5727433
An Experimental Study and Comparison of Topological Peeling and Topological Walk
In this paper, we present an experimental study comparing two algorithms, topological peeling and topological walk, for traversing arrangements of planar lines. Given a set H of n lines and a convex region R on a plane, both topological peeling and topological walk sweep the portion AR of the arrangement of H inside R in O(K + n log(n + r)) time and O(n + r) space, where K is the number of cells of AR and r is the number of boundary vertices of R. In our study, we robustly implemented these two algorithms using the LEDA library. Based on the implementation, we carried out experiments to conduct several comparisons, such as the arrangement traversal fashions, memory consumption, and execution time. In general, topological peeling exhibits a better control on the propagation of its sweeping curve (called the wavefront). For memory consumption, two types of measures, logical and physical memory, were examined. Our experiments showed that although both algorithms use nearly the same amount of logical memory, topological peeling could use twice as much physical memory as topological walk. For execution time, experiments revealed an interesting phenomenon that topological peeling has a 10% to 25% faster execution time than topological walk in most cases. Our analysis of this phenomenon indicates that the execution times of topological peeling and topological walk are both sensitive to the ratio of the lower input lines to all input lines. When the ratio of the lower lines to all input lines is around 85%, the two algorithms have roughly the same amount of execution time. Under this ratio, topological peeling considerably outperforms topological walk; above this ratio, topological walk slightly outperforms topological peeling.
16148439
Solving the brachytherapy seed localization problem using geometric and linear programming techniques
We propose a technique to solve the Brachytherapy Seed Localization problem in prostate brachytherapy. Our algorithm is based on novel geometric approaches to exploit the special structure of the problem combined with a number of key observations which help us formulate it as an Integer Program. We solve the equivalent linear program and present a variant of randomized rounding to yield an integral solution to our problem. The algorithm is efficient and performs extremely well in practice. We discuss in detail the underlying theory and performance evaluations based on our implementation.
42016450
Geometric permutations of higher dimensional spheres
In this paper, we prove that the maximum number of geometric permutations (induced by line transversals) of a set of n pairwise disjoint spheres with a bounded radius ratio in Rd for d ≥ 3 is at most 2⌊√2M⌋+1, where M is the ratio of the largest radius and the smallest radius of the spheres. Setting M to 1, this gives an upper bound of 4 on the maximum number of geometric permutations for congruent spheres in Rd, matching a recently independently discovered result [Y. Zhou, S. Suri, in: Proc. of 12th Annual ACM-SIAM Symp. on Discrete Algorithms, 2001, pp. 234-243] on this case. Our result settles a conjecture in combinatorial geometry.
6260724
Finding an optimal path without growing the tree
For problems on computing an optimal path as well as its length in a certain setting, the "standard" approach for finding an actual optimal path is by building (or "growing") a single-source optimal path tree. In this paper, we study a class of optimal path problems with the following phenomenon: The space complexity of the algorithms for reporting the lengths of single-source optimal paths for these problems is asymptotically smaller than the space complexity of the "standard" tree-growing algorithms for finding actual optimal paths. We present a general and efficient algorithmic paradigm for finding an actual optimal path for such problems without having to grow a single-source optimal path tree. Our paradigm is based on the "marriage-before-conquer" strategy, the prune-and-search technique, and a new data structure called clipped trees. The paradigm enables us to compute an actual path for a number of optimal path problems and dynamic programming problems in computational geometry, graph theory, and combinatorial optimization. Our algorithmic solutions improve the space bounds (in certain cases, the time bounds as well) of the previously best known algorithms, and settle some open problems. Our techniques are likely to be applicable to other problems.
2025275
Optimizing the sum of linear fractional functions and applications
1 I n t r o d u c t i o n . The problem of optimizing the sum of linear fractional functions (SOLF) is defined as follows: max f(Zl,...,Xd)= ~ [ : 7 : ~ d ) (Xl ..... z~t)ES i=1 ~ a r t m e n t of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN 46556, USA, {chen,odaescu,xwu,jxu}@cse.nd.edu. This research was supported in part by the National Science Foundation under Grant CCR-9623585. tThe work of this author was supported in part by a fellowship of the Center for Applied Mathematics, University of Notre Dame, Notre Dame, Indiana, USA. t Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro-ku, Tokyo 152, Japan, daiOis.titech.ac.jp. Grant-in-Aid of Ministry of Science, Culture and Education of Japan, No. 10780274. §Department of Architecture and Architectural Systems, Graduate School of Engineering, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto, 606-8501 Japan, naoki~archi.kyoto-u.ac.jp. Partially supported by Grant-inAid for Scientific Research on Priority Areas (B) by the Ministry of Education, Science, Sports and Culture of Japan. IThe work of this author was supported in part by a fellowship of the Center for Applied Mathematics, and by a summer graduate research fellowship of the Graduate School, University of Notre Dame, Notre Dame, Indiana, USA. such that for each i = 1 , 2 , . . . , m , ni(Xl,...,Xd) and di(Xl,...,Xd) are linear functions in a fixed d-D space ]~d, di (x t , . . . ,Xd) ~ 0 for any (Xl , . . . ,Xd) E S, the feasible domain S is defined by n linear constraints (i.e., half-spaces in Rd), and S ~ ¢. Each linear fractional term ~(~1 ..... ~ ) ddxl,...,z~) is called a ratio. The SOLF problem appears in the algorithmic solutions for several geometric optimization problems. In layered manufacturing, Majhi et al. [17], [18] showed that the length-optimal supports for a simple nonconvex polygon can be obtained by solving the 1-D SOLF problem on an interval. Arkin et al. [1] formulated the problem of finding a minimum-area starshaped or monotone polygon that contains a simple polygon, which appears in material layout [1], [8], [21] (e.g., cloth manufacturing) and manufacturing [18], as one of optimizing the sum of 2-D fractional polynomials of degrees 3 and 2 under linear constraints. We will show that the objective function of this problem can actually be simplified to the 2-D SOLF form. For the problem of finding optimal penetrations among weighted regions [5], we shall show tha t the L1 and Leo versions of this problem have the 2-D SOLF problem as a key subproblem. The SOLF problem also arises in other areas. For example, in combinatorial optimization, a class of problems on finding a structure with the maximum (or minimum) mean-weight cost can be solved by optimizing some linear fractional functions [23]. In operations research, it was shown in [11], [24] that many economic applications (e.g., maximization of productivity, return on investment, and return/risk) can be reduced to solving the SOLF problem. Quite a few solutions have been given for the SOLF problem [1], [11], [15], [17], [18], [25]. It is interesting to note that the SOLF related problems were originally studied in economic applications, where the number of variables is usually much bigger than the number of fractional terms. As a result, many previous SOLF algorithms were designed to target problems with only a few fractional terms (less than 10) for a reasonable
14297439
Geometric permutations of high dimensional spheres
We prove the maximum number of geometric permutations, induced by line transversals to a set of <i>n</i> pairwise disjoint congruent spheres in <i>R<sup>d</sup></i> with <i>d</i> ⪈ 3, is no more than 4 when <i>n</i> is sufficiently large, achieving the best known upper bound for this problem. We also prove the maximum number of geometric permutations of a set of <i>n</i> noncongruent spheres of bounded radius ratio in R<sup>d</sup>, <i>d</i> ⪈ 3, is at most 2<sup>[√2<i>M</i>]+1</sup>, where <i>M</i> is the ratio or the largest radius and the smallest radius. Our result settles a conjecture in combinatorial geometry.
39336188
Local Strongly Arc-Connectivity in Regular Bipartite Digraphs
We show that for any vertex x of a d-regular bipartite digraph there are a vertex y, in the other class of the bipartition, and d (x, y)-paths and d (y, x)-paths such that all 2d of them are pairwise arc-disjoint. This result generalizes a theorem of Hamidoune and Las Vergnas for graphs.
14424091
Sustaining Availability of Web Services under Distributed Denial of Service Attacks
The recent tide of Distributed Denial of Service (DDoS) attacks against high-profile web sites demonstrate how devastating DDoS attacks are and how defenseless the Internet is under such attacks. We design a practical DDoS defense system that can protect the availability of web services during severe DDoS attacks. The basic idea behind our system is to isolate and protect legitimate traffic from a huge volume of DDoS traffic when an attack occurs. Traffic that needs to be protected can be recognized and protected using efficient cryptographic techniques. Therefore, by provisioning adequate resource (e.g., bandwidth) to legitimate traffic separated by this process, we are able to provide adequate service to a large percentage of clients during DDoS attacks. The worst-case performance (effectiveness) of the system is evaluated based on a novel game theoretical framework, which characterizes the natural adversarial relationship between a DDoS adversary and the proposed system. We also conduct a simulation study to verify a key assumption used in the game-theoretical analysis and to demonstrate the system dynamics during an attack.
240982
Data streaming algorithms for estimating entropy of network traffic
Using entropy of traffic distributions has been shown to aid a wide variety of network monitoring applications such as anomaly detection, clustering to reveal interesting patterns, and traffic classification. However, realizing this potential benefit in practice requires accurate algorithms that can operate on high-speed links, with low CPU and memory requirements. In this paper, we investigate the problem of estimating the entropy in a streaming computation model. We give lower bounds for this problem, showing that neither approximation nor randomization alone will let us compute the entropy efficiently. We present two algorithms for randomly approximating the entropy in a time and space efficient manner, applicable for use on very high speed (greater than OC-48) links. The first algorithm for entropy estimation is inspired by the structural similarity with the seminal work of Alon et al. for estimating frequency moments, and we provide strong theoretical guarantees on the error and resource usage. Our second algorithm utilizes the observation that the performance of the streaming algorithm can be enhanced by separating the high-frequency items (or elephants) from the low-frequency items (or mice). We evaluate our algorithms on traffic traces from different deployment scenarios.
8575653
Using Shortest Path Algorithms to Identify Criminal Associations
Frequently in criminal investigations, law enforcement agencies face the problem of identifying associations between a group of entities such as individuals and organizations. In this paper we present a link analysis technique to solve such a problem. This approach uses shortest path algorithms to find the strongest associations between two or more given entities. The experimental results have demonstrated that our approach is potentially useful in terms of quality and efficiency. Specifically, we found that the two-tree Priority-First Search algorithm in most cases was the fastest algorithm to find shortest paths and the paths found consisted of meaningful criminal associations around 80 percent of the time.
19232033
Protein Threading by Linear Programming: Theoretical Analysis and Computational Results
In a previous paper (Xu, Li, Kim, and Xu, Journal of Bioinformatics and Computational Biology, vol. 1, no. 1, pp. 95–117, 2003), we have used an integer programming approach to implement a protein threading program RAPTOR for protein 3D structure prediction, based on the threading model treating pairwise contacts rigorously and allowing variable gaps. We have solved the integer program by the canonical branch-and-bound method. In this paper we present a branch-and-cut method, a careful theoretical analysis of our formulation and why our approach is so effective. The result of cutting plane analysis is that two types of well-known cuts for this problem are already implied in the constraint set, which provides us some intuition that our formulation would be very effective. Experimental results show that for about 99 percent of real threading instances, the linear relaxations of their integer programs solve to integral optimal solutions directly. For the rest one percent of real instances, the integral solutions can be obtained with only several branch nodes. Experimental results also show that no special template or sequence features result in more possibilities of fractional solutions. This indicates that extra effort to seek for cutting planes to strengthen the existing formulation is unnecessary.