How to Couple from the Past Using a Read-Once Source of Randomness

We give a new method for generating perfectly random samples from the stationary distribution of a Markov chain. The method is related to coupling from the past (CFTP), but only runs the Markov chain forwards in time, and never restarts it at previous times in the past. The method is also related to an idea known as PASTA (Poisson arrivals see time averages) in the operations research literature. Because the new algorithm can be run using a read-once stream of randomness, we call it read-once CFTP. The memory and time requirements of read-once CFTP are on par with the requirements of the usual form of CFTP, and for a variety of applications the requirements may be noticeably less. Some perfect sampling algorithms for point processes are based on an extension of CFTP known as coupling into and from the past; for completeness, we give a read-once version of coupling into and from the past, but it remains unpractical. For these point process applications, we give an alternative coupling method with which read-once CFTP may be efficiently used.Comment: 28 pages, 2 figure

Knot Graphs

We consider the equivalence classes of graphs induced by the unsigned versions of the Reidemeister moves on knot diagrams. Any graph which is reducible by some finite sequence of these moves, to a graph with no edges is called a knot graph. We show that the class of knot graphs strictly contains the set of delta-wye graphs. We prove that the dimension of the intersection of the cycle and cocycle spaces is an effective numerical invariant of these classes

Hamilton Cycles in a Class of Random Directed Graphs

AbstractWe prove that almost every 3-in, 3-out digraph is Hamiltonian

Discordant voting processes on finite graphs

We consider an asynchronous voting process on graphs which we call discordant voting, and which can be described as follows. Initially each vertex holds one of two opinions, red or blue say. Neighbouring vertices with different opinions interact pairwise. After an interaction both vertices have the same colour. The quantity of interest is T, the time to reach consensus, i.e. the number of interactions needed for all vertices have the same colour. An edge whose endpoint colours differ (i.e. one vertex is coloured red and the other one blue) is said to be discordant. A vertex is discordant if its is incident with a discordant edge. In discordant voting, all interactions are based on discordant edges. Because the voting process is asynchronous there are several ways to update the colours of the interacting vertices. Push: Pick a random discordant vertex and push its colour to a random discordant neighbour. Pull: Pick a random discordant vertex and pull the colour of a random discordant neighbour. Oblivious: Pick a random endpoint of a random discordant edge and push the colour to the other end point. We show that ET, the expected time to reach consensus, depends strongly on the underlying graph and the update rule. For connected graphs on n vertices, and an initial half red, half blue colouring the following hold. For oblivious voting, ET = n2/4 independent of the underlying graph. For the complete graph Kn, the push protocol has ET = =(n log n), whereas the pull protocol has ET = =(2n). For the cycle Cn all three protocols have ET = =(n2). For the star graph however, the pull protocol has ET = O(n2), whereas the push protocol is slower with ET = =(n2 log n). The wide variation in ET for the pull protocol is to be contrasted with the well known model of synchronous pull voting, for which ET = O(n) on many classes of expanders

The Power of Two Choices in Distributed Voting

Distributed voting is a fundamental topic in distributed computing. In pull voting, in each step every vertex chooses a neighbour uniformly at random, and adopts its opinion. The voting is completed when all vertices hold the same opinion. On many graph classes including regular graphs, pull voting requires $\Theta(n)$ expected steps to complete, even if initially there are only two distinct opinions. In this paper we consider a related process which we call two-sample voting: every vertex chooses two random neighbours in each step. If the opinions of these neighbours coincide, then the vertex revises its opinion according to the chosen sample. Otherwise, it keeps its own opinion. We consider the performance of this process in the case where two different opinions reside on vertices of some (arbitrary) sets $A$ and $B$, respectively. Here, $|A| + |B| = n$ is the number of vertices of the graph. We show that there is a constant $K$ such that if the initial imbalance between the two opinions is ?$\nu_0 = (|A| - |B|)/n \geq K \sqrt{(1/d) + (d/n)}$, then with high probability two sample voting completes in a random $d$ regular graph in $O(\log n)$ steps and the initial majority opinion wins. We also show the same performance for any regular graph, if $\nu_0 \geq K \lambda_2$ where $\lambda_2$ is the second largest eigenvalue of the transition matrix. In the graphs we consider, standard pull voting requires $\Omega(n)$ steps, and the minority can still win with probability $|B|/n$.Comment: 22 page

Parameter estimators of random intersection graphs with thinned communities

This paper studies a statistical network model generated by a large number of randomly sized overlapping communities, where any pair of nodes sharing a community is linked with probability $q$ via the community. In the special case with $q=1$ the model reduces to a random intersection graph which is known to generate high levels of transitivity also in the sparse context. The parameter $q$ adds a degree of freedom and leads to a parsimonious and analytically tractable network model with tunable density, transitivity, and degree fluctuations. We prove that the parameters of this model can be consistently estimated in the large and sparse limiting regime using moment estimators based on partially observed densities of links, 2-stars, and triangles.Comment: 15 page

Boosting search by rare events

Randomized search algorithms for hard combinatorial problems exhibit a large variability of performances. We study the different types of rare events which occur in such out-of-equilibrium stochastic processes and we show how they cooperate in determining the final distribution of running times. As a byproduct of our analysis we show how search algorithms are optimized by random restarts.Comment: 4 pages, 3 eps figures. References update

A proposed intense slow positron source based on 58Co

Positron beams have proven very useful for condensed matter and surface research. The highest intensity of the current operating positron beams is ∼109 slow e+/second. The goal of our proposal is to build an Intense Slow Positron Source (ISPS) demonstration beam (Phase I) of unprecedented brightness at the Idaho National Engineering Laboratory, INEL (up to 1010 slow e+/s at 5 keV over a <0.03 cm. diameter). This Phase I beam will prove the principles necessary to build a larger facility scale ISPS Phase II beam which will have a potential of 1013 e+/s, or ≳1012 e+/s over 0.03 cm. The INEL is an ideal location for the ISPS because of the fast breeder reactor EBR‐II, which is perfectly suited to creating the positron emitting isotope 58Co, and the excellent radioactive materials handling capability and expertise. Sufficient expertise is available at INEL for the construction and operation of a user facility (Phase II).Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/87604/2/289_1.pd

Solving Medium-Density Subset Sum Problems in Expected Polynomial Time: An Enumeration Approach

The subset sum problem (SSP) can be briefly stated as: given a target integer $E$ and a set $A$ containing $n$ positive integer $a_j$, find a subset of $A$ summing to $E$. The \textit{density} $d$ of an SSP instance is defined by the ratio of $n$ to $m$, where $m$ is the logarithm of the largest integer within $A$. Based on the structural and statistical properties of subset sums, we present an improved enumeration scheme for SSP, and implement it as a complete and exact algorithm (EnumPlus). The algorithm always equivalently reduces an instance to be low-density, and then solve it by enumeration. Through this approach, we show the possibility to design a sole algorithm that can efficiently solve arbitrary density instance in a uniform way. Furthermore, our algorithm has considerable performance advantage over previous algorithms. Firstly, it extends the density scope, in which SSP can be solved in expected polynomial time. Specifically, It solves SSP in expected $O(n\log{n})$ time when density $d \geq c\cdot \sqrt{n}/\log{n}$, while the previously best density scope is $d \geq c\cdot n/(\log{n})^{2}$. In addition, the overall expected time and space requirement in the average case are proven to be $O(n^5\log n)$ and $O(n^5)$ respectively. Secondly, in the worst case, it slightly improves the previously best time complexity of exact algorithms for SSP. Specifically, the worst-case time complexity of our algorithm is proved to be $O((n-6)2^{n/2}+n)$, while the previously best result is $O(n2^{n/2})$.Comment: 11 pages, 1 figur