John Noga

Competitive analysis of randomized paging algorithms

The paging problem is defined as follows: we are given a two-level memory system, in which one level is a fast memory, called cache, capable of holding k items, and the second level is an unbounded but slow memory. At each given time step, a request to an item is issued. Given a request to an item p, a miss occurs if p is not present in the fast memory. In response to a miss, we need to choose an item q in the cache and replace it by p. The choice of q needs to be made on-line, without the knowledge of future requests. The objective is to design a replacement strategy with a small number of misses. In this paper we use competitive analysis to study the performance of randomized on-line paging algorithms. Our goal is to show how the concept of work functions, used previously mostly for the analysis of deterministic algorithms, can also be applied, in a systematic fashion, to the randomized case. We present two results: we first show that the competitive ratio of the marking algorithm is exactly 2Hk−1. Previously, it was known to be between Hk and 2Hk. Then we provide a new, Hk-competitive algorithm for paging. Our algorithm, as well as its analysis, is simpler than the known algorithm by McGeoch and Sleator. Another advantage of our algorithm is that it can be implemented with complexity bounds independent of the number of past requests: O(k2logk) memory and O(k2) time per request.

LRU is better than FIFO

Abstract. In the paging problem we have to manage a two-level memory system, in which the first level has short access time but can hold only up to k pages, while the second level is very large but slow. We use competitive analysis to study the relative performance of the two best known algorithms for paging, LRU and FIFO. Sleator and Tarjan proved that the competitive ratio of LRU and FIFO is k . In practice, however, LRU is known to perform much better than FIFO. It is believed that the superiority of LRU can be attributed to locality of reference exhibited in request sequences. In order to study this phenomenon, Borodin et al. [2] refined the competitive approach by introducing the concept of access graphs. They conjectured that the competitive ratio of LRU on each access graph is less than or equal to the competitive ratio of FIFO. We prove this conjecture in this paper.

On-line scheduling of unit time jobs with rejection: minimizing the total completion time

We consider on-line scheduling of unit time jobs on a single machine with job-dependent penalties. The jobs arrive on-line (one by one) and can be either accepted and scheduled, or be rejected at the cost of a penalty. The objective is to minimize the total completion time of the accepted jobs plus the sum of the penalties of the rejected jobs. We give an on-line algorithm for this problem with competitive ratio 12(2+3)~1.86602. Moreover, we prove that there does not exist an on-line algorithm with competitive ratio better than 1.63784.

An online partially fractional knapsack problem

The knapsack problem can and has been used to model many resource sharing problems. The allocation of a portion of a resource to a particular agent provides a benefit to the system, but also blocks other agents from utilizing that portion of the resource. For a problem where the number of agents as well as each agents demand and potential benefit are known prior to any decision being made, the optimal allocation and its value can be calculated. In many situations these values are not known initially, but only learned over time. Online algorithms and competitive analysis are often employed when a problem requires decisions to be made prior to having all information available. In this paper we suggest an online version of the knapsack problem, provide some justification for the model, give the exact competitive ratio for the problem in the deterministic case, and provide bounds on the competitive ratio in the randomized case.

Preemptive scheduling in overloaded systems

The following scheduling problem is studied: We are given a set of tasks with release times, deadlines, and profit rates. The objective is to determine a 1-processor preemptive schedule of the given tasks that maximizes the overall profit. In the standard model, each completed task brings profit, while noncompleted tasks do not. In the metered model, a task brings profit proportional to the execution time even if not completed. For the metered task model, we present an efficient offline algorithm and improve both the lower and upper bounds on the competitive ratio of online algorithms. Furthermore, we prove three lower bound results concerning resource augmentation in both models.

Scheduling with Machine Cost

For most scheduling problems the set of machines is fixed initially and remains unchanged for the duration of the problem. We consider two basic online scheduling problems with the modification that initially the algorithm possesses no machines, but that at any point additional machines may be purchased. Upper and lower bounds on the competitive ratio are shown for both problems.

An optimal online algorithm for scheduling two machines with release times

We present a deterministic online algorithm for scheduling two parallel machines when jobs arrive over time and show that it is (1/2) (5−v5) 1.38198-competitive. The best previously known algorithm is (3/2)-competitive. Our upper bound matches a previously known lower bound, and thus our algorithm has the best possible competitive ratio. We also present a lower bound of 1.21207 on the competitive ratio of any randomized online algorithm for any number of machines. This improves a previous result of 4 − 2v2 ?1.17157. Copyright 2001 Elsevier Science B.V.

The Buffer Minimization Problem for Multiprocessor Scheduling with Conflicts

We consider the problem of scheduling a sequence of tasks in a multi-processor system with conflicts. Conflicting processors cannot process tasks at the same time. At certain times new tasks arrive in the system, where each task specifies the amount of work (processing time) added to each processors workload. Each processor stores this workload in its input buffer. Our objective is to schedule task execution, obeying the conflict constraints, and minimizing the maximum buffer size of all processors. In the off-line case, we prove that, unless P = NP, the problem does not have a polynomial-time algorithm with a polynomial approximation ratio. In the on-line case, we provide the following results: (i) a competitive algorithm for general graphs, (ii) tight bounds on the competitive ratios for cliques and complete k-partite graphs, and (iii) a (Δ/2 + 1)-competitive algorithm for trees, where Δ is the diameter. We also provide some results for small graphs with up to 4 vertices.

Preemptive Scheduling in Overloaded Systems

The following scheduling problem is studied: We are given a set of tasks withrelease times, deadlines, and profit rates. The objective is to determine a 1-processor preemptive schedule of the given tasks that maximizes the overall profit. In the standard model, each completed task brings profit, while non-completed tasks do not. In the metered model, a task brings profit proportional to the execution time even if not completed. For the metered task model, we present an efficient offline algorithm and improve both the lower and upper bounds on the competitive ratio of online algorithms. Furthermore, we prove three lower bound results concerning resource augmentation in both models.

Oblivious medians via online bidding

Following Mettu and Plaxton [22, 21], we study oblivious algorithms for the k-medians problem. Such an algorithm produces an incremental sequence of facility sets. We give improved algorithms, including a (24+e)-competitive deterministic polynomial algorithm and a 2e ≈ 5.44-competitive randomized non-polynomial algorithm. Our approach is similar to that of [18], which was done independently. We then consider the competitive ratio with respect to size. An algorithm is s-size-competitive if, for each k, the cost of Fk is at most the minimum cost of any set of k facilities, while the size of Fk is at most sk. We present optimally competitive algorithms for this problem. Our proofs reduce oblivious medians to the following online bidding problem: faced with some unknown threshold