We address the page migration problem, one of the most classical online problems. In this problem, we are given online requests from nodes on a network for accessing a single page, a data set stored in a node, and asked to determine a node for the page to be stored after each request. Serving a request costs the distance between the request and the page at the point of the request, and migrating the page costs the migration distance multiplied by the page size. The objective is to minimize the total sum of the service and migration costs. This problem is motivated by efficient cache management on multiprocessor systems. In this paper, we prove that no deterministic online page migration algorithm is (3 + o(1))-competitive, where o-notation is with respect to the page size. Our lower bound first breaks the barrier of 3 by an additive constant for arbitrarily large page size, and disproves Black and Sleator's conjecture even in the asymptotic sense.

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This paper addresses the page migration problem: given online requests from nodes on a network for accessing a page stored in a node, output online migrations of the page. Serving a request costs the distance between the request and the page, and migrating the page costs the migration distance multiplied by the page size D ≥ 1. The objective is to minimize the total sum of service costs and migration costs. Black and Sleator conjectured that there exists a 3-competitive deterministic algorithm for every graph. Although the conjecture was disproved for the case D = 1, whether or not an asymptotically (with respect to D) 3-competitive deterministic algorithm exists for every graph is still open. In fact, we did not know if there exists a 3-competitive deterministic algorithm for an extreme case of three nodes with D ≥ 2. As the first step toward an asymptotic version of the Black and Sleator conjecture, we present 3-and (3 + 1/D)-competitive algorithms on three nodes with D = 2 and D ≥ 3, respectively, and a lower bound of 3 + Ω (1/D) that is greater than 3 for every D ≥ 3. In addition to the results on three nodes, we also derive ρ-competitiveness on complete graphs with edge-weights between 1 and 2 − 2/ρ for any ρ ≥ 3, extending the previous 3-competitive algorithm on uniform networks.

This paper addresses the page migration problem: given online requests from nodes on a network for accessing a page stored in a node, output online migrations of the page. Serving a request costs the distance between the request and the page, and migrating the page costs the migration distance multiplied by the page size D ≥ 1. The objective is to minimize the total sum of service costs and migration costs. Black and Sleator conjectured that there exists a 3-competitive deterministic algorithm for every graph. Although the conjecture was disproved for the case D = 1, whether or not an asymptotically (with respect to D) 3-competitive deterministic algorithm exists for every graph is still open. In fact, we did not know if there exists a 3-competitive deterministic algorithm for an extreme case of three nodes with D ≥ 2. As the first step toward an asymptotic version of the Black and Sleator conjecture, we present 3-and (3 + 1/D)-competitive algorithms on three nodes with D = 2 and D ≥ 3, respectively, and a lower bound of 3 + Ω (1/D) that is greater than 3 for every D ≥ 3. In addition to the results on three nodes, we also derive ρ-competitiveness on complete graphs with edge-weights between 1 and 2 − 2/ρ for any ρ ≥ 3, extending the previous 3-competitive algorithm on uniform networks.

SUMMARYIn this paper we consider the VLSI layout (i.e., Manhattan layout) of graphs into grids with minimum width (i.e., the length of the shorter side of a grid) as well as with minimum area. The layouts into minimum area and minimum width are equivalent to those with the largest possible aspect ratio of a minimum area layout. Thus such a layout has a merit that, by "folding" the layout, a layout of all possible aspect ratio can be obtained with increase of area within a small constant factor. We show that an N -vertex tree with layout-width k (i.e., the minimum width of a grid into which the tree can be laid out is k) can be laid out into a grid of area O(N ) and width O(k). For binary tree layouts, we give a detailed trade-off between area and width: an N -vertex binary tree with layout-width k can be laid out into area O( k+α 1+α N ) and width k + α, where α is an arbitrary integer with 0 ≤ α ≤ √ N , and the area is existentially optimal for any k ≥ 1 and α ≥ 0. This implies that α = Ω(k) is essential for a layout of a graph into optimal area. The layouts proposed here can be constructed in polynomial time. We also show that the problem of laying out a given graph G into given area and width, or equivalently, into given length and width is NP-hard even if G is restricted to a binary tree.

We consider the problem of embedding graphs into hypercubes with minimal congestion. Kim and Lai [2] showed that for a given N -vertex graph G and a hypercube it is NP-complete to determine whether G is embeddable in the hypercube with unit congestion, but G can be embedded with unit congestion in a hypercube of dimension 6 log N if the maximum degree of a vertex in G is no more than 6 log N . Bhatt, Chung, Leighton, and Rosenberg [1] showed that every N -vertex binary tree can be embedded in a hypercube of dimension log N with O(1) congestion. In this paper we extend the results above and show the following:• Every N -vertex graph G can be embedded with unit congestion in a hypercube of dimension 2 log N if the maximum degree of a vertex in G is no more than 2 log N .• Every N -vertex binary tree can be embedded in a hypercube of dimension log N with congestion at most 5.The former answers a question posed by Kim and Lai [2]. The latter is the first result that shows a simple embedding of a binary tree into an optimal sized hypercube with explicit small congestion of 5. This partially answers a question posed by Bhatt, Chung, Leighton, and Rosenberg [1]. The embeddings proposed here are quite simple and can be constructed in polynomial time.

Abstract: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 D = 1. All request locations are not known in advance; however, they are sequentially presented in an online fashion. We design a 2.75-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 2.732 for our algorithm. It was already known that 2.5 is a lower bound for this problem.

In this paper we consider the problem of embedding a (guest) graph into a grid with the same number of nodes as those of the guest graph with minimum edge congestion. We show that a graph which has some efficient recursive separators can be embedded into a grid of the same size with small congestion. Our results imply that an N -node planar graph with maximum vertex degree ∆ can be embedded into an N -node grid with congestion O(∆ 2 log N ), and if the graph is a tree, then it can be embedded into an N -node grid with congestion O(∆). The congestion for trees is optimal within a constant factor, and the congestion for planar graphs is optimal within an O(min{∆ 2 √ log N , ∆ log N }) factor.