An efficient Algorithm to partition a Sequence of Integers into Subsets with equal Sums

To partition a sequence of n integers into subsets with prescribed sums is an NP-hard problem in general. In this paper we present an efficient solution for the homogeneous version of this problem; i.e. where the elements in each subset add up to the same sum.Comment: 12 page

Applications of Soft Computing in Mobile and Wireless Communications

Soft computing is a synergistic combination of artificial intelligence methodologies to model and solve real world problems that are either impossible or too difficult to model mathematically. Furthermore, the use of conventional modeling techniques demands rigor, precision and certainty, which carry computational cost. On the other hand, soft computing utilizes computation, reasoning and inference to reduce computational cost by exploiting tolerance for imprecision, uncertainty, partial truth and approximation. In addition to computational cost savings, soft computing is an excellent platform for autonomic computing, owing to its roots in artificial intelligence. Wireless communication networks are associated with much uncertainty and imprecision due to a number of stochastic processes such as escalating number of access points, constantly changing propagation channels, sudden variations in network load and random mobility of users. This reality has fuelled numerous applications of soft computing techniques in mobile and wireless communications. This paper reviews various applications of the core soft computing methodologies in mobile and wireless communications

A pseudo-spectral approach to inverse problems in interface dynamics

An improved scheme for computing coupling parameters of the Kardar-Parisi-Zhang equation from a collection of successive interface profiles, is presented. The approach hinges on a spectral representation of this equation. An appropriate discretization based on a Fourier representation, is discussed as a by-product of the above scheme. Our method is first tested on profiles generated by a one-dimensional Kardar-Parisi-Zhang equation where it is shown to reproduce the input parameters very accurately. When applied to microscopic models of growth, it provides the values of the coupling parameters associated with the corresponding continuum equations. This technique favorably compares with previous methods based on real space schemes.Comment: 12 pages, 9 figures, revtex 3.0 with epsf style, to appear in Phys. Rev.

Sticky Brownian Rounding and its Applications to Constraint Satisfaction Problems

Semidefinite programming is a powerful tool in the design and analysis of approximation algorithms for combinatorial optimization problems. In particular, the random hyperplane rounding method of Goemans and Williamson has been extensively studied for more than two decades, resulting in various extensions to the original technique and beautiful algorithms for a wide range of applications. Despite the fact that this approach yields tight approximation guarantees for some problems, e.g., Max-Cut, for many others, e.g., Max-SAT and Max-DiCut, the tight approximation ratio is still unknown. One of the main reasons for this is the fact that very few techniques for rounding semidefinite relaxations are known. In this work, we present a new general and simple method for rounding semi-definite programs, based on Brownian motion. Our approach is inspired by recent results in algorithmic discrepancy theory. We develop and present tools for analyzing our new rounding algorithms, utilizing mathematical machinery from the theory of Brownian motion, complex analysis, and partial differential equations. Focusing on constraint satisfaction problems, we apply our method to several classical problems, including Max-Cut, Max-2SAT, and MaxDiCut, and derive new algorithms that are competitive with the best known results. To illustrate the versatility and general applicability of our approach, we give new approximation algorithms for the Max-Cut problem with side constraints that crucially utilizes measure concentration results for the Sticky Brownian Motion, a feature missing from hyperplane rounding and its generalization

Mathematics and Morphogenesis of the City: A Geometrical Approach

Cities are living organisms. They are out of equilibrium, open systems that never stop developing and sometimes die. The local geography can be compared to a shell constraining its development. In brief, a city's current layout is a step in a running morphogenesis process. Thus cities display a huge diversity of shapes and none of traditional models from random graphs, complex networks theory or stochastic geometry takes into account geometrical, functional and dynamical aspects of a city in the same framework. We present here a global mathematical model dedicated to cities that permits describing, manipulating and explaining cities' overall shape and layout of their street systems. This street-based framework conciliates the topological and geometrical sides of the problem. From the static analysis of several French towns (topology of first and second order, anisotropy, streets scaling) we make the hypothesis that the development of a city follows a logic of division / extension of space. We propose a dynamical model that mimics this logic and which from simple general rules and a few parameters succeeds in generating a large diversity of cities and in reproducing the general features the static analysis has pointed out.Comment: 13 pages, 13 figure

Convex Dynamics and Applications

This paper proves a theorem about bounding orbits of a time dependent dynamical system. The maps that are involved are examples in convex dynamics, by which we mean the dynamics of piecewise isometries where the pieces are convex. The theorem came to the attention of the authors in connection with the problem of digital halftoning. \textit{Digital halftoning} is a family of printing technologies for getting full color images from only a few different colors deposited at dots all of the same size. The simplest version consist in obtaining grey scale images from only black and white dots. A corollary of the theorem is that for \textit{error diffusion}, one of the methods of digital halftoning, averages of colors of the printed dots converge to averages of the colors taken from the same dots of the actual images. Digital printing is a special case of a much wider class of scheduling problems to which the theorem applies. Convex dynamics has roots in classical areas of mathematics such as symbolic dynamics, Diophantine approximation, and the theory of uniform distributions.Comment: LaTex with 9 PostScript figure