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Division of Electrical Engineering and Computer Science, Kanazawa University, Kanazawa 920-1192, Japan
Author to whom correspondence should be addressed.
Received: 16 June 2016 / Accepted: 18 August 2016 / Published: 23 August 2016
The page migration problem in Euclidean space is revisited. In this problem, online requests occur at any location to access a single page located at a server. Every request must be served, and the server has the choice to migrate from its current location to a new location in space. Each service costs the Euclidean distance between the server and request. A migration costs the distance between the former and the new server location, multiplied by the page size. We study the problem in the uniform model, in which the page has size . All request locations are not known in advance; however, they are sequentially presented in an online fashion. We design a -competitive online algorithm that improves the current best upper bound for the problem with the unit page size. We also provide a lower bound of for our algorithm. It was already known that 2.5 is a lower bound for this problem.
page migration problem; competitive analysis; server problems; online algorithms; adversary model; euclidean space; uniform model
The page migration problem is a classical formulation of efficient memory management in a shared memory multiprocessor system comprising a network of processors having their own local memories. In this problem, we are given a sequence of requests issued by processors to access a single shared data object, called a page. Each request must be served via communication between the processor issuing the request and the processor holding the page at the time of the request. The service for the request costs the distance of the communication. After each service, the page is allowed to migrate from the current processor to a new processor before the next request is issued. The migration of the page costs the distance of the migration multiplied by the page size. The objective of the page migration problem is to minimize the total costs of services and migrations (see Section 2 for a formal definition of this problem). This problem can also be viewed as a formulation of general situations for managing shared information in a distributed network, such as the efficient allocation of each shared data object to be accessed by a distributed program running on the network .
Black and Sleator  first studied online algorithms for the page migration problem via competitive analysis. Some basic notions of competitive analysis are given in Section 2. Chrobak, Larmore, Reingold, and Westbrook  studied this problem on continuous metric spaces as well as networks. They proposed a -competitive randomized algorithm on trees against oblivious adversaries, together with the same lower bound as this competitiveness even at two points, where is a positive integer representing the page size. They also proved that any randomized page migration algorithm on can be derandomized while preserving its competitiveness. As any page migration algorithm on a space can be generalized on Cartesian products of the space , a -competitive deterministic algorithm on under norm can be obtained through generalizing the algorithm on trees in . For general metric spaces, Westbrook  proposed a -competitive randomized algorithm against oblivious adversaries, where is a function defined in  that tends toward as grows large. Derandomizing this algorithm on , we have a -competitive deterministic algorithm on under any norm .
Moreover, is a lower bound for any randomized or deterministic algorithm in the interval . This lower bound is also admitted on under norm. This is because, for the interval on a dimension in , any online algorithm A locating its server in for requests only in has a cost of at least that of a certain algorithm locating its server only in , i.e., projection on the kth coordinate of A’s server location if the projection is in , and the closer endpoint of otherwise. It is a longstanding open question whether we can close the gap of and under norm even with any .
In this paper, we partially answer this question by proposing a -competitive deterministic algorithm on under norm for the case . We note that the competitiveness improves , which is equal to as mentioned in . We also provide a lower bound of for our algorithm. Our algorithm is extremely simple: it maintains the page at the center of two special points, both initially located at the initial location of the server. Upon each request, the algorithm chooses one of the points farther from the requested location and moves the point to the requested location; the other point does not move. This algorithm can be viewed as a natural extension of the -competitive algorithm under norm of . We note that if there exists a -competitive algorithm under norm, then it would have to behave on any line in the same way as the -competitive algorithm of . The setting of the unit page size, i.e., is called the uniform model in the context of data management problems including the page migration problem [1,5,6,7].
Other previous results for the page migration problem are as follows. Bartal, Charikar, and Indyk  proposed a -competitive deterministic algorithm for general networks. For trees and uniform networks, Black and Sleator  presented 3-competitive deterministic algorithms, together with the lower bound of 3 even at two points. The first deterministic lower bound larger than 3 for general networks was , which was improved to by Matsubayashi . Although these lower bounds, larger than , were proved only for , the first lower bound of was recently proved , where notation is with respect to . For three points, -competitive deterministic algorithms with [3,10] and a -competitive deterministic algorithm with  were proposed. The latter algorithm asymptotically matches the lower bound of on three points for every . As for randomized algorithms against adaptive online adversaries, a -competitive algorithm for general metric spaces was proposed in  and the upper bound of is also a lower bound on two points . As for a randomized algorithm against oblivious adversaries, Lund, Reingold, Westbrook, and Yan  proved that the optimal competitiveness of can be achieved even on uniform networks.
The rest of this paper is organized as follows. Some preliminaries are given in Section 2. We define our algorithm on under norm and prove its competitiveness in Section 3. In Section 4, we prove a lower bound of for our algorithm. We conclude the paper with remarks and future works in Section 5.
Let be a metric space with a distance function . For a given initial page location , a sequence of requests , and a page size , the page migration problem is to compute page locations such that the cost function is minimized. We call a server.
An online algorithm must compute without any information regarding , whereas an offline algorithm may compute using the entire sequence of requests . An adversary against an online algorithm A generates a sequence of requests given to A and computes an output sequence for the generated requests. If A is deterministic, then the adversary generates requests using the definition of A, or equivalently, the information of the actual behavior of A, and computes its own output according to an optimal offline algorithm OPT. The deterministic algorithm A is -competitive if for the initial server and any sequence of requests, where and are costs of A and OPT, respectively, and is a constant. For randomized online algorithms, there are two types of adversaries. An adversary is said to be oblivious if it generates requests in advance only using the definition of A, i.e., without any information about the random behavior of A and computes its own output according to OPT. In contrast, an adaptive online adversary generates requests using information of the random behavior of A and computes its own output in an online fashion. The competitiveness of a randomized online algorithm against oblivious or adaptive online adversaries is defined in a similar way to that of a deterministic online algorithm, except that expected values are used for randomized costs.
One commonly used technique for proving the competitiveness of an online algorithm A is to utilize a potential function , which typically maps the situation at a point of time, such as the page locations of A and OPT, to a real value. More specifically, we suitably divide the sequence of the online processes of A and OPT into certain events. Our goal is to define the value of in such a way that the initial value of is at most some constant , is always at least some constant and that for any event, where denotes the change of values by the event. Summing the inequality over all events for an initial server and a request sequence , we have , which means that A is -competitive.
3. Algorithm and Its Upper Bound Analysis
We focus on the page size and the metric space under norm for . We state our deterministic algorithm, called PQ, in the following paragraph. Intuitively, the algorithm determines the location of the page using two locations from previous requests. We prove the competitiveness of , claimed in Theorem 1, using a potential function . As mentioned in Section 2, defining suitable events and showing that for any defined event is sufficient. For the proof of the theorem, we need to show an inequality different from a triangle inequality. We separately provide a technical part for that inequality as Lemma 1. The lemma is about points computed by the algorithm.
This algorithm maintains the server at the center of two pointsand, both of which are initially located at the initial server location. Upon each request at location, if, thenmoves to; otherwise,moves to. The algorithm migrates its server toafterormoves.
For anyandsuch that,, andis the center ofand,is maximized if, or, or.
We may assume without loss of generality that and with . Moreover, we fix and regard as a function of . The aim is to prove that is maximized at or .
It follows that:
Therefore, we have
monotonically increases from to 1 as changes from to . If , then for any . Therefore, is maximized at . Otherwise, since , there exists such that . Since for and for , is maximized at or . ☐
PQ is-competitive forand the page size .
We use the following potential function for OPT’s server location , PQ’s server , and point locations and :
We separate the online events into two parts. The first is to consider only the migration costs incurred by OPT’s server, and the second is to consider the service costs incurred by OPT together with the migration and service costs incurred by PQ. It is sufficient to show that the inequality follows in both parts, upon each request located at .
The migration of OPT’s server from to induces a change of to the total cost of the optimal algorithm OPT but no change to the cost incurred by PQ. The total change of is then . Therefore, it is sufficient to show the following inequality:
This follows by the symmetry of the distance function () and the triangle inequalities and .
We may assume, without loss of generality, that . By this assumption, PQ moves to . Since PQ maintains its server at the center of and , the migration cost incurred by PQ is then . For this part, we have the following equalities:
Therefore, we shall show the following inequality:
Since , it is sufficient to show:
This follows for the cases and , because and , respectively. We assume and . It is sufficient to show that the maximum value of the left-hand side of Equation (1) is less than or equal to zero. If we regard , , , and as vectors in , then at most three vectors of them, say , , and , are independent. Therefore, the points , , , and are on a plane in . Applying Lemma 1 on this plane, the left-hand side of Equation (1) is maximized in one of three situations. Situation 1: , Situation 2: , or Situation 3: . We proceed to show inequality (1) for each of these three situations.
Substitution of by in Equation (1) reveals the following inequality.
By applying and dividing both sides by , it is sufficient to show
Since , Equation (2) can be written as:
By taking the derivative with respect to on the plane containing , , and , the left-hand side of Equation (2) is maximized at and . Therefore, Equation (2) follows.
The inequality in Equation (1) can be rewritten as:
In this situation, we recall that and are all located on the same line segment. Since , and is located at the center of and , it follows that . Since , the inequality in Equation (3) follows.
Since , we rewrite Equation (1) as:
The points and are also located on the same line segment in this situation. By , we have . By we have . In addition, by we have .
Therefore, the proof completes. ☐
4. A Lower Bound for PQ
In this section, we show that the exact competitiveness of the algorithm is greater than . We introduce an adversary through Theorem 2 and prove the existence of such a lower bound. The adversary makes a special sequence of requests on against the proposed algorithm. The requests are given at vertices of a triangle that tends to be equilateral. After the second request, the server repeatedly migrates between the center points of two sides of the triangle.
For a sufficiently large integer, there exist a request sequence, three request locationsandon a plane, and the initial server location, such thatfor .
We describe our approach before more technical discussion in the subsequent paragraphs. In Figure 1, we illustrate the behavior of PQ against requests by our adversary. We assume that is slightly larger than . Points and are the centers of and , and and , respectively. For requests , PQ migrates its server along the dashed line, because does not move from . A request occurs at when the server is at , and a request occurs at when the server is at . Note that , , and tend to be the vertices of an equilateral triangle as approaches .
For , , and , where and , , and are the vertices of an equilateral triangle with a unit side length, the cost of the optimal algorithm is at most by keeping the server at the initial location .
The adversary infinitesimally perturbs the distances by slightly increasing the distance between and . Upon the first request at , PQ serves the request by the cost of and migrates the server to , which is the center of and , by the cost of . This is because point at moves to . The th request at is served with the cost of , causing point at to move to , and the server migrates to , which is at the center of and , with the cost of . The st request at is served with the cost of . The point at moves to ; hence, the server migrates to with the cost of .
We compute the total cost of PQ by using the distances of the unperturbed triangle, since the actual distances (and hence the actual costs of PQ and OPT) may differ by an infinitesimally small amount from these computations. Therefore, we have
then is at least . This completes the proof. ☐
5. Concluding Remarks
In this paper, we proposed a deterministic algorithm for the page migration problem in Euclidean space with a page size . The -competitiveness of the algorithm is an improvement on the former ratio. An adversary was found to express a lower bound of for the algorithm. If possible, one could seek to find an algorithm to cover a larger with better competitiveness than . Another area of improvement is to narrow the upper and lower bounds of the algorithm. Moreover, the generalization of the algorithm under norms other than the Euclidean and Manhattan ones remains an open problem in this research area.
The authors are thankful to the anonymous reviewers who provided valuable and helpful comments. This work was supported by JSPS KAKENHI Grant Number 26330008.
The design of the algorithm and the proofs were all done together by the authors, step by step.
Conflicts of Interest
The authors declare no conflict of interest.
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The request sequence to access the page at the server; .
The request sequence to access the page at the server; .