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 constant d. 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. 1 Introduction Recently much attention has been given to competitive analysis of on-line algorithms [7, 20, 22, 25]. Ro...

. The page migration problem is to manage a globally addressed shared memory in a multiprocessor system. Each physical page of memory is located at a given processor, and memory references to that page by other processors incur a cost proportional to the network distance. At times the page may migrate between processors at cost proportional to the distance times D, a page size factor. The problem is to schedule movements on-line so that the total cost of memory references is within a constant factor c of the best off-line schedule. An algorithm that does so is called c-competitive. Black and Sleator gave 3-competitive deterministic on-line algorithms for uniform networks (complete graphs with unit edge lengths) and for trees with arbitrary edge lengths. No good deterministic algorithm is known for general networks with arbitrary edge lengths. We present randomized algorithms for the migration problem that are both simple and better than 3-competitiveagainst an oblivious adversary. We ...

This thesis presents research done by the author on competitive analysis of on-line problems. An on-line problem is a problem that is given and solved one piece at a time. An on-line strategy for solving such a problem must give the solution to each piece knowing only the current piece and preceding pieces, in ignorance of the pieces to be given in the future. We consider on-line strategies that are competitive (guaranteeing solutions whose costs are within a constant factor of optimal) for several combinatorial optimization problems: paging, weighted caching, the k-server problem, and weighted matching. We introduce variations on the standard model of competitive analysis for paging: allowing randomization, allowing resource-bounded lookahead, and loose competitiveness, in which performance over a range of fast memory sizes is considered and noncompetitiveness is allowed provided the fault rate is insignificant. Each variation leads to substantially better competitive ratios. We prese...

In the offline list update problem, we maintain an unsorted linear list used as a dictionary. Accessing the item at position i in the list costs i units. In order to reduce access cost, we are allowed to update the list at any time by transposing consecutive items at a cost of one unit. Given a sequence of requests one has to serve in turn, we are interested in the minimal cost needed to serve all requests. Little is known about this problem. The best algorithm so far needs exponential time in the number of items in the list. We show that there is no polynomial algorithm unless P = NP.

This paper concerns the online list accessing problem. In the first part of the paper we present two new families of list accessing algorithms. The first family is of optimal, 2-competitive, deterministic online algorithms. This family, called the mri (move-to-recent-item) family, includes as members the well known move-to-front (MTF) algorithm, and the recent, more "conservative" algorithm TIMESTAMP due to Albers. So far move-to-front and TIMESTAMP were the only algorithms known to be optimal in terms of their competitive ratio. This new family contains a sequence of algorithms fA(i)g i1 where A(1) is equivalent to TIMESTAMP and the limit element A(1) is MTF. Further, in this class, for each i, the algorithm A(i) is more conservative than algorithm A(i + 1) in the sense that it is more reluctant to move an accessed item to the front, thus giving a gradual transition from the conservative TIMESTAMP to the "reckless" MTF. The second new family , called the pri (pass-recent-item) family is also infinite and contains TIMESTAMP; We show that most algorithms in this family attain a competitive ratio of 3.

The optimal competitive ratio for a randomized online list update algorithm is known to be at least 1.5 and at most 1.6, but the remaining gap is not yet closed. We present a new lower bound of 1.50084 for the partial cost model. The construction is based on game trees with incomplete information, which seem to be generally useful for the competitive analysis of online algorithms.

. The list update problem, a well-studied problem in dynamic data structures, can be described abstractly as a metrical task system. In this paper, we prove that a generic metrical task system algorithm, called the work function algorithm, has constant competitive ratio for list update. In the process, we present a new formulation of the well-known "list factoring" technique in terms of a partial order on the elements of the list. This approach leads to a new simple proof that a large class of online algorithms, including Move-To-Front, is (2 \Gamma 1=k)-competitive. 1 Introduction 1.1 Motivation The list accessing or list update problem is one of the most well-studied problems in competitive analysis [1],[2],[3],[4],[5]. The problem consists of maintaining a set S of items in an unsorted linked list, for example as a data structure for implementation of a dictionary. The data structure must support three types of requests: ACCESS(x), INSERT(x) and DELETE(x), where x is the n...

It is well-established that input sequences for paging and list update have locality of reference. In this paper we analyze the performance of algorithms for these problems in terms of the amount of locality in the input sequence. We define a measure for locality that is based on Denning’s working set model and express the performance of well known algorithms in term of this parameter. This introduces parameterized-style analysis to online algorithms. The idea is that rather than normalizing the performance of an online algorithm by an (optimal) offline algorithm, we explicitly express the behavior of the algorithm in terms of two more natural parameters: the size of the cache and Denning’s working set measure. This technique creates a performance hierarchy of paging algorithms which better reflects their intuitive relative strengths. It also reflects the intuition that a larger cache leads to a better performance. We obtain similar separation for list update algorithms. Lastly, we show that, surprisingly, certain randomized algorithms which are superior to MTF in the classical model are not so in the parameterized case, which matches experimental results.

We present an optimal on-line algorithm for the List Update Problem when the request sequence has some particular structure. 1 Introduction 1.1 The General Setting We investigate the List Update Problem. The problem can be briefly stated as follows : There are n different items in a doubly-linked list. Every item has its own unique key. Access requests are coming for these items, one at a time, in the form of their key values. We satisfy the request by finding an item of the corresponding key value in the doubly-linked list. If the position of the item requested is i in the list, then the cost of this operation is i units. While satisfying such a request, we are allowed to move the requested item in any of the first i positions for no cost. The goal is to satisfy all requests at minimum possible cost. In the off-line setting, we know the request sequence in advance. On the other hand, in the on-line setting, we have to satisfy a request without the knowledge of any future request. ...

On-line analysis models a player A (randomized or deterministic) who makes immediate responses to incoming elements of an input sequence s = a 1 : : : a r . In this paper a 1 ; : : : ; a r are interpreted as elements offered without repetition to the player from a fixed universe and the player's response to each a i is a single bit interpreted as pick/not-to-pick. Before seeing the stream s, the player is given a monotone system M of sets indicating all sets of elements that she is allowed to hold at any time during the game and the player's objective is to pick as many elements as possible under these constraints. We assign the performance measure e(M) = maxA min s E(jA(s)j)=OPT (s) to every monotone set system M, where E() is the expectation over the player's random choices and OPT (s) = maxM2M jM "sj. Note that for every M, e(M) 2 [0; 1], and it is easy to show that a system has performance one iff it is a matroid. This model generalizes the apparently unrelated models in [2] and [3].