14106112
Mixed Pfair/ERfair scheduling of asynchronous periodic tasks
In this paper, we prove that a simplified variant of the PD Pfair algorithm, called PD/sup 2/, is optimal for scheduling any mix of early-release and non-early-release asynchronous tasks on a multiprocessor. This result breaks new ground by incorporating both early-release and non-early-release tasks under a common framework. In addition, all prior work on optimal multiprocessor Pfair scheduling algorithms has been limited to synchronous periodic task systems.
12321308
Efficient gossip-based aggregate computation
Recently, there has been a growing interest in gossip-based protocols that employ randomized communication to ensure robust information dissemination. In this paper, we present a novel gossip-based scheme using which all the nodes in an n-node overlay network can compute the common aggregates of MIN, MAX, SUM, AVERAGE, and RANK of their values using O(n log log n) messages within O(log n log log n) rounds of communication. To the best of our knowledge, ours is the first result that shows how to compute these aggregates with high probability using only O(n log log n) messages. In contrast, the best known gossip-based algorithm for computing these aggregates requires O(nlog n) messages and O(log n) rounds. Thus, our algorithm allows system designers to trade off a small increase in round complexity with a significant reduction in message complexity. This can lead to dramatically lower network congestion and longer node lifetimes in wireless and sensor networks, where channel bandwidth and battery life are severely constrained.
16475170
Optimal rate-based scheduling on multiprocessors
The PD2 Pfair/ERfair scheduling algorithm is the most efficient known algorithm for optimally scheduling periodic tasks on multiprocessors. In this paper, we prove that PD2 is also optimal for scheduling "rate-based" tasks whose processing steps may be highly jittered. The rate-based task model we consider generalizes the widely-studied sporadic task model.
150268
Integrating a periodic and recurrent tasks on fair-scheduled multiprocessors
We propose two server implementations for multiplexing aperiodic and recurrent real-time tasks in fair-scheduled multiprocessor systems. We also provide admission-control tests for scheduling hard aperiodic tasks. Further, we consider several complexities, most of which arise because of the parallelism available in multiprocessor systems, and present techniques to handle them. Finally, we present experimental results that demonstrate the effectiveness of our implementations.
5246711
Dependent rounding in bipartite graphs
We combine the pipage rounding technique of Ageev & Sviridenko with a recent rounding method developed by Srinivasan (2001), to develop a new randomized rounding approach for fractional vectors defined on the edge-sets of bipartite graphs. We show various ways of combining this technique with other ideas, leading to the following applications: richer random-graph models for graphs with a given degree-sequence; improved approximation algorithms for: (i) throughput-maximization in broadcast scheduling, (ii) delay-minimization in broadcast scheduling, and (iii) capacitated vertex cover; fair scheduling of jobs on unrelated parallel machines. A useful feature of our method is that it lets us prove certain (probabilistic) per-user fairness properties.
38873931
Distributions on level-sets with applications to approximation algorithms
We consider a family of distributions on fixed-weight vectors in {0, 1}/sup t/; these distributions enjoy certain negative correlation properties and also satisfy pre-specified conditions on their marginal distributions. We show the existence of such families, and present a linear-time algorithm to sample from them. This yields improved approximation algorithms for the following problems: (a) low-congestion multi-path routing; (b) maximum coverage versions of set cover; (c) partial vertex cover problems for bounded-degree graphs; and (d) the Group Steiner Tree problem. For (a) and (b), the improvement is in the approximation ratio; for (c), we show how to speedup existing approximation algorithms while preserving the best-known approximation ratio; we also improve the approximation ratio for certain families of instances of unbounded degree. For (d), we derive an approximation algorithm whose approximation guarantee is at least as good as what is known; our algorithm is shown to have a better approximation guarantee for the worst known input families for existing algorithms.
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 $FewP\subseteq C_{\!\!=}P$.
9234553
Lower Bounds on the Deterministic and Quantum Communication Complexities of Hamming-Distance Problems
Alice and Bob want to know if two strings of length n are almost equal. That is, do they differ on at mosta bits? Let 0≤a≤n–1. We show that any deterministic protocol, as well as any error-free quantum protocol (C* version), for this problem requires at least n–2 bits of communication. We show the same bounds for the problem of determining if two strings differ in exactlya bits. We also prove a lower bound of n/2–1 for error-free Q* quantum protocols. Our results are obtained by employing basic tools from combinatorics and calculus to lower-bound the ranks of the appropriate matrices.
5089320
Better approximation guarantees for job-shop scheduling
Job-shop scheduling is a classical NP-hard problem. Shmoys, Stein, and Wein presented the first polynomial-time approximation algorithm for this problem that has a good (polylogarithmic) approximation guarantee. We improve the approximation guarantee of their work and present further improvements for some important NP-hard special cases of this problem (e.g., in the preemptive case where machines can suspend work on operations and later resume). We also present NC algorithms with improved approximation guarantees for some NP-hard special cases.