Journal of the ACM (JACM)

Zhang, Yuanlin; Yap, Roland H. C.

doi: 10.1145/765568.765569pmid: N/A

Darwiche, Adnan

doi: 10.1145/765568.765570pmid: N/A

We present a new approach to inference in Bayesian networks, which is based on representing the network using a polynomial and then retrieving answers to probabilistic queries by evaluating and differentiating the polynomial. The network polynomial itself is exponential in size, but we show how it can be computed efficiently using an arithmetic circuit that can be evaluated and differentiated in time and space linear in the circuit size. The proposed framework for inference subsumes one of the most influential methods for inference in Bayesian networks, known as the tree-clustering or jointree method, which provides a deeper understanding of this classical method and lifts its desirable characteristics to a much more general setting. We discuss some theoretical and practical implications of this subsumption.

Reed, Bruce

doi: 10.1145/765568.765571pmid: N/A

Let H n be the height of a random binary search tree on n nodes. We show that there exist constants α = 4.311… and β = 1.953… such that E ( H n ) = α ln n − β ln ln n + O (1), We also show that Var ( H n ) = O (1).

Drmota, Michael

doi: 10.1145/765568.765572pmid: N/A

It is shown that all centralized absolute moments E | H n − E H n | α (α ≥ 0) of the height H n of binary search trees of size n and of the saturation level H n ′ are bounded. The methods used rely on the analysis of a retarded differential equation of the form Φ′( u ) = −α −2 Φ( u /α) 2 with α > 1. The method can also be extended to prove the same result for the height of m -ary search trees. Finally the limiting behaviour of the distribution of the height of binary search trees is precisely determined.

Bilardi, Gianfranco; Pingali, Keshav

doi: 10.1145/765568.765573pmid: N/A

The Static Single Assignment (SSA) form is a program representation used in many optimizing compilers. The key step in converting a program to SSA form is called φ-placement. Many algorithms for φ-placement have been proposed in the literature, but the relationships between these algorithms are not well understood.In this article, we propose a framework within which we systematically derive (i) properties of the SSA form and (ii) φ-placement algorithms. This framework is based on a new relation called merge which captures succinctly the structure of a program's control flow graph that is relevant to its SSA form. The φ-placement algorithms we derive include most of the ones described in the literature, as well as several new ones. We also evaluate experimentally the performance of some of these algorithms on the SPEC92 benchmarks.Some of the algorithms described here are optimal for a single variable. However, their repeated application is not necessarily optimal for multiple variables. We conclude the article by describing such an optimal algorithm, based on the transitive reduction of the merge relation, for multi-variable φ-placement in structured programs. The problem for general programs remains open.