Online Algorithms

Investigators: Spyros Angelopoulos, Ho-Leung Chan, Nicole Megow, Pascal Schweitzer, and Rob van Stee

Stochastic Online Scheduling with Precedence Constraints. We consider the preemp-tive and non-preemppreemp-tive problems of scheduling jobs with precedence constraints on parallel machines with the objective to minimize the sum of (weighted) completion times. We inves-tigate an online model in which the scheduler learns about a job when all its predecessors have completed. For scheduling on a single machine, we present in [16] online algorithms with matching lower and upper bounds of Θ(n) and Θ(√

n) for jobs with general and equal weights, respectively. We also derive corresponding results on parallel identical machines.

Our result for arbitrary job weights holds even in the more general stochastic online scheduling model where, in addition to the limited information about the job set, processing times are uncertain. For a large class of processing time distributions, we derive also an improved performance guarantee if weights are equal.

Online Unit Clustering. We continue the study of the online unit clustering problem [13], introduced by Chan and Zarrabi-Zadeh [10]. We design a deterministic algorithm with a competitive ratio of 7/4 for the one-dimensional case. This is the first deterministic algo-rithm that beats the bound of 2. It also has a better competitive ratio than the previous randomized algorithms. Moreover, we provide the first non-trivial deterministic lower bound (1.6), improve the randomized lower bound to 1.5, and prove the first lower bounds for higher dimensions.

We also study several variants and generalizations of the online unit clustering problem, which are inspired by variants of packing and scheduling problems in the literature [12].

In particular, we present optimal online algorithms for clustering with rejection, weighted clustering, and clustering with temporary request points. In addition, we present algorithms and lower bounds for clustering with cardinality constraints and clustering with resource augmentation.

The Online Steiner Tree Problem in Directed Graphs. Steiner tree problems occupy a central place in the area of approximation and online algorithms. Apart from their theoretical significance, Steiner tree formulations are also useful in modeling multicast communication in networks. Most of the existing research on Steiner trees assumes undirected graphs. In contrast, however, typical communication networks consist of links asymmetric in the char-acteristics of their antiparallel links (such as bandwidth and latency), and thus are better modeled by directed graphs.

The above considerations motivated the definition of edge asymmetry α, originally due to Ramanathan [17], as the maximum ratio of the weight of antiparallel links in the graph. Ra-manathan’s work is focused on approximation algorithms (i.e., off-line algorithms). Faloutsos et gal. [14], followed by Angelopoulos [1] considered the online variant of the Steiner tree problem in graphs of bounded asymmetry.

In recent work [2], we improved the upper bound on the competitive ratio of a simple greedy algorithm to O(minn

maxn

α^{log k}_{log α}, α_{log log k}^{log k} o , ko

), where k is the number of terminals.

Since [14] and [1] imply a lower bound of Ω algorithms, this bound is near-tight. The analysis is based on identifying “hard” input graphs (i.e., graphs on which the algorithm does not perform well), and which, surprisingly, have a relatively simple structure.

Bijective and Average Analysis of Online Algorithms. It has long been known that for some fundamental online problems, competitive analysis is not consistent with empirical evaluation. The most notable example is the paging problem: there exists a vast class of algorithms, ranging from extremely naive and inefficient strategies (such as Flush-When-Full) to strategies of excellent performance in practice (such as Least-Recently-Used (LRU)) all of which attain the same competitive ratio. A similar situation arises in the list update problem:

in particular, under a natural cost formulation, all algorithms have the same asymptotic, non-constant competitive ratio.

In order to bridge this gap between theoretical analysis and empirical performance, An-gelopoulos, Dorrigiv and L´opez-Ortiz [3] introduced bijective analysis as a natural and intu-itive framework for comparing the performance of paging strategies. Given two algorithms A

and B, denote by A(σ) and B(σ) the cost incurred by the algorithms on a request sequence σ. Let also In denote the set of all request sequences of size n. We say that A is no worse than B according to bijective analysis, if for all n ≥ n0 (for some constant n0), there exists a permutation π : I_n→ I_n such that A(σ) ≤ B(π(σ)). A relaxed version of bijective analysis is average analysis in which one compares the average cost incurred by two algorithms over all request sequences of the same size.

In [4] the same authors extended the results of [3] to the list update problem. More specif-ically [4] shows that any two (natural) algorithms for list update are equivalent according to bijective analysis. The central result in this work shows that the Move-to-Front algorithm (MTF) is superior to all other algorithms according to average analysis, once locality of reference is considered. To our knowledge, this was the first study of the effect of locality of reference in list update.

In [5], Angelopoulos and Schweitzer extended the results of [3] and [4] using the more powerful technique of bijective analysis. More precisely, we showed that LRU and MTF are the unique optimal online algorithms at the presence of locality of reference. This establishes a theoretical justification of the superiority of these two algorithms, in a strong sense.

Weighted Flow Time. We considered the classic online scheduling problem of minimizing the total weighted flow time on a single machine with preemptions. Here, each job j has an arbitrary arrival time r_j, weight w_j and size p_j, and given a schedule its flow time is defined as the duration of time since its arrival until it completes its service requirement.

The first non-trivial algorithms with poly-logarithmic competitive ratio for this problem were obtained relatively recently [11, 8], and it was widely believed that the problem admits a constant factor competitive algorithm. In [6], we showed an ω(1) lower bound on the competitive ratio of any deterministic online algorithm. Our result was based on a gap amplification technique for online algorithms. Starting with a trivial lower bound of 1, we gave a procedure to improve the lower bound sequentially, while ensuring at each step that the size of the instance increases relatively modestly. Future work includes closing the gap between the upper and lower bounds and investigating the possibility of O(1)-competitive randomized algorithms.

Energy Efficient Scheduling. Energy consumption has become a key issue in the design of microprocessors. Major chip manufacturers, such as Intel, AMD and IBM, now produce chips with dynamically scalable speeds, and produce associated software that enables an operating system to manage power by scaling processor speed. Within the last few years there has been a significant amount of research on the scheduling problems that arise in this setting (see, e.g., [15] for a survey).

All of the theoretical speed scaling research to date has assumed that the power function, which expresses the power consumption P as a function of the processor speed s, is of the form P = s^α, where α > 1 is some constant. Motivated in part by technological advances, in [7], we initiated the study of speed scaling with arbitrary power functions. We considered the problem of minimizing the total flow time plus energy. Our main result was a (3 + )-competitive algorithm for this problem, that holds for essentially any power function. We also gave a (2 + )-competitive algorithm for the objective of fractional weighted flow plus

energy. Even for power functions of the form s^α, it was not previously known how to obtain competitiveness independent of α for these problems. We also introduced a model of allowable speeds that generalizes all known models in the literature.

We then studied online non-clairvoyant speed scaling to minimize total flow time plus energy. By non-clairvoyant, it means the size of a job is not known until it is completed. In [9], we first considered the traditional model where the power function is P (s) = s^α. We gave a non-clairvoyant algorithm that was shown to be O(α³)-competitive. We then showed an Ω(α^1/3−) lower bound on the competitive ratio of any non-clairvoyant algorithm. We also showed that there are power functions for which no non-clairvoyant algorithm can be O(1)-competitive. Note that modern processors usually have multicores and large scale server farms can have millions of processors. The most interesting future work is to extend the study of energy efficient scheduling to the multiprocessor setting.

References

• [1] S. Angelopoulos. Improved bounds for the online steiner tree problem in graphs of bounded edge-asymmetry. In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, New Orleans, Louisiana, USA, 2007, pp. 248–257. SIAM.

• [2] S. Angelopoulos. A near-tight bound for the online steiner tree problem in graphs of bounded asymmetry. In 16th Annual European Symposium on Algorithms (ESA 2008), Karlsruhe, 2008, LNCS 5193, pp. 76–87. Springer.

• [3] S. Angelopoulos, R. Dorrigiv, and A. L´opez-Ortiz. On the separation and equivalence of pag-ing strategies. In Proceedpag-ings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, New Orleans, Louisiana, USA, 2007, pp. 229–237. SIAM.

• [4] S. Angelopoulos, R. Dorrigiv, and A. Lopez-Ortiz. List update with locality of reference. In 8th Latin American Symposium on Theoretical Informatics (LATIN 2008), Buzios, Brasil, 2008, LNCS 4957, pp. 399–410. Springer.

• [5] S. Angelopoulos and P. Schweitzer. Paging and list update under bijective analysis. In C. Math-ieu, ed., 20th ACM-SIAM Symposium on Discrete Algorithms (SODA 2009), New York, 2009, pp. 1136–1145. ACM press.

• [6] N. Bansal and H.-L. Chan. Weighted flow time does not admit o(1)-competitive algorithms.

In C. Mathieu, ed., Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), New York, USA, 2009, pp. 1238–1244. ACM Press.

• [7] N. Bansal, H.-L. Chan, and K. Pruhs. Speed scaling with an arbitrary power function. In C. Mathieu, ed., Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Al-gorithms (SODA), New York, USA, 2009, pp. 693–701. ACM Press.

[8] N. Bansal and K. Dhamdhere. Minimizing weighted flow time. ACM Transactions on Algorithms, (SODA 2002 and 2003 special issue), 3(4), 2007.

• [9] H.-L. Chan, J. Edmonds, T.-W. Lam, L.-K. Lee, A. Marchetti-Spaccamela, and K. Pruhs. Non-clairvoyant speed scaling for flow and energy. In S. Albers and J.-Y. Marion, eds., Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS), Freiburg, Germany, 2009, pp. 255–264. IBFI.

[10] T. M. Chan and H. Zarrabi-Zadeh. A randomized algorithm for onine unit clustering. In Proc.

4th Workshop on Approximation and Online Algorithms (WAOA 2006), 2007, LNCS 4368, pp.

121–131. To appear in Theory of Computing Systems (doi:10.1007/s00224-007-9085-7).

[11] C. Chekuri, S. Khanna, and A. Zhu. Algorithms for minimizing weighted flow time. In Proceed-ings on 33rd Annual ACM Symposium on Theory of Computing (STOC), 2001, pp. 84–93.

• [12] L. Epstein, A. Levin, and R. van Stee. Online unit clustering: Variations on a theme. Theoretical Computer Science, 407(1-3):85–96, 2008.

• [13] L. Epstein and R. van Stee. On the online unit clustering problem. ACM Transactions on Algorithms, 2009. To appear.

[14] M. Faloutsos, R.Pankaj, and K. C. Sevcik. The effect of asymmetry on the on-line multicast routing problem. Int. J. Foundations of Computer Science, 13(6):889–910, 2002.

[15] S. Irani and K. R. Pruhs. Algorithmic problems in power management. SIGACT News, 36(2):63–

76, 2005.

[16] N. Megow and T. Vredeveld. Stochastic online scheduling with precedence constraints. Submit-ted, 2008.

[17] S. Ramanathan. Multicast tree generation in networks with asymmetric links. IEEE/ACM Transactions on Networking, 4(4):558–568, 1996.