The grid M(r,s) is the graph whose vertex set is the set of pairs on nonnegative integers,, in which there is an edge between vertices (i,j) and (k,l) if either |i−k|=1 and j=l or i=k and |j−l|=1. An embedding of the graph G into the graph H is an injective mapping f from the vertices of G to the vertices of H, together with a mapping which assigns to each edge of G a path between f(u) and f(v) in H. Let G and H be two simple, undirected graphs.
We also show that an embedding of a complete binary tree of odd height into its optimal grid, which is obtained from the embedding into the optimal extended grid by a simple transformation, has edge-congestion 2, dilation 22, and it creates in the grid far fewer edges of congestion 2 than the embedding from , This embedding has edge-congestion 1, dilation 2 + 1, and its average dilation is less than 1.12. In addition, our result shows that a path of length k is unit-length embeddable on G, between its two given vertices s and t, if \(k\le L\) and \(k\equiv L (\mathrm, in which there is an edge between vertices (i, j) and (k, l) if and only if |I – k| ≤ 1 and |j-l| ≤ 1.In this paper, we give an embedding of any complete binary tree of odd height into its optimal square extended grid. We show that a cycle of length k is unit-length embeddable on a solid grid graph G if k is an even integer between four and the length of the longest cycle of G. In this paper, we study the problem of embedding paths and cycles on solid grid graphs. grids with added diagonals.ĭespite many algorithms for embedding graphs on unbounded grids, only a few results on embedding graphs on restricted grids have been published. We also give bounds for the point expansion and expansion problems for layouts of Th into extended grids, i.e. That is, we give efficient layouts of complete binary trees into square grids, making improvements upon the previous work of others.
We show constructively that the minimum possible expansion ratio over all layouts of Th is bounded above by 1.4656 for sufficiently large h. Define the expansion ratio of such a layout to be n2/n(Th)=n2/(2h+1−1). 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. 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. 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. Suppose that T is a tree which is a subdivision of Th and is also isomorphic to a subgraph of M. 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. Let Th be the complete binary tree of height h.
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