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

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 Δ ≥ 2, and with a node-separator of size s, where s is a function such that s(n) = O(n
^α
) with 0 ≤ α <; 1, can be embedded with a dilation of O(N
^1/d
log Δ / log N) into a grid with a fixed dimension d ≥ 2, at least N nodes, and constant aspect ratio. This dilation matches a trivial existential lower bound. A remarkable merit of this embedding is that it can be used to bound the dilation of another embedding algorithm. Combining with a previous embedding algorithm bounding the edge-congestion, we obtain an edge-congestion of O(Δ) as well as the dilation O(N
^1/d
log Δ / log N) if d > 1/(1 - α). This congestion achieves constant ratio approximation. For d ≤ 1/(1 - α), we present a trade-off between tight upper bounds of dilation and edge-congestion. Specifically, we prove a dilation of O(N
^1/d
log Δ / ϵ log N) and an edge-congestion of O(Δ(N
^{α - 1+1/d+ϵ}
+ log N)) for any 1/log N ≤ ϵ <; 1 - α. These dilation and edge-congestion match existential lower bounds for ϵ = Ω (1) and ϵ = 1/log N, respectively. Besides, there exists a guest graph for which better dilation and edge-congestion cannot simultaneously be obtained for ϵ = log log N / log N. 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.