On the relationship between parallel computation and graph embeddings

Ajay Kumar Gupta, Purdue University

Abstract

The problem of efficiently simulating an algorithm designed for an n-processor parallel machine G on an m-processor parallel machine H with $n > m$ arises when parallel algorithms designed for an ideal size machine are simulated on existing machines which are of a fixed size. In this thesis, we study this problem when every processor of H takes over the function of a number of processors in G, and we phrase the simulation problem as a graph embedding problem. We present new embeddings that address relevant issues arising from the parallel computation environment. The main focus of our work centers around embedding complete binary trees into smaller-sized binary trees, butterflies, and hypercubes. We also consider simultaneous embeddings of r source machines into a single hypercube. Constant factors play a crucial role in our embeddings since they are not only important in practice but also lead to interesting theoretical problems. All of our embeddings minimize dilation and load, which are the conventional cost measures in graph embeddings and determine the maximum amount of time required to simulate one step of G on H. Our embeddings also optimize a new cost measure called ($\alpha,\beta$)-utilization which characterizes how evenly the processors of H are used by the processors of G. Ideally, the utilization should be balanced (i.e., every processor of H simulates at most $\lceil {n\over m}\rceil$ processors of G) and the ($\alpha,\beta$)-utilization measures how far off from a balanced utilization the embedding is. We present embeddings for the situation when some processors of G have different capabilities (e.g. memory or I/O) than others and the processors with different capabilities are to be distributed uniformly among the processors of H. We show that placing such conditions on an embedding results in an increase in some of the cost measures. Our complete binary tree embeddings also minimize the level-load which measures the load over the edges of H when only two consecutive levels of tree G are active at any time. In order to achieve a balanced utilization (or one close to it) without causing an increase in other cost measures, our embeddings first obtain an initial embedding with unbalanced ($\alpha$,$\beta$)-utilization, $\alpha < 1$, and then this embedding is refined to obtain a balanced utilization. The refinement strategies are the heart of the embeddings.

Degree

Ph.D.

Advisors

Hambrusch, Purdue University.

Subject Area

Computer science

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