September 2, 2017

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25–36, 1999. c Springer-Verlag Berlin Heidelberg 1999 26 Federico Bassetti et al. and data structures in the context of memory access times. Recently hierarchical memory models have been proposed [7,8,9,10,11] and efficient algorithms for a number of basic problems have been proposed under these models. 2 Background and Relationship to Previous Work Relaxation is a well-known technique in the solution of many numerical problems. We recall the ideas from [16,14] on graph covers based methods for efficient implementation of linear relaxation problems.

Tiling. In general tiling maps a n-deep nested loop structure into a 2n deep nested loop structure with only a small fixed number of iterations. Graphically, tiling can be viewed as way to break the iteration space into smaller blocks with the following conditions: 1. The blocks are disjoint. 2. e. if p ≺ q, then p is executed before q. Stated another way, the complete ordering for scheduling the execution of the iterations should respect the partial ordering given by the dependence graph. 3. g.

Neighborhood Cover: Given a graph G(V, E), a τ -neighborhood cover G = {G1 , . . , Gk } is set of subgraphs Gi , 1 ≤ i ≤ k, such that ∀v ∈ V , there exists a Gi with the property that N τ (v) ⊆ Vi . Given a graph G(V, E), a sparse τ -neighborhood cover for G is a τ -neighborhood cover with the additional properties ∀i, 1 ≤ i ≤ k, |Gi | ≤ M , and k = O(|E|/M ). Note that a vertex v can belong to more than one subgraph Gi but for the purposes of computing its value there is at least one subgraph Gv that may (τ ) be regarded its “home”, meaning that the state xv at a vertex v at time τ can be calculated by just looking at the graph Gv .

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