Bounding Dilation and Edge-Congestion of Separator-Based Graph Embeddings into Grids

Abstract

In this paper, we address a classical problem of embedding a guest graph with minimum dilation into a multidimensional grid of the same size as that of the guest graph. This problem has applications such as efficient VLSI layout and parallel computation. We propose a relatively simple embedding bounding dilation based on graph separators. Specifically, we prove that any graph with N nodes, maximum node degree ? = 2, and with a node-separator of size s, where s is a function such that s(n) = O(na) with 0 = a 1/(1 -- a). This congestion achieves constant ratio approximation. For d = 1/(1 -- a), we present a trade-off between tight upper bounds of dilation and edge-congestion. Specifically, we prove a dilation of O(N1/d log?/elogN) and an edge-congestion of O(?(Na -- 1+1/d+e+logN)) for any 1/logN= e < 1 -- a. These dilation and edge-congestion match existential lower bounds for e = O(1) and e = 1/logN, respectively. Besides, there exists a guest graph for which better dilation and edge-congestion cannot simultaneously be obtained for e = loglogN/logN. This is the first observation that minimizing both dilation and edge-congestion is generally impossible in embeddings into grids. The above results improve or generalize several previous results.