D. T. Lee

Two algorithms for constructing a Delaunay triangulation

This paper provides a unified discussion of the Delaunay triangulation. Its geometric properties are reviewed and several applications are discussed. Two algorithms are presented for constructing the triangulation over a planar set ofN points. The first algorithm uses a divide-and-conquer approach. It runs inO(N logN) time, which is asymptotically optimal. The second algorithm is iterative and requiresO(N2) time in the worst case. However, its average case performance is comparable to that of the first algorithm.

Securing web application code by static analysis and runtime protection

Security remains a major roadblock to universal acceptance of the Web for many kinds of transactions, especially since the recent sharp increase in remotely exploitable vulnerabilities have been attributed to Web application bugs. Many verification tools are discovering previously unknown vulnerabilities in legacy C programs, raising hopes that the same success can be achieved with Web applications. In this paper, we describe a sound and holistic approach to ensuring Web application security. Viewing Web application vulnerabilities as a secure information flow problem, we created a lattice-based static analysis algorithm derived from type systems and typestate, and addressed its soundness. During the analysis, sections of code considered vulnerable are instrumented with runtime guards, thus securing Web applications in the absence of user intervention. With sufficient annotations, runtime overhead can be reduced to zero. We also created a tool named.WebSSARI (Web application Security by Static Analysis and Runtime Inspection) to test our algorithm, and used it to verify 230 open-source Web application projects on SourceForge.net, which were selected to represent projects of different maturity, popularity, and scale. 69 contained vulnerabilities. After notifying the developers, 38 acknowledged our findings and stated their plans to provide patches. Our statistics also show that static analysis reduced potential runtime overhead by 98.4%.

Medial Axis Transformation of a Planar Shape

The medial axis transformation is a means first proposed by Blum to describe a shape. In this paper we present a 0(n log n) algorithm for computing the medial axis of a planar shape represented by an n-edge simple polygon. The algorithm is an improvement over most previously known results interms of both efficiency and exactness and has been implemented in Fortran. Some computer-plotted output of the program are also shown in the paper.

Travel-time prediction with support vector regression

Travel time is a fundamental measure in transportation. Accurate travel-time prediction also is crucial to the development of intelligent transportation systems and advanced traveler information systems. We apply support vector regression (SVR) for travel-time prediction and compare its results to other baseline travel-time prediction methods using real highway traffic data. Since support vector machines have greater generalization ability and guarantee global minima for given training data, it is believed that SVR will perform well for time series analysis. Compared to other baseline predictors, our results show that the SVR predictor can significantly reduce both relative mean errors and root-mean-squared errors of predicted travel times. We demonstrate the feasibility of applying SVR in travel-time prediction and prove that SVR is applicable and performs well for traffic data analysis.

Computational complexity of art gallery problems

We study the computational complexity of the art gallery problem originally posed by Klee, and its variations. Specifically, the problem of determining the minimum number of vertex guards that can see an n -wall simply connected art gallery is shown to be NP-hard. The proof can be modified to show that the problems of determining the minimum number of edge guards and the minimum number of point guards in a simply connected polygonal region are also NP-hard. As a byproduct, the problem of decomposing a simple polygon into a minimum number of star-shaped polygons such that their union is the original polygon is also shown to be NP-hard.

Euclidean Shortest Paths in the Presence of Rectilinear Barriers

In this paper we address the problem of constructing a Euclidean shortest path between two specified points (source, destination) in the plane, which avoids a given set of barriers. This problem had been solved earlier for polygonal obstacles with the aid of the visibility graph. This approach however, has an Ω(n2) time lower bound, if n is the total number of vertices of the obstacles. Our goal is to find interesting cases for which the solution can be obtained without the explicit construction of the entire visibility graph. The two cases are (i) the path must lie within an n-vertex simple polygon; (ii) the obstacles are n disjoint and parallel line segments. In both instances greedy O(n log n) time algorithms can be developed which solve the problems by constructing the shortest-path tree from the source to all the vertices of the obstacles and to the destination.

On k-Nearest Neighbor Voronoi Diagrams in the Plane

The notion of Voronoi diagram for a set of N points in the Euclidean plane is generalized to the Voronoi diagram of order k and an iterative algorithm to construct the generalized diagram in 0(k2N log N) time using 0(k2(N − k)) space is presented. It is shown that the k-nearest neighbor problem and other seemingly unrelated problems can be solved efficiently with the diagram.

A simple on-line bin-packing algorithm

The one-dimensional on-line bin-packing problem is considered, A simple <italic>O</italic>(1)-space and <italic>O</italic>(<italic>n</italic>)-time algorithm, called HARMONIC<italic><subscrpt>M</subscrpt></italic>, is presented. It is shown that this algorithm can achieve a worst-case performance ratio of less than 1.692, which is better than that of the <italic>O</italic>(<italic>n</italic>)-space and <italic>O</italic>(<italic>n</italic> log <italic>n</italic>)-time algorithm FIRST FIT. Also shown is that 1.691 … is a lower bound for <italic>all</italic> <italic>0</italic>(1)-space on-line bin-packing algorithms. Finally a revised version of HARMONIC<italic><subscrpt>M </subscrpt></italic>, an <italic>O</italic>(<italic>n</italic>)-space and <italic>O</italic>(<italic>n</italic>)- time algorithm, is presented and is shown to have a worst-case performance ratio of less than 1.636. d

Generalization of Voronoi Diagrams in the Plane

In this paper we study the Voronoi diagram for a set of N line segments and circles in the Euclidean plane. The diagram is a generalization of the Voronoi diagram for a set of points in the plane and has applications in wire layout, facility location, clustering and contouring problems. We present an

Generalized delaunay triangulation for planar graphs

O(N(\log N)^2 )