A factor graph is a bipartite graph that expresses how a "global" function of many variables factors into a product of "local" functions. Factor graphs subsume many other graphical models including Bayesian networks, Markov random fields, and Tanner graphs. Following one simple computational rule, the sum-product algorithm operates in factor graphs to compute---either exactly or approximately---various marginal functions by distributed message-passing in the graph. A wide variety of algorithms developed in artificial intelligence, signal processing, and digital communications can be derived as specific instances of the sum-product algorithm, including the forward/backward algorithm, the Viterbi algorithm, the iterative "turbo" decoding algorithm, Pearl's belief propagation algorithm for Bayesian networks, the Kalman filter, and certain fast Fourier transform algorithms.

Effective memory hierarchy utilization is critical to the performance of modern multiprocessor architectures. We havedeveloped the first compiler system that fully automatically parallelizes sequential programs and changes the original array layouts to improve memory system performance. Our optimization algorithm consists of two steps. The first step chooses the parallelization and computation assignment such that synchronization and data sharing are minimized. The second step then restructures the layout of the data in the shared address space with an algorithm that is based on a new data transformation framework. We ran our compiler on a set of application programs and measured their performance on the Stanford DASH multiprocessor. Our results show that the compiler can effectively optimize parallelism in conjunction with memory subsystem performance. 1 Introduction In the last decade, microprocessor speeds have been steadily improving at a rate of 50% to 100% every year[16]. Meanwh...

We consider the indexable dictionary problem, which consists of storing a set S ⊆ {0,...,m − 1} for some integer m, while supporting the operations of rank(x), which returns the number of elements in S that are less than x if x ∈ S, and −1 otherwise; and select(i) which returns the i-th smallest element in S. We give a data structure that supports both operations in O(1) time on the RAM model and requires B(n,m)+ o(n)+O(lg lg m) bits to store a set of size n, where B(n,m) = ⌈ lg ( m) ⌉ n is the minimum number of bits required to store any n-element subset from a universe of size m. Previous dictionaries taking this space only supported (yes/no) membership queries in O(1) time. In the cell probe model we can remove the O(lg lg m) additive term in the space bound, answering a question raised by Fich and Miltersen, and Pagh. We present extensions and applications of our indexable dictionary data structure, including: • an information-theoretically optimal representation of a k-ary cardinal tree that supports standard operations in constant time, • a representation of a multiset of size n from {0,...,m − 1} in B(n,m+n) + o(n) bits that supports (appropriate generalizations of) rank and select operations in constant time, and • a representation of a sequence of n non-negative integers summing up to m in B(n,m + n) + o(n) bits that supports prefix sum queries in constant time. 1

An algorithm for de6nite hypergeometric summation is given. It is based, in a non-obvious way, on Gosper's algorithm for definite hypergeometric summation, and its theoretical justification relies on Bernstein's theory of holonomic systems. 1.

Many applications compute aggregate functions...

The function class #P lacks an important closure property: it is not closed under subtraction. To remedy this problem, we introduce the function class GapP as a natural alternative to #P. GapP is the closure of #P under subtraction, and has all the other useful closure properties of #P as well. We show that most previously studied counting classes, including PP, C=P, and Mod k P, are "gap-definable," i.e., definable using the values of GapP functions alone. We show that there is a smallest gap-definable class, SPP, which is still large enough to contain Few. We also show that SPP consists of exactly those languages low for GapP, and thus SPP languages are low for any gap-definable class. These results unify and improve earlier disparate results of Cai & Hemachandra [7] and Kobler, Schoning, Toda, & Tor'an [15]. We show further that any countable collection of languages is contained in a unique minimum gap-definable class, which implies that the gap-definable classes form a lattice un...

We propose a new randomized algorithm for maintaining a set of clusters among moving nodes in the plane. Given a specified cluster radius, our algorithm selects and maintains a variable subset of the nodes as cluster centers. This subset has the property that (1) balls of the given radius centered at the chosen nodes cover all the others and (2) the number of centers selected is a constant-factor approximation of the minimum possible. As the nodes move, an event-based kinetic data structure updates the clustering as necessary. This kinetic data structure is shown to be responsive, efficient, local, and compact. The produced cover is also smooth, in the sense that wholesale cluster re-arrangements are avoided. The algorithm can be implemented without exact knowledge of the node positions, if each node is able to sense its distance to other nodes up to the cluster radius. Such a kinetic clustering can be used in numerous applications where mobile devices must be interconnected into an ad-hoc network to collaboratively perform some tasks. 1

We define and analyze quantum computational variants of random walks on one-dimensional lattices. In particular, we analyze a quantum analog of the symmetric random walk, which we call the Hadamard walk. Several striking differences between the quantum and classical cases are observed. For example, when unrestricted in either direction, the Hadamard walk has position that is nearly uniformly distributed in the range [\Gamma t= p

: Let V ` f0; 1g n have Vapnik-Chervonenkis dimension d. Let M(k=n;V ) denote the cardinality of the largest W ` V such that any two distinct vectors in W differ on at least k indices. We show that M(k=n;V ) (cn=(k + d)) d for some constant c. This improves on the previous best result of ((cn=k) log(n=k)) d . This new result has applications in the theory of empirical processes. 1 The author gratefully acknowledges the support of the Mathematical Sciences Research Institute at UC Berkeley and ONR grant N00014-91-J-1162. 1 1 Statement of Results Let n be natural number greater than zero. Let V ` f0; 1g n . For a sequence of indices I = (i 1 ; . . . ; i k ), with 1 i j n, let V j I denote the projection of V onto I, i.e. V j I = f(v i 1 ; . . . ; v i k ) : (v 1 ; . . . ; v n ) 2 V g: If V j I = f0; 1g k then we say that V shatters the index sequence I. The Vapnik-Chervonenkis dimension of V is the size of the longest index sequence I that is shattered by V [VC71] (t...

The impact of cache interferences on program performance (particularly numerical codes, which heavily use the memory hierarchy) remains unknown. The general knowledge is that cache interferences are highly irregular, in terms of occurrence and intensity. In this paper, the different types of cache interferences that can occur in numerical loop nests are identified. An analytical method is developed for detecting the occurrence of interferences and, more important, for computing the number of cache misses due to interferences. Simulations and experiments on real machines show that the model is generally accurate and that most interference phenomena are captured. Experiments also show that cache interferences can be intense and frequent. Certain parameters such as array base addresses or dimensions can have a strong impact on the occurrence of interferences. Modifying these parameters only can induce global execution time variations of 30% and more. Applications of these modeling techniq...