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Cristopher Moore

Publications and source records attributed to Cristopher Moore.

At least 127 records · Page 7Linked to original sources
arXiv2005

A Continuous-Discontinuous Second-Order Transition in the Satisfiability of Random Horn-SAT Formulas

We compute the probability of satisfiability of a class of random Horn-SAT formulae, motivated by a connection with the nonemptiness problem of finite tree automata. In particular, when the maximum clause length is 3, this model displays a curve in its parameter space along which the probability of satisfiability is discontinuous, ending in a second-order phase transition where it becomes continuous. This is the first case in which a phase transition of this type has been rigorously established for a random constraint satisfaction problem.

math.PR↗
arXiv2005

Hiding Satisfying Assignments: Two are Better than One

The evaluation of incomplete satisfiability solvers depends critically on the availability of hard satisfiable instances. A plausible source of such instances consists of random k-SAT formulas whose clauses are chosen uniformly from among all clauses satisfying some randomly chosen truth assignment A. Unfortunately, instances generated in this manner tend to be relatively easy and can be solved efficiently by practical heuristics. Roughly speaking, as the formula's density increases, for a number of different algorithms, A acts as a stronger and stronger attractor. Motivated by recent results on the geometry of the space of satisfying truth assignments of random k-SAT and NAE-k-SAT formulas, we introduce a simple twist on this basic model, which appears to dramatically increase its hardness. Namely, in addition to forbidding the clauses violated by the hidden assignment A, we also forbid the clauses violated by its complement, so that both A and complement of A are satisfying. It appears that under this "symmetrization'' the effects of the two attractors largely cancel out, making it much harder for algorithms to find any truth assignment. We give theoretical and experimental evidence supporting this assertion.

cs.AI↗
arXiv2005

Generating Hard Satisfiable Formulas by Hiding Solutions Deceptively

To test incomplete search algorithms for constraint satisfaction problems such as 3-SAT, we need a source of hard, but satisfiable, benchmark instances. A simple way to do this is to choose a random truth assignment A, and then choose clauses randomly from among those satisfied by A. However, this method tends to produce easy problems, since the majority of literals point toward the ``hidden'' assignment A. Last year, Achlioptas, Jia and Moore proposed a problem generator that cancels this effect by hiding both A and its complement. While the resulting formulas appear to be just as hard for DPLL algorithms as random 3-SAT formulas with no hidden assignment, they can be solved by WalkSAT in only polynomial time. Here we propose a new method to cancel the attraction to A, by choosing a clause with t > 0 literals satisfied by A with probability proportional to q^t for some q < 1. By varying q, we can generate formulas whose variables have no bias, i.e., which are equally likely to be true or false; we can even cause the formula to ``deceptively'' point away from A. We present theoretical and experimental results suggesting that these formulas are exponentially hard both for DPLL algorithms and for incomplete algorithms such as WalkSAT.

cs.AI↗
arXiv2005

The Power of Strong Fourier Sampling: Quantum Algorithms for Affine Groups and Hidden Shifts

Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a Hidden Subgroup problem, in which an unknown subgroup H of a group G must be determined from a uniform superposition on a left coset of H. These hidden subgroup problems are typically solved by Fourier sampling. When G is nonabelian, two important variants of Fourier sampling have been identified: the weak standard method, where only representation names are measured, and the strong standard method, where full measurement (i.e., the row and column of the representation, in a suitably chosen basis) occurs. It has remained open whether the strong standard method is indeed stronger. In this article, we settle this question in the affirmative. We show that hidden subgroups H of the q-hedral groups, i.e., semidirect products Z_q \ltimes Z_p where q | (p-1), and in particular the affine groups A_p, can be information-theoretically reconstructed using the strong standard method. Moreover, if |H| = p/ \polylog(p), these subgroups can be fully reconstructed with a polynomial amount of quantum and classical computation. We show that, for some q, neither the ``forgetful'' abelian method nor measuring in a random basis succeeds, even information-theoretically. Thus, at least for some groups, it is crucial to use the full power of representation theory: namely, to measure the high-dimensional representations in an adapted basis that respects the group's subgroup structure. We apply our algorithm for the hidden subgroup problem to new families of cryptographically motivated Hidden Shift problems, generalizing work of van Dam, Hallgren and Ip on shifts of multiplicative characters.

quant-ph↗
arXiv2004

Finding community structure in very large networks

The discovery and analysis of community structure in networks is a topic of considerable recent interest within the physics community, but most methods proposed so far are unsuitable for very large networks because of their computational cost. Here we present a hierarchical agglomeration algorithm for detecting community structure which is faster than many competing algorithms: its running time on a network with n vertices and m edges is O(m d log n) where d is the depth of the dendrogram describing the community structure. Many real-world networks are sparse and hierarchical, with m ~ n and d ~ log n, in which case our algorithm runs in essentially linear time, O(n log^2 n). As an example of the application of this algorithm we use it to analyze a network of items for sale on the web-site of a large online retailer, items in the network being linked if they are frequently purchased by the same buyer. The network has more than 400,000 vertices and 2 million edges. We show that our algorithm can extract meaningful communities from this network, revealing large-scale patterns present in the purchasing habits of customers.

cond-mat.stat-mech↗
arXiv2004

Why Mapping the Internet is Hard

Despite great effort spent measuring topological features of large networks like the Internet, it was recently argued that sampling based on taking paths through the network (e.g., traceroutes) introduces a fundamental bias in the observed degree distribution. We examine this bias analytically and experimentally. For classic random graphs with mean degree c, we show analytically that traceroute sampling gives an observed degree distribution P(k) ~ 1/k for k < c, even though the underlying degree distribution is Poisson. For graphs whose degree distributions have power-law tails P(k) ~ k^-alpha, the accuracy of traceroute sampling is highly sensitive to the population of low-degree vertices. In particular, when the graph has a large excess (i.e., many more edges than vertices), traceroute sampling can significantly misestimate alpha.

cond-mat.dis-nn↗
arXiv2004

The Chromatic Number of Random Regular Graphs

Given any integer d >= 3, let k be the smallest integer such that d < 2k log k. We prove that with high probability the chromatic number of a random d-regular graph is k, k+1, or k+2, and that if (2k-1) \log k < d < 2k \log k then the chromatic number is either k+1 or k+2.

cond-mat.dis-nn↗
arXiv2004

Traceroute sampling makes random graphs appear to have power law degree distributions

The topology of the Internet has typically been measured by sampling traceroutes, which are roughly shortest paths from sources to destinations. The resulting measurements have been used to infer that the Internet's degree distribution is scale-free; however, many of these measurements have relied on sampling traceroutes from a small number of sources. It was recently argued that sampling in this way can introduce a fundamental bias in the degree distribution, for instance, causing random (Erdos-Renyi) graphs to appear to have power law degree distributions. We explain this phenomenon analytically using differential equations to model the growth of a breadth-first tree in a random graph G(n,p=c/n) of average degree c, and show that sampling from a single source gives an apparent power law degree distribution P(k) ~ 1/k for k < c.

cond-mat.dis-nn↗
arXiv2004

Quantum and Stochastic Branching Programs of Bounded Width

In this paper we show that one qubit polynomial time computations are at least as powerful as $\NC^1$ circuits. More precisely, we define syntactic models for quantum and stochastic branching programs of bounded width and prove upper and lower bounds on their power. We show any $\NC^1$ language can be accepted exactly by a width-2 quantum branching program of polynomial length, in contrast to the classical case where width 5 is necessary unless $\NC^1=\ACC$. This separates width-2 quantum programs from width-2 doubly stochastic programs as we show the latter cannot compute the middle bit of multiplication. Finally, we show that bounded-width quantum and stochastic programs can be simulated by classical programs of larger but bounded width, and thus are in $\NC^1$. The change in the revised version is the addition of the syntactic condition.

quant-ph↗
arXiv2003

How Do Networks Become Navigable?

Networks created and maintained by social processes, such as the human friendship network and the World Wide Web, appear to exhibit the property of navigability: namely, not only do short paths exist between any pair of nodes, but such paths can easily be found using only local information. It has been shown that for networks with an underlying metric, algorithms using only local information perform extremely well if there is a power-law distribution of link lengths. However, it is not clear why or how real networks might develop this distribution. In this paper we define a decentralized ``rewiring'' process, inspired by surfers on the Web, in which each surfer attempts to travel from their home page to a random destination, and updates the outgoing link from their home page if this journey takes too long. We show that this process does indeed cause the link length distribution to converge to a power law, achieving a routing time of O(log^2 n) on networks of size n. We also study finite-size effects on the optimal exponent, and show that it converges polylogarithmically slowly as the lattice size goes to infinity.

cond-mat↗
arXiv2003

Random k-SAT: Two Moments Suffice to Cross a Sharp Threshold

Many NP-complete constraint satisfaction problems appear to undergo a "phase transition'' from solubility to insolubility when the constraint density passes through a critical threshold. In all such cases it is easy to derive upper bounds on the location of the threshold by showing that above a certain density the first moment (expectation) of the number of solutions tends to zero. We show that in the case of certain symmetric constraints, considering the second moment of the number of solutions yields nearly matching lower bounds for the location of the threshold. Specifically, we prove that the threshold for both random hypergraph 2-colorability (Property B) and random Not-All-Equal k-SAT is 2^{k-1} ln 2 -O(1). As a corollary, we establish that the threshold for random k-SAT is of order Theta(2^k), resolving a long-standing open problem.

cond-mat.stat-mech↗
arXiv2003

Generic Quantum Fourier Transforms

The quantum Fourier transform (QFT) is the principal algorithmic tool underlying most efficient quantum algorithms. We present a generic framework for the construction of efficient quantum circuits for the QFT by ``quantizing'' the separation of variables technique that has been so successful in the study of classical Fourier transform computations. Specifically, this framework applies the existence of computable Bratteli diagrams, adapted factorizations, and Gel'fand-Tsetlin bases to offer efficient quantum circuits for the QFT over a wide variety a finite Abelian and non-Abelian groups, including all group families for which efficient QFTs are currently known and many new group families. Moreover, the method gives rise to the first subexponential-size quantum circuits for the QFT over the linear groups GL_k(q), SL_k(q), and the finite groups of Lie type, for any fixed prime power q.

quant-ph↗
arXiv2003

What Is a Macrostate? Subjective Observations and Objective Dynamics

We consider the question of whether thermodynamic macrostates are objective consequences of dynamics, or subjective reflections of our ignorance of a physical system. We argue that they are both; more specifically, that the set of macrostates forms the unique maximal partition of phase space which 1) is consistent with our observations (a subjective fact about our ability to observe the system) and 2) obeys a Markov process (an objective fact about the system's dynamics). We review the ideas of computational mechanics, an information-theoretic method for finding optimal causal models of stochastic processes, and argue that macrostates coincide with the ``causal states'' of computational mechanics. Defining a set of macrostates thus consists of an inductive process where we start with a given set of observables, and then refine our partition of phase space until we reach a set of states which predict their own future, i.e. which are Markovian. Macrostates arrived at in this way are provably optimal statistical predictors of the future values of our observables.

cond-mat.stat-mech↗
arXiv2003

The Hidden Subgroup Problem in Affine Groups: Basis Selection in Fourier Sampling

Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which a subgroup H of a group G must be determined from a quantum state y uniformly supported on a left coset of H. These hidden subgroup problems are then solved by Fourier sampling: the quantum Fourier transform of y is computed and measured. When the underlying group is non-Abelian, two important variants of the Fourier sampling paradigm have been identified: the weak standard method, where only representation names are measured, and the strong standard method, where full measurement occurs. It has remained open whether the strong standard method is indeed stronger, that is, whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak, or any other known, method. In this article, we settle this question in the affirmative. We show that hidden subgroups of semidirect products of Z_p by Z_q, where q divides (p-1) and q = p / polylog(p), can be efficiently determined by the strong standard method. Furthermore, the weak standard method and the ``forgetful'' Abelian method are insufficient for these groups. We extend this to an information-theoretic solution for the hidden subgroup problem over semidirect products of Z_p by \Z_q where q divides (p-1) and, in particular, the Affine groups A_p. Finally, we prove a closure property for the class of groups over which the hidden subgroup problem can be solved efficiently.

quant-ph↗
arXiv2002

The Asymptotic Order of the k-SAT Threshold

Form a random k-SAT formula on n variables by selecting uniformly and independently m=rn clauses out of all 2^k (n choose k) possible k-clauses. The Satisfiability Threshold Conjecture asserts that for each k there exists a constant r_k such that, as n tends to infinity, the probability that the formula is satisfiable tends to 1 if r < r_k and to 0 if r > r_k. It has long been known that 2^k / k < r_k < 2^k. We prove that r_k > 2^{k-1} \ln 2 - d_k, where d_k \to (1+\ln 2)/2. Our proof also allows a blurry glimpse of the ``geometry'' of the set of satisfying truth assignments, and a nearly exact location of the threshold for Not-All-Equal (NAE) k-SAT.

cond-mat.stat-mech↗
arXiv2001

Counting, Fanout, and the Complexity of Quantum ACC

We propose definitions of $\QAC^0$, the quantum analog of the classical class $\AC^0$ of constant-depth circuits with AND and OR gates of arbitrary fan-in, and $\QACC[q]$, the analog of the class $\ACC[q]$ where $\Mod_q$ gates are also allowed. We prove that parity or fanout allows us to construct quantum $\MOD_q$ gates in constant depth for any $q$, so $\QACC[2] = \QACC$. More generally, we show that for any $q,p > 1$, $\MOD_q$ is equivalent to $\MOD_p$ (up to constant depth). This implies that $\QAC^0$ with unbounded fanout gates, denoted $\QACwf^0$, is the same as $\QACC[q]$ and $\QACC$ for all $q$. Since $\ACC[p] \ne \ACC[q]$ whenever $p$ and $q$ are distinct primes, $\QACC[q]$ is strictly more powerful than its classical counterpart, as is $\QAC^0$ when fanout is allowed. This adds to the growing list of quantum complexity classes which are provably more powerful than their classical counterparts. We also develop techniques for proving upper bounds for $\QACC^0$ in terms of related language classes. We define classes of languages $\EQACC$, $\NQACC$ and $\BQACC_{\rats}$. We define a notion of $\log$-planar $\QACC$ operators and show the appropriately restricted versions of $\EQACC$ and $\NQACC$ are contained in $¶/\poly$. We also define a notion of $\log$-gate restricted $\QACC$ operators and show the appropriately restricted versions of $\EQACC$ and $\NQACC$ are contained in $\TC^0$.

quant-ph↗
arXiv2001

Quantum Walks on the Hypercube

Recently, it has been shown that one-dimensional quantum walks can mix more quickly than classical random walks, suggesting that quantum Monte Carlo algorithms can outperform their classical counterparts. We study two quantum walks on the n-dimensional hypercube, one in discrete time and one in continuous time. In both cases we show that the quantum walk mixes in (π/4)n steps, faster than the O(n log n) steps required by the classical walk. In the continuous-time case, the probability distribution is {\em exactly} uniform at this time. More importantly, these walks expose several subtleties in the definition of mixing time for quantum walks. Even though the continuous-time walk has an O(n) instantaneous mixing time at which it is precisely uniform, it never approaches the uniform distribution when the stopping time is chosen randomly as in [AharonovAKV2001]. Our analysis treats interference between terms of different phase more carefully than is necessary for the walk on the cycle; previous general bounds predict an exponential, rather than linear, mixing time for the hypercube.

quant-ph↗