Hsueh-I Lu

Approximating Maximum Leaf Spanning Trees in Almost Linear Time

Given an undirected graph, finding a spanning tree of the graph with the maximum number of leaves is MAX SNP-complete. In this paper we give a new greedy 3-approximation algorithm for maximum leaf spanning trees. The running timeO((m+n)?(m,n)) required by our algorithm, wheremis the number of edges andnis the number of nodes, is almost linear in the size of the graph. We also demonstrate that our analysis of the performance of the greedy algorithm is tight via an example.

An Optimal Algorithm for the Maximum-Density Segment Problem

We address a fundamental problem arising from analysis of biomolecular sequences. The input consists of two numbers wmin and wmax and a sequence S of n number pairs (ai,wi) with wi > 0. Let segment S(i,j) of S be the consecutive subsequence of S between indices i and j. The density of S(i,j) is d(i,j) = (ai + ai + 1 + cdots + aj)/(wi + wi + 1 + cdots + wj)

Orderly Spanning Trees with Applications

. The maximum-density segment problem is to find a maximum-density segment over all segments S(i,j) with wmin leq wi + wi + 1 + cdots + wj leq wmax. The best previously known algorithm for the problem, due to Goldwasser, Kao, and Lu [Proceedings of the Second International Workshop on Algorithms in Bioinformatics, R. Guigo and D. Gusfield, eds., Lecture Notes in Comput. Sci. 2452, Springer-Verlag, New York, 2002, pp. 157--171], runs in O(n log(wmax- wmin+1)) time. In the present paper, we solve the problem in O(n) time. Our approach bypasses the complicated right-skew decomposition, introduced by Lin, Jiang, and Chao [J. Comput. System Sci., 65 (2002), pp. 570--586]. As a result, our algorithm has the capability to process the input sequence in an online manner, which is an important feature for dealing with genome-scale sequences. Moreover, for a type of input sequences S representable in O(m) space, we show how to exploit the sparsity of S and solve the maximum-density segment problem for S in O(m) time.

Balanced parentheses strike back

We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes 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 orderly pair for any connected planar graph G, consisting of an embedded planar graph H isomorphic to G, and an orderly spanning tree of H. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyders realizer theorem, (2) the first algorithm for computing an area-optimal 2-visibility drawing of a planar graph, and (3) the most compact known encoding of a planar graph with O(1)-time query support. All algorithms in this paper run in linear time.

Improved Compact Visibility Representation of Planar Graph via Schnyder's Realizer

An ordinal tree is an arbitrary rooted tree where the children of each node are ordered. Succinct representations for ordinal trees with efficient query support have been extensively studied. The best previously known result is due to Geary et al. [2004b, pages 1--10]. The number of bits required by their representation for an n-node ordinal tree T is 2n + o(n), whose first-order term is information-theoretically optimal. Their representation supports a large set of O(1)-time queries on T. Based upon a balanced string of 2n parentheses, we give an improved 2n + o(n)-bit representation for T. Our improvement is two-fold: First, the set of O(1)-time queries supported by our representation is a proper superset of that supported by the representation of Geary, Raman, and Raman. Second, it is also much easier for our representation to support new queries by simply adding new auxiliary strings.

Compact floor-planning via orderly spanning trees

Let G be an n-node planar graph. In a visibility representation of G, each node of G is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of G are vertically visible to each other. In the present paper we give the best known compact visibility representation of G. Given a canonical ordering of the triangulated G, our algorithm draws the graph incrementally in a greedy manner. We show that one of three canonical orderings obtained from Schnyders realizer for the triangulated G yields a visibility representation of G no wider than

Power-saving scheduling for weakly dynamic voltage scaling devices

leftlfloor{frac{22n-40}{15}}rightrfloor

Linear-time compression of bounded-genus graphs into information-theoretically optimal number of bits

. 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 Kants open question about whether

Approximation Algorithms for Multiprocessor Energy-Efficient Scheduling of Periodic Real-Time Tasks with Uncertain Task Execution Time

leftlfloor{frac{3n-6}{2}}rightrfloor

Improved Compact Routing Tables for Planar Networks via Orderly Spanning Trees

is a worst-case lower bound on the required width. Also, if G has no degree-three (respectively, degree-five) internal node, then our visibility representation for G is no wider than