6305531
Shortest path queries in planar graphs
The problem of processing shortest pa th queries in graphs arises in application areas such as intelligent t ranspor ta t ion system (ITS), geographic information system (GIS), and robotics. In this paper, we present an efficient algorithmic solution for processing shortest pa th queries in undirected planar graphs with non-negative edge weights. Previous algorithms for this problem on planar graphs all have a tradeoff between the query t ime and the da ta structure space used by the solutions. The previously best known trade-off on an n-vertex planar graph G in the worst case is O(v/-r) t ime for a pa th length query and O(v/-r + L) t ime for report ing an actual shortest path, with a da ta structure of O(n2/x/~) space, where r is an integer parameter with 1 < r _< n and L is the number of edges on the output shortest path. We present a new scheme, called frame search, that exploits the graph planari ty in a novel fashion. Wi th this scheme, we build an improved da ta structure of O(n + pv/-r + p2/r) space (in O(n + p2/v/~ + pr 3/4) time), where p is the minimum cardinality of a face-covering of the planar embedding of G with 1 < p < O(n), and r is an integer parameter with 1 < r < p. This da ta structure enables us to process each length query in O(v/-rlog r + a(n)) t ime and report an actual shortest path in an addit ional O(L) time, where a(n) is the inverse of Ackermann's function. In the worst case, our approach reduces the previously best query t ime by a factor of up to O(n 1/4) if the same amount of space is used. Our technique can also be applied to improve the previous query algorithms for some geometric shortest pa th problems on the plane.
29728094
Graph Bandwidth of Weighted Caterpillars
Graph bandwidth minimization (GBM) is a classical and challenging problem in graph algorithms and combinatorial optimization. Most of existing researches on this problem have focused on unweighted graphs. In this paper, we study the bandwidth minimization problem of weighted caterpillars, and propose several algorithms for solving various types of caterpillars and general graphs. More specifically, we show that the GBM problem on caterpillars with hair-length at most 2 and the GBM problem on star-shape caterpillars are NP-complete, and give a lower bound of the graph bandwidth for general weighted graphs. For caterpillars with hair-length at most 1, we present an O(n log n log(nwmax))-time algorithm to compute an optimal bandwidth layout, where n is the total number of vertices in the graph and wmax is the maximum wedge weight. For caterpillars with hair-length at most k, we give a k-approximation algorithm. For arbitrary caterpillars and general graphs, we give a heuristic algorithm and some experimental results. Experiments show that the solutions obtained by our heuristic algorithm are roughly within a factor of c log(n) of the lower bound for a small number c, which is consistent with the inapproximability results of this problem (i.e., no constant approximation for the GBM problem unless P = NP). © 2006 Published by Elsevier B.V.
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.
21098955
Topological Peeling and Implementation
We present a new approach, called topological peeling, and its implementation for traversing a portion AR of the arrangement formed by n lines within a convex region R on the plane. Topological peeling visits the cells of AR in a fashion of propagating a "wave" of a special shape (called a double-wriggle curve) starting at a single source point. This special traversal fashion enables us to solve several problems (e.g., computing shortest paths) on planar arrangements to which previously best known arrangement traversal techniques such as topological sweep and topological walkm ay not be directly applicable. Our topological peeling algorithm takes O(K + n log(n + r)) time and O(n + r) space, where K is the number of cells in AR and r is the number of boundary vertices of R. Comparing with topological walk, topological peeling uses a simpler and more efficient way to sweep different types of lines, and relies heavily on exploring small local structures, rather than a much larger global structure. Experiments show that, on average, topological peeling outperforms topological walkb y 10 - 15% in execution time.
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.
28240465
Motion Tracking and Intensity Surface Recovery in Microscopic Nuclear Images
Current techniques for microscopic imaging do not provide necessary spatial and temporal resolutions for real time visualization of the nucleus. Images can only be acquired in time lapse mode, leading to significant loss of information between image frames. Such data, if available, can be extremely helpful in the study of nuclear organization and function. In this paper, we present a gamut of geometric-technique-based approaches for solving the problem. Our techniques, working together, can effectively recover complicated motion and deformation as well as the change of intensity surfaces from pairs of images in a microscopic image sequence, and has low time complexity, particularly desirable by many biological applications where large amount of DNA need to be processed. These techniques are also readily applicable to other types of images for reconstructing motion and intensity surfaces of deformable objects.
5147011
Wide Diameters of Cartesian Product Graphs and Digraphs
In graph theory and study of fault tolerance and transmission delay of networks, connectivity and diameter of a graph are two very important parameters and have been deeply studied by many authors. Wide diameter combining connectivity with diameter is a more important parameter to measure fault tolerance and efficiency of parallel processing computer networks and has received much attention in the recent years. Diameter with width k of a graph G is defined as the minimum integer d for which between any two distinct vertices in G there exist at least k internally disjoint paths of length at most d. In the present paper, the tight upper bounds of wide diameter of the Cartesian product graphs are obtained. Some known results can be deduced or improved from ours.
10184291
Efficient and scalable query routing for unstructured peer-to-peer networks
Searching for content in peer-to-peer networks is an interesting and challenging problem. Queries in Gnutella-like unstructured systems that use flooding or random walk to search must visit O(n) nodes in a network of size n, thus consuming significant amounts of bandwidth. In this paper, we propose a query routing protocol that allows low bandwidth consumption during query forwarding using a low cost mechanism to create and maintain information about nearby objects. To achieve this, our protocol maintains a lightweight probabilistic routing table at each node that suggests the location of each object in the network. Following the corresponding routing table entries, a query can reach the destination in a small number of hops with high probability. However, maintaining routing tables in a large and highly dynamic network requires non-traditional mechanisms. We design a novel data structure called an exponentially decaying bloom filter (EDBF) that encodes such probabilistic routing tables in a highly compressed manner, and allows for efficient aggregation and propagation. The search primitives provided by our system can be used to search for single keys or multiple keywords with equal ease. Analytical modeling of our design predicts significant improvements in search efficiency, verified through extensive simulations in which we observed an order of magnitude reduction in query path length over previous proposals.
17686330
Protein Threading by Linear Programming
Protein three-dimensional structure prediction through threading approach has been extensively studied and various models and algorithms have been proposed. In order to further explore ways to improve accuracy and efficiency of the threading process, this paper investigates the effectiveness of a new method: protein threading via linear programming. Based on the contact map model of protein 3D structure, we formulate the protein threading problem as a large scale integer programming problem, then relax to a linear programming problem, and finally solve the integer program by a branch-and-bound method. The final solution is optimal with respect to energy functions incorporating pairwise interaction and allowing variable gaps. The algorithm has been implemented as software package RAPTOR--RApid Protein Threading predictOR. Experimental results for fold recognition show that RAPTOR significantly outperforms other programs at the fold similarity level. The RAPTOR webserver is at http://www.cs.uwaterloo.ca/-j3xu/RAPTOR_form.htm.
1942335
On restricted edge-connectivity of graphs
This paper considers the concept of restricted edge-connectivity, and relates that to the edge-degree of a connected graph. The author gives some necessary conditions for a graph whose restricted edge-connectivity is smaller than its minimum edge-degree, then uses these conditions to show some large classes of graphs, such as all connected edge-transitive graphs except a star, and all connected vertex-transitive graphs with odd order or without triangles, have equality of the restricted edge-connectivity and the minimun edge-degree.
26863394
A Firewalling Scheme for Securing MPOA-Based Enterprise Networks
A well-known security problem with MPOA is that cut-through connections generally bypasses firewall routers if there are any. None of the previously proposed approaches solved the problem properly. In this paper, we propose a novel firewalling scheme for MPOA that nicely fixes the security hole. Our firewalling scheme has three outstanding advantages that make it ideal for securing MPOA-based enterprise networks. First, based on our novel concept of "logical chokepoints", our firewalling scheme does not require the existence of physical chokepoints inside the network. Second, the scheme is nicely embedded into the MPOA protocol so that its cost, performance overhead, and protocol complexity are reduced to a minimum. Third, the scheme is centrally administered so that it scales well to very large networks.