Given a set of n nonnegative weighted circular-arcs on a unit circle, and an integer k, the k Best Cuts for Circular-Arcs problem, abbreviated as k-BCCA problem, is to find a placement of k points, called cuts, on the circle such that the total weight of the arcs that contain at least one cut is maximized. We first solve a simpler version using dynamic programming, the k Best Cuts for Intervals (k-BCI) problem in O(kn + n log n) time and O(kn) space. The algorithm is then extended to solve a variation, called the k-restricted BCI problem, and the space complexity of the k-BCI problem can be improved to O(n). Based on these results, we then show that the k-BCCA problem can be solved in O(I(k; n) + n log n) time, where I(k; n) is the time complexity of the k-BCI problem. As a by-product, the k Maximum Cliques Cover problem, (k ? 1) for the circular-arc graphs can be solved in O(I(k; n) + n log n) time. Supported in part by the National Science Foundation under the Grants CCR-890181...

The pairwise nearest neighbor (PNN) method, also known as Ward's method belongs to the class of agglomerative clustering methods. The PNN method generates hierarchical clustering using a sequence of merge operations until the desired number of clusters is obtained. This method selects the cluster pair to be merged so that it increases the given objective function value least. The main drawback of the PNN method is its slowness because the time complexity of the fastest known exact implementation of the PNN method is lower bounded by �(N 2), where N is the number of data objects. We consider several speed-up methods for the PNN method in the first publication. These methods maintain the precision of the method. Another method for speeding-up the PNN method is investigated in the second publication, where we utilize a k-neighborhood graph for reducing distance calculations and operations. A remarkable speed-up is achieved at the cost of slight increase in distortion. The PNN method can also be adapted for multilevel thresholding, which can be seen as

When restricted to cost arrays possessing the sum Monge property, many combinatorial optimization problems with sum objective functions become significantly easier to solve. (An array A = fa[i; j]g possesses the sum Monge property if for all i ! k and j ! `, a[i; j] + a[k; `] a[i; `] + a[k; j].) Examples include the usual sum-objective-function versions of the assignment problem, the transportation problem, the traveling-salesman problem, and several shortest-path problems. Furthermore, the more general algebraic assignment and transportation problems, which are formulated in terms of an ordered commutative semigroup (H; ; ), are similarly easier to solve given cost arrays possessing the corresponding algebraic Monge property, which requires that for all i ! k and j ! `, a[i; j] a[k; `] a[i; `] a[k; j]. In this paper, we show that Monge-array results for two sum-of-edge-costs shortest-path problems can likewise be extended to a general algebraic setting, provided the problems' ordered commutative semigroup (H; ; ) satisfies one additional restriction. Specifically, we require that for all a; b; c 2 H , a OE b implies c a OE c b. In addition to this general result, we also show how our algorithms can be modified to solve certain bottleneck shortest-path problems, even though the ordered commutative semigroup (!; max; ) naturally associated with bottleneck problems does not satisfy our additional restriction. The bottleneck shortest-path problems we can solve are those with cost arrays possessing what we call the strict bottleneck Monge property, which requires that for all i ! k and j ! `, either maxfa[i; j]; a[k; `]g ! maxfa[i; `]; a[k; j]g or both maxfa[i; j]; a[k; `]g = maxfa[i; `]; a[k; j]g and minfa[i; j]; a[k; `]g minfa[i; `]; a[k; j]g. Fin...

Abstract- Shortest Path problems are among the most studied network flow optimization problems with interesting applications in a wide range of fields. One such application is in the field of GPS routing systems. These systems need to quickly solve large shortest path problems but are typically embedded in devices with limited memory and external storage. Conventional techniques for solving shortest paths within large networks cannot be used as they are either too slow or require huge amounts of storage. In this project we have tried to reduce the runtime of conventional techniques by exploiting the physical structure of the road network and using network pre-processing techniques. Our algorithms may not guarantee optimal results but can offer significant savings in terms of memory requirements and processing speed. Our work uses heuristic estimates to bind the search and directs it towards a destination. We also associate a radius with each node that gives a measure of importance for roads in the network. The farther we get from either the origin or destination the more selective we become about the roads we travel with greater importance (i.e. roads with larger radii).By using these techniques we were able to dramatically reduce the runtime performance compared to conventional techniques while still maintaining an acceptable level of accuracy. Keywords- Routing, Shortest Path, Network, Radius. I

Given a probability distribution over a set of n words to be transmitted, the Huffman Coding problem is to find a minimal-cost prefix free code for transmitting those words. The basic Huffman coding problem can be solved in O(nlog n) time but variations are more difficult. One of the standard techniques for solving these variations utilizes a top-down dynamic programming approach. In this paper we show that this approach is amenable to dynamic programming speedup techniques, permitting a speedup of an order of magnitude for many algorithms in the literature for such variations as mixed radix, reserved length and one-ended coding. These speedups are immediate implications of a general structural property that permits batching together the calculation of many DP entries. 1

Abstract. In this paper we address the problem of building a compressed self-index that, given a distribution for the pattern queries and a bound on the space occupancy, minimizes the expected query-time within that index-space bound. We solve this problem by exploiting a reduction to the problem of finding a minimum weight K-linkpathina particular Directed Acyclic Graph. Interestingly enough, our solution is independent of the underlying compressed index in use. Our experiments compare this optimal strategy with several other standard approaches, showing its effectiveness in practice. 1

An instance of a p-median problem gives n demand points. The ob-jective is to locate p supply points in order to minimize the total distance of the demand points to their nearest supply point. P-median is polynomially solvable in one dimension but NP-hard in two or more dimensions, when either the Euclidean or the rectilinear distance measure is used. In this thesis, we treat the p-median problem under a new distance measure, the directional rec-tilinear distance, which requires the nearest supply point for a given demand point to lie above and to the right of it. This restriction has applications to multiprocessor scheduling of periodic tasks as well as to traffic quantization and Quality of Service scheduling in packet-switched computer networks. We show that the directional p-median problem is polynomially solvable in one dimension and give two algorithms. We prove the problem NP-hard in two or more dimensions and then present an efficient heuristic to solve it. Compared to the robust Teitz & Bart heuristic for p-median, our heuristic enjoys substan-tial speedup while sacrificing little in terms of solution quality, making it an ideal choice for our target applications with thousands of demand points.