241 research outputs found
Replacement Paths via Row Minima of Concise Matrices
Matrix is {\em -concise} if the finite entries of each column of
consist of or less intervals of identical numbers. We give an -time
algorithm to compute the row minima of any -concise matrix.
Our algorithm yields the first -time reductions from the
replacement-paths problem on an -node -edge undirected graph
(respectively, directed acyclic graph) to the single-source shortest-paths
problem on an -node -edge undirected graph (respectively, directed
acyclic graph). That is, we prove that the replacement-paths problem is no
harder than the single-source shortest-paths problem on undirected graphs and
directed acyclic graphs. Moreover, our linear-time reductions lead to the first
-time algorithms for the replacement-paths problem on the following
classes of -node -edge graphs (1) undirected graphs in the word-RAM model
of computation, (2) undirected planar graphs, (3) undirected minor-closed
graphs, and (4) directed acyclic graphs.Comment: 23 pages, 1 table, 9 figures, accepted to SIAM Journal on Discrete
Mathematic
Improved Compact Visibility Representation of Planar Graph via Schnyder's Realizer
Let be an -node planar graph. In a visibility representation of ,
each node of is represented by a horizontal line segment such that the line
segments representing any two adjacent nodes of are vertically visible to
each other. In the present paper we give the best known compact visibility
representation of . Given a canonical ordering of the triangulated , our
algorithm draws the graph incrementally in a greedy manner. We show that one of
three canonical orderings obtained from Schnyder's realizer for the
triangulated yields a visibility representation of no wider than
. Our easy-to-implement O(n)-time algorithm bypasses the
complicated subroutines for four-connected components and four-block trees
required by the best previously known algorithm of Kant. Our result provides a
negative answer to Kant's open question about whether is a
worst-case lower bound on the required width. Also, if has no degree-three
(respectively, degree-five) internal node, then our visibility representation
for is no wider than (respectively, ).
Moreover, if is four-connected, then our visibility representation for
is no wider than , matching the best known result of Kant and He. As a
by-product, we obtain a much simpler proof for a corollary of Wagner's Theorem
on realizers, due to Bonichon, Sa\"{e}c, and Mosbah.Comment: 11 pages, 6 figures, the preliminary version of this paper is to
appear in Proceedings of the 20th Annual Symposium on Theoretical Aspects of
Computer Science (STACS), Berlin, Germany, 200
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 , consisting of a plane graph of , and an
orderly spanning tree of . 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 , and (3) the best known
encodings of 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
Computing the Girth of a Planar Graph in Linear Time
The girth of a graph is the minimum weight of all simple cycles of the graph.
We study the problem of determining the girth of an n-node unweighted
undirected planar graph. The first non-trivial algorithm for the problem, given
by Djidjev, runs in O(n^{5/4} log n) time. Chalermsook, Fakcharoenphol, and
Nanongkai reduced the running time to O(n log^2 n). Weimann and Yuster further
reduced the running time to O(n log n). In this paper, we solve the problem in
O(n) time.Comment: 20 pages, 7 figures, accepted to SIAM Journal on Computin
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