A k-server problem with parallel requests and unit distances

In this paper we consider k-server problems with parallel requests where several servers can also be located on one point. We will distinguish the surplus-situation where the request can be completely fulfilled by means of the k servers and the scarcity-situation where the request cannot be completely met. We use the method of the potential function by Bartal and Grove [2] in order to prove that a corresponding Harmonic algorithm is competitive for the more general k-server problem in the case of unit distances. For this purpose we partition the set of points in relation to the online and offline servers? positions and then use detailed considerations related to sets of certain partitions.     

Online chasing problems for regular polygons

We consider a server location problem with only one server to move. In this paper we assume that a request is given as a region and that the service can be done anywhere inside the region. Namely, for each request an online algorithm chooses an arbitrary point in the region and moves the server there. Note that if every request is a single point and the server must exactly go there in the given order as conventional server problems, there is no choice for the online player and the problem is trivial. Our main result shows that if the region is a regular n-gon, the competitive ratio of the greedy algorithm is for odd n, and for even n. In particular for a square region, the greedy algorithm turns out to be optimal.     

Randomized competitive algorithms for the list update problem

We prove upper and lower bounds on the competitiveness of randomized algorithms for the list update problem of Sleator and Tarjan. We give a simple and elegant randomized algorithm that is more competitive than the best previous randomized algorithm due to Irani. Our algorithm uses randomness only during an initialization phase, and from then on runs completely deterministically. It is the first randomized competitive algorithm with this property to beat the deterministic lower bound. We generalize our approach to a model in which access costs are fixed but update costs are scaled by an arbitrary constantd. We prove lower bounds for deterministic list update algorithms and for randomized algorithms against oblivious and adaptive on-line adversaries. In particular, we show that for this problem adaptive on-line and adaptive off-line adversaries are equally powerful.A preliminary version of these results appeared in a joint paper with S. Irani in theProceedings of the 2nd Symposium on Discrete Algorithms, 1991 [17].This research was partially supported by NSF Grants CCR-8808949 and CCR-8958528.This research was partially supported by NSF Grant CCR-9009753.This research was supported in part by the National Science Foundation under Grant CCR-8658139, by DIMACS, a National Science Foundation Science and Technology center, Grant No. NSF-STC88-09648.     

The ski-rental problem with multiple discount options

We propose the ski-rental problem with multiple discount options in which there are n options to rent an equipment. Every option has a rental duration; the longer the duration, the more the discount. This generalizes the standard ski-rental problem where one can only rent or buy. We present a 4-competitive on-line algorithm for our problem and show that no deterministic algorithm can have a smaller competitive ratio when n is large enough. Comparing with the original ski-rental problem, the competitive ratio becomes larger when there are more options.     

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.     

On Capital Investment

We deal with the problem of making capital investments in machines for manufacturing a product. Opportunities for investment occur over time, every such option consists of a capital cost for a new machine and a resulting productivity gain, i.e., a lower production cost for one unit of product. The goal is that of minimizing the total production costs and capital costs when future demand for the product being produced and investment opportunities are unknown. This can be viewed as a generalization of the ski-rental problem and related to the mortgage problem [3]. If all possible capital investments obey the rule that lower production costs require higher capital investments, then we present an algorithm with constant competitive ratio. If new opportunities may be strictly superior to previous ones (in terms of both capital cost and production cost), then we give an algorithm which is O (min{1+log C , 1+log log P , 1+log M }) competitive, where C is the ratio between the highest and the lowest capital costs, P is the ratio between the highest and the lowest production costs, and M is the number of investment opportunities. We also present a lower bound on the competitive ratio of any on-line algorithm for this case, which is Ω (min{log C , log log P / log log log P , log M / log log M }). This shows that the competitive ratio of our algorithm is tight (up to constant factors) as a function of C , and not far from the best achievable as a function of P and M . Received February 6, 1997; revised November 17, 1997.     

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.     

On the two-dimensional cow search problem

We investigate the problem introduced by Baeza-Yates et al. [R.A. Baeza-Yates, J.C. Culberson, G.J.E. Rawlins, Searching with uncertainty, Research report, University of Waterloo, 1987]: given a plane and a horizontal or a vertical line in unknown location give a strategy to find this line. We use a competitive analysis to measure the performance of this strategy. This problem is one of the first generalisations of the cow search problem.     

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.     

Bounded families for the on-line t-relaxed coloring

We introduce a new generalization of the on-line coloring game. We define the concept of bounded family for on-line t-relaxed colorings. This extends the concept of on-line competitive coloring algorithms to t-relaxed colorings. We characterize the trees T for which the family of T-free graphs is bounded and show that the corresponding bounding function is linear.     

A strongly competitive randomized paging algorithm

Thepaging problem is that of deciding which pages to keep in a memory ofk pages in order to minimize the number of page faults. We develop thepartitioning algorithm, a randomized on-line algorithm for the paging problem. We prove that its expected cost on any sequence of requests is within a factor ofH _k of optimum. (H _k is thekth harmonic number, which is about ln(k).) No on-line algorithm can perform better by this measure. Our result improves by a factor of two the best previous algorithm.Partial support for D. D. Sleator was provided by DARPA, ARPA Order 4976, Amendment 20, monitored by the Air Force Avionics Laboratory under Contract F33615-87-C-1499, and by the National Science Foundation under Grant CCR-8658139.     

On the power of randomization in on-line algorithms

Against in adaptive adversary, we show that the power of randomization in on-line algorithms is severely limited! We prove the existence of an efficient simulation of randomized on-line algorithms by deterministic ones, which is best possible in general. The proof of the upper bound is existential. We deal with the issue of computing the efficient deterministic algorithm, and show that this is possible in very general cases.A previous version of this paper appeared in the22nd ACM STOC Conference Proceedings. Part of this research was performed while A. Borodin and A. Wigderson were visitors at the International Computer Science Institute, and while G. Tardos was a visitor at the Hebrew University.     

Weighted nearest neighbor algorithms for the graph exploration problem on cycles

In the graph exploration problem, a searcher explores the whole set of nodes of an unknown graph. We assume that all the unknown graphs are undirected and connected. The searcher is not aware of the existence of an edge until he/she visits one of its endpoints. The searcher's task is to visit all the nodes and go back to the starting node by traveling a tour as short as possible. One of the simplest strategies is the nearest neighbor algorithm (NN), which always chooses the unvisited node nearest to the searcher's current position. The weighted NN (WNN) is an extension of NN, which chooses the next node to visit by using the weighted distance. It is known that WNN with weight 3 is 16-competitive for planar graphs. In this paper we prove that NN achieves the competitive ratio of 1.5 for cycles. In addition, we show that the analysis for the competitive ratio of NN is tight by providing an instance for which the bound of 1.5 is attained, and NN is the best for cycles among WNN with all possible weights. Furthermore, we prove that no online algorithm to explore cycles is better than 1.25-competitive.     

The online Prize-Collecting Traveling Salesman Problem

We study the online version of the Prize-Collecting Traveling Salesman Problem (PCTSP), a generalization of the Traveling Salesman Problem (TSP). In the TSP, the salesman has to visit a set of cities while minimizing the length of the overall tour. In the PCTSP, each city has a given weight and penalty, and the goal is to collect a given quota of the weights of the cities while minimizing the length of the tour plus the penalties of the cities not in the tour. In the online version, cities are disclosed over time. We give a 7/3-competitive algorithm for the problem, which compares with a lower bound of 2 on the competitive ratio of any deterministic algorithm. We also show how our approach can be combined with an approximation algorithm in order to obtain an O(1)-competitive algorithm that runs in polynomial time.     

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.     

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.     

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.     

LRU Is Better than FIFO

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. Received June 2, 1997; revised January 28, 1998.     

On the Influence of Lookahead in Competitive Paging Algorithms

We introduce a new model of lookahead for on-line paging algorithms and study several algorithms using this model. A paging algorithm is n-line with strong lookahead l if it sees the present request and a sequence of future requests that contains l pairwise distinct pages. We show that strong lookahead has practical as well as theoretical importance and improves the competitive factors of on-line paging algorithms. This is the first model of lookahead having such properties. In addition to lower bounds we present a number of deterministic and randomized on-line paging algorithms with strong lookahead which are optimal or nearly optimal. Received April 8, 1994; revised May 15, 1995.     

Online hypergraph coloring

In this paper we investigate the online hypergraph coloring problem. In this online problem the algorithm receives the vertices of the hypergraph in some order v₁,…,vn and it must color vi by only looking at the subhypergraph Hi=(Vi,Ei) where Vi={v₁,…,vi} and Ei contains the edges of the hypergraph which are subsets of Vi. We show that there exists no online hypergraph coloring algorithm with sublinear competitive ratio. Furthermore we investigate some particular classes of hypergraphs (k-uniform hypergraphs, hypergraphs with bounded matching number, projective planes), we analyse the online algorithm FF and give matching lower bounds for these classes.