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Nicholas Pippenger

Publications and source records attributed to Nicholas Pippenger.

31 records · Page 2Linked to original sources
arXiv2010

A Census of Vertices by Generations in Regular Tessellations of the Plane

We consider regular tessellations of the plane as infinite graphs in which $q$ edges and $q$ faces meet at each vertex, and in which $p$ edges and $p$ vertices surround each face. For $1/p + 1/q = 1/2$, these are tilings of the Euclidean plane; for $1/p + 1/q < 1/2 $, they are tilings of the hyperbolic plane. We choose a vertex as the origin, and classify vertices into generations according to their distance (as measured by the number of edges in a shortest path) from the origin. For all $p\ge 3$ and $q \ge 3$ with $1/p + 1/q \le 1/2 $, we determine the rational generating function giving the number of vertices in each generation.

math.CO↗
arXiv2010

Local versus Global Search in Channel Graphs

Previous studies of search in channel graphs has assumed that the search is global; that is, that the status of any link can be probed by the search algorithm at any time. We consider for the first time local search, for which only links to which an idle path from the source has already been established may be probed. We show that some well known channel graphs may require exponentially more probes, on the average, when search must be local than when it may be global.

cs.CC↗
arXiv2009

Large Deviations and Moments for the Euler Characteristic of a Random Surface

We study random surfaces constructed by glueing together $N/k$ filled $k$-gons along their edges, with all $(N-1)!! = (N-1)(N-3)...3\cdot 1$ pairings of the edges being equally likely. (We assume that lcm $\{2,k\}$ divides $N$.) The Euler characteristic of the resulting surface is related to the number of cycles in a certain random permutation of $\{1, ..., N\}$. Gamburd has shown that when 2 lcm $\{2,k\}$ divides $N$, the distribution of this random permutation converges to that of the uniform distribution on the alternating group $A_N$ in the total-variation distance as $N\to\infty$. We obtain large-deviations bounds for the number of cycles that, together with Gamburd's result, allow us to derive sharp estimates for the moments of the number of cycles. These estimates allow us to confirm certain cases of conjectures made by Pippenger and Schleich.

math.PR↗
arXiv2007

Fault Tolerance in Cellular Automata at High Fault Rates

A commonly used model for fault-tolerant computation is that of cellular automata. The essential difficulty of fault-tolerant computation is present in the special case of simply remembering a bit in the presence of faults, and that is the case we treat in this paper. We are concerned with the degree (the number of neighboring cells on which the state transition function depends) needed to achieve fault tolerance when the fault rate is high (nearly 1/2). We consider both the traditional transient fault model (where faults occur independently in time and space) and a recently introduced combined fault model which also includes manufacturing faults (which occur independently in space, but which affect cells for all time). We also consider both a purely probabilistic fault model (in which the states of cells are perturbed at exactly the fault rate) and an adversarial model (in which the occurrence of a fault gives control of the state to an omniscient adversary). We show that there are cellular automata that can tolerate a fault rate $1/2 - ξ$ (with $ξ>0$) with degree $O((1/ξ^2)\log(1/ξ))$, even with adversarial combined faults. The simplest such automata are based on infinite regular trees, but our results also apply to other structures (such as hyperbolic tessellations) that contain infinite regular trees. We also obtain a lower bound of $Ω(1/ξ^2)$, even with purely probabilistic transient faults only.

math.PR↗
arXiv2007

Attribute Estimation and Testing Quasi-Symmetry

A Boolean function is symmetric if it is invariant under all permutations of its arguments; it is quasi-symmetric if it is symmetric with respect to the arguments on which it actually depends. We present a test that accepts every quasi-symmetric function and, except with an error probability at most delta>0, rejects every function that differs from every quasi-symmetric function on at least a fraction epsilon>0 of the inputs. For a function of n arguments, the test probes the function at O((n/epsilon)\log(n/delta)) inputs. Our quasi-symmetry test acquires information concerning the arguments on which the function actually depends. To do this, it employs a generalization of the property testing paradigm that we call attribute estimation. Like property testing, attribute estimation uses random sampling to obtain results that have only "one-sided'' errors and that are close to accurate with high probability.

cs.CC↗
arXiv2005

The Linking Probability of Deep Spider-Web Networks

We consider crossbar switching networks with base $b$ (that is, constructed from $b\times b$ crossbar switches), scale $k$ (that is, with $b^k$ inputs, $b^k$ outputs and $b^k$ links between each consecutive pair of stages) and depth $l$ (that is, with $l$ stages). We assume that the crossbars are interconnected according to the spider-web pattern, whereby two diverging paths reconverge only after at least $k$ stages. We assume that each vertex is independently idle with probability $q$, the vacancy probability. We assume that $b\ge 2$ and the vacancy probability $q$ are fixed, and that $k$ and $l = ck$ tend to infinity with ratio a fixed constant $c>1$. We consider the linking probability $Q$ (the probability that there exists at least one idle path between a given idle input and a given idle output). In a previous paper it was shown that if $c\le 2$, then the linking probability $Q$ tends to 0 if $0 1$. This is done by using generating functions and complex-variable techniques to estimate the second moments of various random variables involved in the analysis of the networks.

math.PR↗
arXiv2004

Random cyclations

Consider n unit intervals, say [1,2], [3,4], ..., [2n-1,2n]. Identify their endpoints in pairs at random, with all (2n-1)!! = (2n-1) (2n-3) ... 3 1 pairings being equally likely. The result is a collection of cycles of various lengths, and we investigate the distribution of these lengths. The distribution is similar to that of the distribution of the lengths of cycles in a random permutation, but it also exhibits some striking differences.

math.CO↗
arXiv2004

The average amount of information lost in multiplication

We show that if X and Y are integers independently and uniformly distributed in the set {1, ..., N}, then the information lost in forming their product (which is given by the equivocation H(X,Y | XY)), is of order log log N. We also prove two extremal results regarding cases in which X and Y are not necessarily independently or uniformly distributed. First, we note that the information lost in multiplication can of course be 0. We show that the condition H(X,Y | XY) = 0 implies that 2log_2 N - H(X, Y) is of order at least log log N. Furthermore, if X and Y are independent and uniformly distributed on disjoint sets of primes, it is possible to have H(X,Y | XY) = 0 with log_2 N - H(X) and log_2 N - H(Y) each of order at most log log N. Second, we show that however X and Y are distributed, H(X,Y | XY) is of order at most log N/log log N. Furthermore, there are distributions (in which X and Y are independent and uniformly distributed over sets of numbers having only small and distinct prime factors) for which H(X,Y | XY) is of order log log N.

math.PR↗
arXiv2003

Topological Characteristics of Random Surfaces Generated by Cubic Interactions

We consider random topologies of surfaces generated by cubic interactions. Such surfaces arise in various contexts in 2-dimensional quantum gravity and as world-sheets in string theory. Our results are most conveniently expressed in terms of a parameter h = n/2 + χ, where n is the number of interaction vertices and χis the Euler characteristic of the surface. Simulations and results for similar models suggest that Ex[h] = log (3n) + γ+ O(1/n) and Var[h] = log (3n) + γ- π^2/6 + O(1/n). We prove rigourously that Ex[h] = log n + O(1) and Var[h] = O(log n). We also derive results concerning a number of other characteristics of the topology of these random surfaces.

gr-qc↗
arXiv2003

The Boolean Functions Computed by Random Boolean Formulas OR How to Grow the Right Function

Among their many uses, growth processes (probabilistic amplification), were used for constructing reliable networks from unreliable components, and deriving complexity bounds of various classes of functions. Hence, determining the initial conditions for such processes is an important and challenging problem. In this paper we characterize growth processes by their initial conditions and derive conditions under which results such as Valiant's (1984) hold. First, we completely characterize growth processes that use linear connectives. Second, by extending Savický's (1990) analysis, via ``Restriction Lemmas'', we characterize growth processes that use monotone connectives, and show that our technique is applicable to growth processes that use other connectives as well. Additionally, we obtain explicit bounds on the convergence rates of several growth processes, including the growth process studied by Savický (1990).

cs.DM↗
arXiv2000

Characterizations of 1-Way Quantum Finite Automata

The 2-way quantum finite automaton introduced by Kondacs and Watrous can accept non-regular languages with bounded error in polynomial time. If we restrict the head of the automaton to moving classically and to moving only in one direction, the acceptance power of this 1-way quantum finite automaton is reduced to a proper subset of the regular languages. In this paper we study two different models of 1-way quantum finite automata. The first model, termed measure-once quantum finite automata, was introduced by Moore and Crutchfield, and the second model, termed measure-many quantum finite automata, was introduced by Kondacs and Watrous. We characterize the measure-once model when it is restricted to accepting with bounded error and show that, without that restriction, it can solve the word problem over the free group. We also show that it can be simulated by a probabilistic finite automaton and describe an algorithm that determines if two measure-once automata are equivalent. We prove several closure properties of the classes of languages accepted by measure-many automata, including inverse homomorphisms, and provide a new necessary condition for a language to be accepted by the measure-many model with bounded error. Finally, we show that piecewise testable languages can be accepted with bounded error by a measure-many quantum finite automaton, in the process introducing new construction techniques for quantum automata.

quant-ph↗
arXiv1998

The Computational Complexity of Knot and Link Problems

We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, {\sc unknotting problem} is in {\bf NP}. We also consider the problem, {\sc unknotting problem} of determining whether two or more such polygons can be split, or continuously deformed without self-intersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in {\bf PSPACE}. We also give exponential worst-case running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.

math.GT↗