14266211
Efficient algorithms for minimum congestion hypergraph embedding in a cycle
The minimum congestion hypergraph embedding in a cycle (MCHEC) problem is to embed the hyperedges of a hypergraph as paths in a cycle with the same node set such that the maximum congestion (the maximum number of paths that use any single edge in the cycle) is minimized. The MCHEC problem has many applications, including optimizing communication congestions in computer networks and parallel computing. The problem is NP-hard. In this paper, we give a 1.8-approximation algorithm for the MCHEC problem. This improves the previous 2-approximation results. Our algorithm has the optimal time complexity O(mn) for a hypergraph with m hyperedges and n nodes. We also propose an algorithm which finds an embedding with the optimal congestion L* for the MCHEC problem in O(n(nL*)/sup L*/) time. This improves the previous O((mn)/sup L*+1/) time algorithm.
14047084
Efficient Algorithm for Embedding Hypergraphs in a Cycle
The problem of Minimum Congestion Hypergraph Embedding in a Cycle (MCHEC) is to embed the hyperedges of a hypergraph as paths in a cycle such that the maximum congestion (the maximum number of paths that use any single link in the cycle) is minimized. This problem has many applications, including minimizing communication congestions in computer networks and parallel computations. The MCHEC problem is NP-hard. We give a 1.8-approximation algorithm for the problem. This improves the previous 2-approximation results. The algorithm has the optimal O(mn) time for the hypergraph with m hyperedges and n nodes.
2617111
Efficient Algorithms for Traffic Grooming in SONET/WDM Networks
In SONET/WDM optical networks, a wavelength channel is shared by multiple low-rate traffic demands. The multiplexing is known as traffic grooming and carried out by SONET add-drop multiplexers (SADM). A key optimization problem in traffic grooming is to minimize the number of SADMs. This optimization problem is challenging and NP-hard even for unidirectional SONET/WDM rings (UPSR) with symmetric unitary traffic demands. In this paper, we give a linear time heuristic algorithm for this NP-hard problem. Empirical results show that the algorithm outperforms previous algorithms. The algorithm uses the minimum number of wavelengths, which are also precious resources in optical networks. An important subclass of the symmetric unitary traffic pattern is the regular traffic pattern, where each network node appears in exactly r symmetric demands. The regular traffic pattern is a generalization of the well known all-to-all traffic pattern, in which r = n - 1 for a network of n nodes. We prove that the optimization problem remains NP-hard for the regular traffic pattern on the UPSR. We also propose an algorithm for this problem with a better upper bound on the number of used SADMs than previous algorithms. This algorithm always uses the minimum number of wavelengths as well
2273352
Near-Linear Time Approximation Algorithms for Curve Simplification
AbstractWe consider the problem of approximating a polygonal
 curve P under a given error criterion by another
 polygonal curve P’ whose vertices are a
 subset of the vertices of P. The goal is to minimize
 the number of vertices of P’ while ensuring that the
 error between P’ and P is below a certain threshold.
 We consider two different error measures: 
 Hausdorff and Frechet. For both error criteria, we present
 near-linear time approximation algorithms that, given a
 parameter ε > 0, compute a simplified polygonal
 curve P’ whose error is less than ε and
 size at most the size of an optimal simplified
 polygonal curve with error ε/2. We consider
 monotone curves in ℝ2 in the case of the Hausdorff error measure
 under the uniform distance metric
 and arbitrary curves in any dimension for the Frechet error measure
 under Lp metrics. 
 We present
 experimental results demonstrating that our algorithms
 are simple and fast, and produce close to optimal
 simplifications in practice.
2560769
Invariance Properties of Random Sequences
We present invariance characterizations of different types of random sequences. We correct Schnorr’s original, incorrect characterization of Martin-Lof random sequences, compare it with Schnorr’s corresponding characterization of his own randomness concept, and give a similar, new chararacterization of Kurtz random sequences. That is, we show that an infinite sequence ξ is Kurtz random if and only if for every partial, computable, measure-invariant function Φ : Σ → Σ the sequence Φ(ξ) is not recursive.
509991
Brain Surface Conformal Parameterization with Algebraic Functions
In medical imaging, parameterized 3D surface models are of great interest for anatomical modeling and visualization, statistical comparisons of anatomy, and surface-based registration and signal processing. Here we introduce a parameterization method based on algebraic functions. By solving the Yamabe equation with the Ricci flow method, we can conformally map a brain surface to a multi-hole disk. The resulting parameterizations do not have any singularities and are intrinsic and stable. To illustrate the technique, we computed parameterizations of several types of anatomical surfaces in MRI scans of the brain, including the hippocampi and the cerebral cortices with various landmark curves labeled. For the cerebral cortical surfaces, we show the parameterization results are consistent with selected landmark curves and can be matched to each other using constrained harmonic maps. Unlike previous planar conformal parameterization methods, our algorithm does not introduce any singularity points. It also offers a method to explicitly match landmark curves between anatomical surfaces such as the cortex, and to compute conformal invariants for statistical comparisons of anatomy.
14088697
Approximate shortest paths in anisotropic regions
Our goal is to find an approximate shortest path for a point robot moving in a planar subdivision with <i>n</i> vertices. Let ρ ≥ 1 be a real number. Distances in each face of this subdivision are measured by a convex distance function whose unit disk is contained in a concentric unit Euclidean disk, and contains a concentric Euclidean disk with radius 1/ρ. Different convex distance functions may be used for different faces, and obstacles are allowed. These convex distance functions may be asymmetric. For all ε ∈ (0, 1), and for any two points <i>v<inf>s</inf></i> and <i>v<inf>d</inf></i>, we give an algorithm that finds a path from <i>v<inf>s</inf></i> to <i>v<inf>d</inf></i> whose cost is at most (1 + ε) times the minimum cost. Our algorithm runs in <i>O</i> (ρ<sub>2</sub>logρ/ε<sup>2</sup>n<sup>3</sup> log (ρn/ε)) time. This bound does not depend on any other parameters; in particular, it does not depend on the minimum angle in the subdivision. We give applications to two special cases that have been considered before: the weighted region problem and motion planning in the presence of uniform flows. For the weighted region problem with weights in [1, ρ] ∪ {∞}, the time bound of our algorithm improves to <i>O</i> (ρ<sub>2</sub>logρ/εn<sup>3</sup> log (ρn/ε)).
18269009
Maximum Flows and Critical Vertices in AND/OR Graphs
We will study this problem and present an algorithm for finding the minimum-time-cost solution graph in an AND/OR graph. We will also study the following problems which often appear in industry when using AND/OR graphs to model manufacturing processes or to model problem solving processes: finding maximum (additive and nonadditive) flows and critical vertices in an AND/OR graph. Though there are well known polynomial time algorithms for the corresponding problems in the traditional graph theory, we will show that generally it is NP-hard to find a non-additive maximum flow in an AND/OR graph, and it is both NP-hard and coNP-hard to find a set of critical vertices in an AND/OR graph. We will also present a polynomial time algorithm for finding a maximum additive flow in an AND/OR graph, and discuss the relative complexity of these problems.