Separator-Based Graph Embedding into Multidimensional Grids with Small Edge-Congestion

We study the problem of embedding a guest graph with minimum edge-congestion into a multidimensional grid with the same size as that of the guest graph. Based on a wellknown notion of graph separators, we show that an embedding with a smaller edgecongestion can be obtained if the guest graph has a smaller separator, and if the host grid has a higher but constant dimension. Specifically, we prove that any graph with N nodes, maximum node degree ∆, and with a node-separator of size O(n) (0 ≤ α < 1) can be embedded into a grid of a fixed dimension d ≥ 2 with at least N nodes, with an edge-congestion of O(∆) if d > 1/(1−α), O(∆ log N) if d = 1/(1−α), and O(∆Nα−1+ 1 d ) if d < 1/(1 − α). This edge-congestion achieves constant ratio approximation if d > 1/(1 − α), and matches an existential lower bound within a constant factor if d ≤ 1/(1 − α). Our result implies that if the guest graph has an excluded minor of a fixed size, such as a planar graph, then we can obtain an edge-congestion of O(∆ log N) for d = 2 and O(∆) for any fixed d ≥ 3. Moreover, if the guest graph has a fixed treewidth, such as a tree, an outerplanar graph, and a series-parallel graph, then we can obtain an edge-congestion of O(∆) for any fixed d ≥ 2. To design our embedding algorithm, we introduce edge-separators bounding expansion, such that in partitioning a graph into isolated nodes using edge-separators recursively, the number of outgoing edges from a subgraph to be partitioned in a recursive step is bounded. We present an algorithm to construct an edge-separator with expansion of O(∆n) from a nodeseparator of size O(n).