Separator-Based Graph Embedding into Higher-Dimensional Grids with Small Congestion

金沢大学理工研究域電子情報学系We study the problem of embedding a guest graph into an optimally-sized grid with minimum edge-congestion. Based on a wellknown notion of graph separator, we prove that any guest graph can be embedded with a smaller edge-congestion as the guest graph has a smaller separator, and as the host grid has a higher dimension. Our results imply the following: An N-node planar graph with maximum node degree Δ can be embedded into an N-node d-dimensional grid with an edge-congestion of O(Δ2 logN) if d = 2, O(Δ2 log logN) if d = 3, and O(Δ2) otherwise. An N-node graph with maximum node degree Δ and a treewidth O(1), such as a tree, an outerplanar graph, and a series-parallel graph, can be embedded into an N-node d-dimensional grid with an edge-congestion of O(Δ) for d ≥ 2

Separator-based graph embedding into higher-dimensional grids with small congestion

金沢大学理工研究域電子情報学系We study the problem of embedding a guest graph into an optimally-sized grid with minimum edge-congestion. Based on a wellknown notion of graph separator, we prove that any guest graph can be embedded with a smaller edge-congestion as the guest graph has a smaller separator, and as the host grid has a higher dimension. Our results imply the following: An N-node planar graph with maximum node degree Δ can be embedded into an N-node d-dimensional grid with an edge-congestion of O(Δ2 logN) if d = 2, O(Δ2 log logN) if d = 3, and O(Δ2) otherwise. An N-node graph with maximum node degree Δ and a treewidth O(1), such as a tree, an outerplanar graph, and a series-parallel graph, can be embedded into an N-node d-dimensional grid with an edge-congestion of O(Δ) for d ≥ 2

Expansion of layouts of complete binary trees into grids

AbstractLet Th be the complete binary tree of height h. Let M be the infinite grid graph with vertex set Z2, where two vertices (x1,y1) and (x2,y2) of M are adjacent if and only if |x1−x2|+|y1−y2|=1. Suppose that T is a tree which is a subdivision of Th and is also isomorphic to a subgraph of M. Motivated by issues in optimal VLSI design, we show that the point expansion ratio n(T)/n(Th)=n(T)/(2h+1−1) is bounded below by 1.122 for h sufficiently large. That is, we give bounds on how many vertices of degree 2 must be inserted along the edges of Th in order that the resulting tree can be laid out in the grid. Concerning the constructive end of VLSI design, suppose that T is a tree which is a subdivision of Th and is also isomorphic to a subgraph of the n×n grid graph. Define the expansion ratio of such a layout to be n2/n(Th)=n2/(2h+1−1). We show constructively that the minimum possible expansion ratio over all layouts of Th is bounded above by 1.4656 for sufficiently large h. That is, we give efficient layouts of complete binary trees into square grids, making improvements upon the previous work of others. We also give bounds for the point expansion and expansion problems for layouts of Th into extended grids, i.e. grids with added diagonals

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 well-known notion of graph separators, we show that an embedding with a smaller edge-congestion 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 NN nodes, maximum node degree ΔΔ, and with a node-separator of size ss, where ss is a function such that s(n)=O(nα)s(n)=O(nα) with 0≤α1/(1−α)d>1/(1−α), O(ΔlogN)O(ΔlogN) if d=1/(1−α)d=1/(1−α), and View the MathML sourceO(ΔNα−1+1d) if d1/(1−α)d>1/(1−α), and matches an existential lower bound within a constant factor if d≤1/(1−α)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(ΔlogN)O(ΔlogN) for d=2d=2 and O(Δ)O(Δ) for any fixed d≥3d≥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(Δ)O(Δ) for any fixed d≥2d≥2. To design our embedding algorithm, we introduce edge-separators bounding extension , 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 extension of O(Δnα)O(Δnα) from a node-separator of size O(nα)O(nα)