Counting Integer flows in Networks

This paper discusses new analytic algorithms and software for the enumeration of all integer flows inside a network. Concrete applications abound in graph theory \cite{Jaeger}, representation theory \cite{kirillov}, and statistics \cite{persi}. Our methods clearly surpass traditional exhaustive enumeration and other algorithms and can even yield formulas when the input data contains some parameters. These methods are based on the study of rational functions with poles on arrangements of hyperplanes

Optimization of Free Space Optical Wireless Network for Cellular Backhauling

With densification of nodes in cellular networks, free space optic (FSO) connections are becoming an appealing low cost and high rate alternative to copper and fiber as the backhaul solution for wireless communication systems. To ensure a reliable cellular backhaul, provisions for redundant, disjoint paths between the nodes must be made in the design phase. This paper aims at finding a cost-effective solution to upgrade the cellular backhaul with pre-deployed optical fibers using FSO links and mirror components. Since the quality of the FSO links depends on several factors, such as transmission distance, power, and weather conditions, we adopt an elaborate formulation to calculate link reliability. We present a novel integer linear programming model to approach optimal FSO backhaul design, guaranteeing $K$-disjoint paths connecting each node pair. Next, we derive a column generation method to a path-oriented mathematical formulation. Applying the method in a sequential manner enables high computational scalability. We use realistic scenarios to demonstrate our approaches efficiently provide optimal or near-optimal solutions, and thereby allow for accurately dealing with the trade-off between cost and reliability

The Number of Nowhere-Zero Flows on Graphs and Signed Graphs

A nowhere-zero $k$-flow on a graph $\Gamma$ is a mapping from the edges of $\Gamma$ to the set \{\pm1, \pm2, ..., \pm(k-1)\} \subset \bbZ such that, in any fixed orientation of $\Gamma$, at each node the sum of the labels over the edges pointing towards the node equals the sum over the edges pointing away from the node. We show that the existence of an \emph{integral flow polynomial} that counts nowhere-zero $k$-flows on a graph, due to Kochol, is a consequence of a general theory of inside-out polytopes. The same holds for flows on signed graphs. We develop these theories, as well as the related counting theory of nowhere-zero flows on a signed graph with values in an abelian group of odd order. Our results are of two kinds: polynomiality or quasipolynomiality of the flow counting functions, and reciprocity laws that interpret the evaluations of the flow polynomials at negative integers in terms of the combinatorics of the graph.Comment: 17 pages, to appear in J. Combinatorial Th. Ser.

Enumerating contingency tables via random permanents

Given m positive integers R=(r_i), n positive integers C=(c_j) such that sum r_i = sum c_j =N, and mn non-negative weights W=(w_{ij}), we consider the total weight T=T(R, C; W) of non-negative integer matrices (contingency tables) D=(d_{ij}) with the row sums r_i, column sums c_j, and the weight of D equal to prod w_{ij}^{d_{ij}}. We present a randomized algorithm of a polynomial in N complexity which computes a number T'=T'(R,C; W) such that T' < T < alpha(R, C) T' where alpha(R,C) = min{prod r_i! r_i^{-r_i}, prod c_j! c_j^{-c_j}} N^N/N!. In many cases, ln T' provides an asymptotically accurate estimate of ln T. The idea of the algorithm is to express T as the expectation of the permanent of an N x N random matrix with exponentially distributed entries and approximate the expectation by the integral T' of an efficiently computable log-concave function on R^{mn}. Applications to counting integer flows in graphs are also discussed.Comment: 19 pages, bounds are sharpened, references are adde

Distinct counting with a self-learning bitmap

Counting the number of distinct elements (cardinality) in a dataset is a fundamental problem in database management. In recent years, due to many of its modern applications, there has been significant interest to address the distinct counting problem in a data stream setting, where each incoming data can be seen only once and cannot be stored for long periods of time. Many probabilistic approaches based on either sampling or sketching have been proposed in the computer science literature, that only require limited computing and memory resources. However, the performances of these methods are not scale-invariant, in the sense that their relative root mean square estimation errors (RRMSE) depend on the unknown cardinalities. This is not desirable in many applications where cardinalities can be very dynamic or inhomogeneous and many cardinalities need to be estimated. In this paper, we develop a novel approach, called self-learning bitmap (S-bitmap) that is scale-invariant for cardinalities in a specified range. S-bitmap uses a binary vector whose entries are updated from 0 to 1 by an adaptive sampling process for inferring the unknown cardinality, where the sampling rates are reduced sequentially as more and more entries change from 0 to 1. We prove rigorously that the S-bitmap estimate is not only unbiased but scale-invariant. We demonstrate that to achieve a small RRMSE value of $\epsilon$ or less, our approach requires significantly less memory and consumes similar or less operations than state-of-the-art methods for many common practice cardinality scales. Both simulation and experimental studies are reported.Comment: Journal of the American Statistical Association (accepted

Virtual-Mobile-Core Placement for Metro Network

Traditional highly-centralized mobile core networks (e.g., Evolved Packet Core (EPC)) need to be constantly upgraded both in their network functions and backhaul links, to meet increasing traffic demands. Network Function Virtualization (NFV) is being investigated as a potential cost-effective solution for this upgrade. A virtual mobile core (here, virtual EPC, vEPC) provides deployment flexibility and scalability while reducing costs, network-resource consumption and application delay. Moreover, a distributed deployment of vEPC is essential for emerging paradigms like Multi-Access Edge Computing (MEC). In this work, we show that significant reduction in networkresource consumption can be achieved as a result of optimal placement of vEPC functions in metro area. Further, we show that not all vEPC functions need to be distributed. In our study, for the first time, we account for vEPC interactions in both data and control planes (Non-Access Stratum (NAS) signaling procedure Service Chains (SCs) with application latency requirements) using a detailed mathematical model