On-Line File Caching

Abstract. Consider the following file caching problem: in response to a sequence of requests for files, where each file has a specified size and retrieval cost , maintain a cache of files of total size at most some specified k so as to minimize the total retrieval cost. Specifically, when a requested file is not in the cache, bring it into the cache and pay the retrieval cost, and remove other files from the cache so that the total size of files remaining in the cache is at most k . This problem generalizes previous paging and caching problems by allowing objects of arbitrary size and cost, both important attributes when caching files for world-wide-web browsers, servers, and proxies. We give a simple deterministic on-line algorithm that generalizes many well-known paging and weighted-caching strategies, including least-recently-used, first-in-first-out, flush-when-full, and the balance algorithm. On any request sequence, the total cost incurred by the algorithm is at most k/(k-h+1) times the minimum possible using a cache of size h ≤ k . For any algorithm satisfying the latter bound, we show it is also the case that for most choices of k , the retrieval cost is either insignificant or at most a constant (independent of k ) times the optimum. This helps explain why competitive ratios of many on-line paging algorithms have been typically observed to be constant in practice.     

On-Line and Off-Line Approximation Algorithms for Vector Covering Problems

This paper deals with vector covering problems in d -dimensional space. The input to a vector covering problem consists of a set X of d -dimensional vectors in [0,1] ^d . The goal is to partition X into a maximum number of parts, subject to the constraint that in every part the sum of all vectors is at least one in every coordinate. This problem is known to be NP-complete, and we are mainly interested in its on-line and off-line approximability. For the on-line version, we construct approximation algorithms with worst case guarantee arbitrarily close to 1/(2d) in d≥ 2 dimensions. This result contradicts a statement of Csirik and Frenk in [5] where it is claimed that, for d≥ 2 , no on-line algorithm can have a worst case ratio better than zero. Moreover, we prove that, for d≥ 2 , no on-line algorithm can have a worst case ratio better than 2/(2d+1) . For the off-line version, we derive polynomial time approximation algorithms with worst case guarantee Θ(1/ log d) . For d=2 , we present a very fast and very simple off-line approximation algorithm that has worst case ratio 1/2 . Moreover, we show that a method from the area of compact vector summation can be used to construct off-line approximation algorithms with worst case ratio 1/d for every d≥ 2 . Received November 1996; revised March 1997.     

On-Line Multi-Threaded Paging

Abstract. In this paper we introduce a generalization of Paging to the case where there are many threads of requests. This models situations in which the requests come from more than one independent source. Hence, apart from deciding how to serve a request, at each stage it is necessary to decide which request to serve among several possibilities. Four different on-line problems arise depending on whether we consider fairness restrictions or not, with finite or infinite input sequences. We study all of them, proving lower and upper bounds for the competitiveness of on-line algorithms. The main competitiveness results presented in this paper state that when no fairness restrictions are imposed it is possible to obtain competitive algorithms for finite and infinite inputs. On the other hand, for the fair case in general there exist no competitive algorithms. In addition, we consider three definitions of competitiveness for infinite inputs. One of them forces algorithms to behave efficiently at every finite stage, while the other two aim at comparing the algorithms' steady-state performances. A priori, the three definitions seem different. We study them and find, however, that they are essentially equivalent. This suggests that the competitiveness results that we obtain reflect the intrinsic difficulty of the problem and are not a consequence of a too strict definition of competitiveness.     

On-Line Competitive Algorithms for Call Admission in Optical Networks

We study the on-line call admission problem in optical networks. We present a general technique that allows us to reduce the problem of call admission and wavelength selection to the call admission problem. We then give randomized algorithms with logarithmic competitive ratios for specific topologies in switchless and reconfigurable optical networks. We conclude by considering full duplex communications. Received August 16, 1997; revised May 11, 1998.     

Improved Bounds for On-Line Load Balancing

We consider the following load balancing problem. Jobs arrive on-line and must be assigned to one of m machines thereby increasing the load on that machine by a certain weight. Jobs also depart on-line. The goal is to minimize the maximum load on any machine, the load being defined as the sum of the weights of the jobs assigned to the machine divided by the machine capacity. The scheduler also has the option of preempting a job and reassigning it to another machine. Whenever a job is assigned or reassigned to a machine, the on-line algorithm incurs a reassignment cost depending on the job. For arbitrary reassignment costs and identical machines, we present an on-line algorithm with a competitive ratio of 3.5981 against current load , i.e., the maximum load at any time is less than 3.5981 times the lowest achievable load at that time. Our algorithm also incurs a reassignment cost less than 6.8285 times the cost of assigning all the jobs. For arbitrary reassignment costs and related machines we present an algorithm with a competitive ratio of 32 and a reassignment factor of 79.4. We also describe algorithms with better performance guarantees for some special cases of the problem. Received August 24, 1996; revised August 6, 1997.     

Algorithms for the On-Line Travelling Salesman 1

In this paper the problem of efficiently serving a sequence of requests presented in an on-line fashion located at points of a metric space is considered. We call this problem the On-Line Travelling Salesman Problem (OLTSP). It has a variety of relevant applications in logistics and robotics. We consider two versions of the problem. In the first one the server is not required to return to the departure point after all presented requests have been served. For this problem we derive a lower bound on the competitive ratio of 2 on the real line. Besides, a 2.5 -competitive algorithm for a wide class of metric spaces, and a 7/3 -competitive algorithm for the real line are provided. For the other version of the problem, in which returning to the departure point is required, we present an optimal 2 -competitive algorithm for the above-mentioned general class of metric spaces. If in this case the metric space is the real line we present a 1.75 -competitive algorithm that compares with a \approx 1.64 lower bound. Received November 12, 1997; revised June 8, 1998.     

Distributed and On-Line Routing on Tori

Abstract. In this paper we study the problem of assigning paths to packets on N \times N tori in an on-line and distributed fashion. By on-line we mean that the routing decisions must be made without any knowledge of future requests. Being distributed is an equally important feature of our design, for such algorithms need not know the global configuration of the network in the process of routing packets. We use the technique of competitive analysis to measure the performance of our design. In addition to showing an Ω(log N) lower bound on the competitive ratio, we present both deterministic and randomized algorithms which are O(log N) competitive with respect to the maximum load (i.e., congestion ) on communication links.     

An On-Line Multi-CBR Agent Dispatching Algorithm

Case-based reasoning (CBR) is an effective and fast problem-solving methodology, which solves new problems by remembering and adaptation of past cases. With the increasing requests for useful references for all kinds of problems and from different locations, keeping a single CBR system seems to be outdated and not practical. Multi-CBR agents located in different places are of great support to fast meet these requests. In this paper, the architecture of a multi-CBR agent system is proposed, where each CBR agent locates at different places, and is assumed to have the same ability to deal with new problem independently. When requests in a request queue are coming one by one from different places, we propose a new policy of agent dispatching to satisfy the request queue. Throughout the paper, we assume that the system must solve the coming request by considering only past requests. In this context, the performance of traditional greedy algorithms is not satisfactory. We apply a new but simple approach – competitive algorithm for on-line problem (called ODAL) to determine the dispatching policy to keep comparative low cost. The corresponding on-line dispatching algorithm is proposed and the competitive ratio is given. Based on the competitive algorithm, the dispatching of multi-CBR agents is optimized.     

Competitive Optimal On-Line Leasing

Consider an on-line player who needs some equipment (e.g., a computer) for an initially unknown number of periods. At the start of each period it is determined whether the player will need the equipment during the current period and the player has two options: to pay a leasing fee c and rent the equipment for the period, or to buy it for a larger amount P . The total cost incurred by the player is the sum of all leasing fees and perhaps one purchase. The above problem, called the leasing problem (in computer science folklore it is known as the ski-rental problem), has received considerable attention in the economic literature. Using the competitive ratio as a performance measure this paper is concerned with determining the optimal competitive on-line policy for the leasing problem. For the simplest version of the leasing problem (as described above) it is known and readily derived that the optimal deterministic competitive performance is achieved by leasing for the first k-1 times and then buying, where k = P/c . This strategy pays at most 2-1/k times the optimal off-line cost. When considering alternative financial transactions one must consider their net present value. Thus, accounting for the interest rate is an essential feature of any reasonable financial model. In this paper we determine both deterministic and randomized optimal on-line leasing strategies while accounting for the interest rate factor. It is shown here, for the leasing problem, that the interest rate factor reduces the uncertainty involved. We find that the optimal deterministic competitive ratio is 1 + (1+i)(1-1/k)(1 - k(i/1+i)) , a decreasing function of the interest i (for all reasonable values of i ). For some applications, realistic values of interest rates result in relatively low competitive ratios. By using randomization the on-line player can further boost up the performance. In particular, against an oblivious adversary the on-line player can attain a strictly smaller competitive ratio of 2 - ( (k/(k-1)) ^γ - 2 )/( (k/(k-1)) ^γ -1 ) where γ = ln ( 1 - k(1 - 1/(1+i)) ) / ln(1/(1+i)) . Here again, this competitive ratio strictly decreases with i . We also study the leasing problem against a distributional adversary called ``Nature.' This adversary chooses the probability distribution of the number of leasing periods and announces this distribution before the on-line player chooses a strategy. Although at the outset this adversary appears to be weaker than the oblivious adversary, it is shown that the optimal competitive ratio against Nature equals the optimal ratio against the oblivious adversary. Received October 8, 1997; revised February 8, 1998.     

On-Line Resource Management with Application to Routing and Scheduling

We propose a framework to model on-line resource management problems based on an on-line version of positive linear programming. We consider both min cost problems and max benefit problems and propose logarithmic competitive algorithms that are optimal up to a constant factor. The proposed framework provides a general methodology that applies to a wide class of on-line problems: shop scheduling, packet routing, and in general a class of packing and assignment problems. Previously studied problems as on-line multiprocessor scheduling and on-line virtual circuit routing can also be modeled within this framework. Received December 18, 1996; revised March 2, 1997.     

On-Line Load Balancing and Network Flow

In this paper we study two problems that can be viewed as on-line games on a dynamic bipartite graph. The first problem is on-line load balancing with preemption. A centralized scheduler must assign tasks to servers, processing on-line a sequence of task arrivals and departures. Each task is restricted to run on some subset of the servers. The scheduler attempts to keep the load well-balanced. If preemptive reassignments are disallowed, Azar et al. [3] proved a lower bound of Ω(n ^1/2 ) on the ratio between the maximum load achieved by an on-line algorithm and the optimum off-line maximum load. We show that this ratio can be greatly reduced by an efficient scheduler using only a small amount of rescheduling. We then apply these ideas to network flow. Cheriyan and Hagerup [6] introduced an on-line game on a bipartite graph as a fundamental step in improving algorithms for computing the maximum flow in networks. They described a randomized strategy to play the game. King et al. [11] studied a modified version of this game, called ``node kill,' and gave a deterministic strategy. We obtain an improved deterministic algorithm for the node kill game (and hence for maximum flow) in all but the sparsest graphs. The running time achieved is O(mn log _m/n n+n ² log ^2+ε n) , compared with King et al.'s O(mn+n ^2+ε ) . These problems combine a demand for good competitive ratios with more traditional requirements of implementation efficiency. Our solutions deal with the tradeoffs between these measures. Received March 15, 1997; revised April 20, 1997.     

A new measure for the study of on-line algorithms

An accepted measure for the performance of an on-line algorithm is the competitive ratio introduced by Sleator and Tarjan. This measure is well motivated and has led to the development of a mathematical theory for on-line algorithms.We investigate the behavior of this measure with respect to memory needs and benefits of lookahead and find some counterintuitive features. We present lower bounds on the size of memory devoted to recording the past. It is also observed that the competitive ratio reflects no improvement in the performance of an on-line algorithm due to any (finite) amount of lookahead.We offer an alternative measure that exhibits a different and, in some respects, more intuitive behavior. In particular, we demonstrate the use of our new measure by analyzing the tradeoff between the amortized cost of on-line algorithms for the paging problem and the amount of lookahead available to them. We also derive on-line algorithms for theK-server problem on any bounded metric space, which, relative to the new measure, are optimal among all on-line algorithms (up to a factor of 2) and are within a factor of 2K from the optimal off-line performance.     

Simple Algorithms for the On-Line Multidimensional Dictionary and Related Problems

The on-line multidimensional dictionary problem consists of executing on-line any sequence of the following operations: INSERT(p) , DELETE(p) , and MEM-BER-SHIP(p) , where p is any (ordered) d -tuple (or string with d elements, or points in d -space where the dimensions have been ordered). We introduce a clean structure based on balanced binary search trees, which we call multidimensional balanced binary search trees, to represent the set of d -tuples. We present algorithms for each of the above operations that take O(d + log n) time, where n is the current number of d -tuples in the set, and each INSERT and DELETE operation requires no more than a constant number of rotations. Our structure requires dn words to represent the input, plus O(n) pointers and data indicating the first component where pairs of d -tuples differ. This information, which can be easily updated, enables us to test for MEMBERSHIP efficiently. Other operations that can be performed efficiently in our multidimensional balanced binary search trees are: print in lexicographic order (O(dn) time), find the (lexicographically) smallest or largest d -tuple (O( log n) time), and concatenation (O(d + log n) time). Finding the (lexicographically) k th smallest or largest d -tuple can also be implemented efficiently (O( log n) time), at the expense of adding an integer value at each node. Received June 13, 1997; revised September 3, 1998.     

On-Line Edge-Coloring with a Fixed Number of Colors

Abstract. We investigate a variant of on-line edge-coloring in which there is a fixed number of colors available and the aim is to color as many edges as possible. We prove upper and lower bounds on the performance of different classes of algorithms for the problem. Moreover, we determine the performance of two specific algorithms, First-Fit and Next-Fit . Specifically, algorithms that never reject edges that they are able to color are called fair algorithms. We consider the four combinations of fair/ not fair and deterministic/ randomized. We show that the competitive ratio of deterministic fair algorithms can vary only between approximately 0.4641 and 1/2, and that Next-Fit is worst possible among fair algorithms. Moreover, we show that no algorithm is better than 4/7-competitive. If the graphs are all k -colorable, any fair algorithm is at least 1/2-competitive. Again, this performance is matched by Next-Fit while the competitive ratio for First-Fit is shown to be k/(2k-1) , which is significantly better, as long as k is not too large.     

New lower bounds for online k-server routing problems

In a k-server routing problem k?1 servers move in a metric space in order to visit specified points or carry objects from sources to destinations. In the online version requests arrive online while the servers are traveling. Two classical objective functions are to minimize the makespan, i.e., the time when the last server has completed its tour (k-Traveling Salesman Problem or k-tsp) and to minimize the sum of completion times (k-Traveling Repairman Problem or k-trp). Both problems, the k-tsp and the k-trp have been studied from a competitive analysis point of view, where the cost of an online algorithm is compared to that of an optimal offline algorithm. However, the gap between the obtained competitive ratios and the corresponding lower bounds have mostly been quite large for k>1, in particular for randomized algorithms against an oblivious adversary.We reduce a number of gaps by providing new lower bounds for randomized algorithms. The most dramatic improvement is in the lower bound for the k-Dial-a-Ride-Problem (the k-trp when objects need to be carried) from to 3 which is currently also the best lower bound for deterministic algorithms.     

The orthogonal CNN problem

The online CNN problem had no known competitive algorithms for a long time. Sitters, Stougie and de Paepe showed that there exists a competitive online algorithm for this problem. However, both their algorithm and analysis are quite complicated, and above all, their upper bound for the competitive ratio is 10⁵. In this paper, we examine why this problem seems so difficult. To this end we introduce a nontrivial restriction, orthogonality, against this problem and show that it decreases the competitive ratio dramatically, down to at most 9.     

Competitive randomized algorithms for nonuniform problems

Competitive analysis is concerned with comparing the performance of on-line algorithms with that of optimal off-line algorithms. In some cases randomization can lead to algorithms with improved performance ratios on worst-case sequences. In this paper we present new randomized on-line algorithms for snoopy caching and the spin-block problem. These algorithms achieve competitive ratios approachinge/(e–1) 1.58 against an oblivious adversary. These ratios are optimal and are a surprising improvement over the best possible ratio in the deterministic case, which is 2. We also consider the situation when the request sequences for these problems are generated according to an unknown probability distribution. In this case we show that deterministic algorithms that adapt to the observed request statistics also have competitive factors approachinge/(e–1). Finally, we obtain randomized algorithms for the 2-server problem on a class of isosceles triangles. These algorithms are optimal against an oblivious adversary and have competitive ratios that approache/(e–1). This compares with the ratio of 3/2 that can be achieved on an equilateral triangle.Supported in part by the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), an NSF Science and Technology Center funded under NSF Contract STC-88-09648 and supported by the New Jersey Commission on Science and Technology.     

Packing Random Intervals On-Line

Starting at time 0, unit-length intervals arrive and are placed on the positive real line by a unit-intensity Poisson process in two dimensions; the probability of an interval arriving in the time interval [t,t+ t] with its left endpoint in [y,y+ y] is t y + o( t y ). Fix x 0. An arriving interval is accepted if and only if it is contained in [0,x] and overlaps no interval already accepted. We study the number N _x (t) of intervals accepted during [0,t] . By Laplace-transform methods, we derive large-x estimates of EN _x (t) and VarN _x (t) with error terms exponentially small in x uniformly in t (0,T) , where T is any fixed positive constant. We prove that, as , EN _x (t) , VarN _x (t) , uniformly in t (0,T) , where and are given by explicit, albeit complicated formulas. Using these asymptotic estimates we show that N _x (t) satisfies a central limit theorem, i.e., for any fixed t where (0,1) is a standard normal random variable, and denotes convergence in distribution. This stochastic, on-line interval packing problem generalizes the classical parking problem, the latter corresponding only to the absorbing states of the interval packing process, where successive packed intervals are separated by gaps less than 1 in length. We verify that, as , (t) and (t) converge to _* = 0.748 . . . and _* = 0.03815 . . ., the constants of Rényi and Mackenzie for the parking problem. Thus, by comparison with the parking analysis in a single space variable, ours is a transient analysis involving both a time and a space variable. Our interval packing problem has applications similar to those of the parking problem in the physical sciences, but the primary source of our interest is the modeling of reservation systems, especially those designed for multimedia communication systems to handle high-bandwidth, real-time demands. Received December 4, 1997; revised February 15, 1998.     

On-line scheduling of parallel jobs in a list

We study an on-line parallel job scheduling problem, where jobs arrive one by one. A parallel job may require a number of machines for its processing at the same time. Upon arrival of a job, its processing time and the number of requested machines become known, and it must be scheduled immediately without any knowledge of future jobs. We present a 7-competitive on-line algorithm, which improves the previous upper bound of 12 by Johannes (J. Sched. 9:433–452, 2006). Furthermore, we investigate a special case in which the largest processing time is known beforehand. A preliminary version of this paper appeared in Proceedings of the 11th Colloquium on Structural Information and Communication Complexity (SIROCCO’04, pp. 279-290). Research of D. Ye was supported by NSFC (10601048). Research of G. Zhang was supported by NSFC (60573020).     

Optimal randomized algorithm for a generalized ski-rental with interest rate

We introduce the continuously compounded interest rate into a generalized ski-rental problem with two options: either pay some rental proportionally to the usage time (the rent option), or buy the equipment and then pay some reduced rental proportionally to the usage time (the generalized buy option). Under the framework of competitive analysis, a randomized algorithm for the modified model is constructed and then is proved optimal by Yao?s Lemma, and thus the optimal competitive ratio is obtained. The result shows that the introduction of the interest rate puts off the optimal purchase date and diminishes the uncertainty involved in the decision making.