Orderly Spanning Trees with Applications

We introduce and study the {\em orderly spanning trees} of plane graphs. This algorithmic tool generalizes {\em canonical orderings}, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an {\em orderly pair} for any connected planar graph $G$, consisting of a plane graph $H$ of $G$, and an orderly spanning tree of $H$. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyder's Realizer Theorem, (2) the first area-optimal 2-visibility drawing of $G$, and (3) the best known encodings of $G$ with O(1)-time query support. All algorithms in this paper run in linear time.Comment: 25 pages, 7 figures, A preliminary version appeared in Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001), Washington D.C., USA, January 7-9, 2001, pp. 506-51

Succinct Representations of Permutations and Functions

We investigate the problem of succinctly representing an arbitrary permutation, \pi, on {0,...,n-1} so that \pi^k(i) can be computed quickly for any i and any (positive or negative) integer power k. A representation taking (1+\epsilon) n lg n + O(1) bits suffices to compute arbitrary powers in constant time, for any positive constant \epsilon <= 1. A representation taking the optimal \ceil{\lg n!} + o(n) bits can be used to compute arbitrary powers in O(lg n / lg lg n) time. We then consider the more general problem of succinctly representing an arbitrary function, f: [n] \rightarrow [n] so that f^k(i) can be computed quickly for any i and any integer power k. We give a representation that takes (1+\epsilon) n lg n + O(1) bits, for any positive constant \epsilon <= 1, and computes arbitrary positive powers in constant time. It can also be used to compute f^k(i), for any negative integer k, in optimal O(1+|f^k(i)|) time. We place emphasis on the redundancy, or the space beyond the information-theoretic lower bound that the data structure uses in order to support operations efficiently. A number of lower bounds have recently been shown on the redundancy of data structures. These lower bounds confirm the space-time optimality of some of our solutions. Furthermore, the redundancy of one of our structures "surpasses" a recent lower bound by Golynski [Golynski, SODA 2009], thus demonstrating the limitations of this lower bound.Comment: Preliminary versions of these results have appeared in the Proceedings of ICALP 2003 and 2004. However, all results in this version are improved over the earlier conference versio

Compact Floor-Planning via Orderly Spanning Trees

Floor-planning is a fundamental step in VLSI chip design. Based upon the concept of orderly spanning trees, we present a simple O(n)-time algorithm to construct a floor-plan for any n-node plane triangulation. In comparison with previous floor-planning algorithms in the literature, our solution is not only simpler in the algorithm itself, but also produces floor-plans which require fewer module types. An equally important aspect of our new algorithm lies in its ability to fit the floor-plan area in a rectangle of size (n-1)x(2n+1)/3. Lower bounds on the worst-case area for floor-planning any plane triangulation are also provided in the paper.Comment: 13 pages, 5 figures, An early version of this work was presented at 9th International Symposium on Graph Drawing (GD 2001), Vienna, Austria, September 2001. Accepted to Journal of Algorithms, 200