approach.
39527744
Efficient Algorithms for the Minimum Weighted Dominating Clique Problem on Permutation Graphs
Given a graph G=(V, E) with real weights assigned to its vertices, a clique of G that also dominates its vertex set V, is called a dominating clique (DC) of G. Given a permutation graph G with all its vertices having nonnegative weights, and its permutation representation, the problem addressed in this paper is that of finding any minimum weight DC of G. We improve the existing O(|V|2) algorithm for this problem to O(|V|log|V|). The space complexity of our algorithm is O(|V|). We also present a |V| processor, O(log|V|) time, O(|V|log|V|) space parallel EREW PRAM algorithm for this problem.
15627466
Testing parallel random number generators
Monte Carlo computations are considered easy to parallelize. However, the results can be adversely affected by defects in the parallel pseudorandom number generator used. A parallel pseudorandom number generator must be tested for two types of correlations--(i) intra-stream correlation, as for any sequential generator, and (ii) inter-stream correlation for correlations between random number streams on different processes. Since bounds on these correlations are difficult to prove mathematically, large and thorough empirical tests are necessary. Many of the popular pseudorandom number generators in use today were tested when computational power was much lower, and hence they were evaluated with much smaller test sizes.This paper describes several tests of pseudorandom number generators, both statistical and application-based. We show defects in several popular generators. We describe the implementation of these tests in the SPRING [ACM Trans. Math. Software 26 (2000) 436; SPRNG--scalable parallel random number generators. SPRNG 1.0--http://www.ncsa.uiuc.edu/ Apps/SPRNG; SPRNG 2.0--http://sprng.cs.fsu.edu] test suite and also present results for the tests conducted on the SPRNG generators. These generators have passed some of the largest empirical random number tests.
5519685
Efficient scheduling of soft real-time applications on multiprocessors
In this paper, we consider fair scheduling of soft real-time applications on multiprocessors using the earliest pseudo deadline first (EPDF) Pfair algorithm. Our main contributions are twofold. First, we establish a condition for ensuring a tardiness of at most one quantum under EPDF. This condition is very liberal and should often hold in practice. Second, we present simulation results involving randomly-generated task sets, including those that do not satisfy our condition. In these experiments, deadline misses were rare, and no misses by more than one quantum ever occurred.
11619174
Provable algorithms for parallel sweep scheduling on unstructured meshes
We present provably efficient parallel algorithms for sweep scheduling on unstructured meshes. Sweep scheduling is a commonly used technique in radiation transport problems, and involves inverting an operator by iteratively sweeping across a mesh. Each sweep involves solving the operator locally at each cell. However, each direction induces a partial order in which this computation can proceed. On a distributed computing system, the goal is to schedule the computation, so that the length of the schedule is minimized. Several heuristics have been proposed for this problem; but none of the heuristics have worst case performance guarantees. We present a simple, almost linear time randomized algorithm which (provably) gives a schedule of length at most O(log/sup 2/n) times the optimal schedule for instances with n cells, when the communication cost is not considered, and a slight variant, which coupled with a much more careful analysis, gives a schedule of (expected) length O(logmlogloglogm) times the optimal schedule for m processors. These are the first such provable guarantees for this problem. We also design a priority based list schedule using these ideas, with the same theoretical guarantee, but much better performance in practice. We complement our theoretical results with extensive empirical analysis. The results show that (i) our algorithm performs very well and has significantly better performance guarantee in practice and (ii) the algorithm compares favorably with other natural and efficient parallel algorithms proposed in the literature [S. Pautz, (2002), S. Plimpton et al., (2001)].
38425944
Scheduling on Unrelated Machines Under Tree-Like Precedence Constraints
We present polylogarithmic approximations for the R|prec|Cmax and R|prec|∑jwjCj problems, when the precedence constraints are “treelike” – i.e., when the undirected graph underlying the precedences is a forest. We also obtain improved bounds for the weighted completion time and flow time for the case of chains with restricted assignment – this generalizes the job shop problem to these objective functions. We use the same lower bound of “congestion+dilation”, as in other job shop scheduling approaches. The first step in our algorithm for the R|prec|Cmax problem with treelike precedences involves using the algorithm of Lenstra, Shmoys and Tardos to obtain a processor assignment with the congestion + dilation value within a constant factor of the optimal. We then show how to generalize the random delays technique of Leighton, Maggs and Rao to the case of trees. For the weighted completion time, we show a certain type of reduction to the makespan problem, which dovetails well with the lower bound we employ for the makespan problem. For the special case of chains, we show a dependent rounding technique which leads to improved bounds on the weighted completion time and new bicriteria bounds for the flow time.
2075186
An assessment of submissions made to the Predictive Toxicology Evaluation Challenge
Constructing "good" models for chemical carcinogenesis was identified in IJCAI-97 as providing a substantial challenge to "knowledge discovery" programs. Attention was drawn to a comparative exercise which called for predictions on the outcome of 30 rodent carcinogenicity bioassays. This-the Predictive Toxicology Evaluation (or PTE) Challenge - was seen to provide AI programs with an opportunity to participate in an enterprise of scientific merit, and a yardstick for comparison against strong competition. Here we provide an assessment of the machine learning (ML) submissions made. Models submitted are assessed on: (1) their accuracy, in comparison to models developed with expert collaboration; and (2) their explanatory value for toxicology. The principal findings were: (a) using structural information available from a standard modelling package, layman-devised features, and outcomes of established biological tests, results from MLderived models were at least as good as those with expert-derived techniques. This was surprising; (b) the combined use of structural and biological features by ML-derived models was unusual, and suggested new avenues for toxicology modelling. This was also unexpected; and (c) significant effort was required to interpret the output of even the most "symbolic" of ML-derived models. Much of this could have been alleviated with measures for converting the results into a more "toxicology-friendly" form. As it stands, their absence is sufficient to prevent a whole-hearted acceptance of these promising methods by toxicologists. This suggests that ML techniques have been able to respond -not fully, but nevertheless substantially -to the PTE Challenge.