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Optimized blist form (OBF
"... Abstract—Any Boolean expressions may be converted into positiveform, which has only union and intersection operators. Let E be a positiveform expression with n literals. Assume that the truthvalues of the literals are read one at a time. The numbers s(n) of steps (operations) and b(n) of working ..."
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Abstract—Any Boolean expressions may be converted into positiveform, which has only union and intersection operators. Let E be a positiveform expression with n literals. Assume that the truthvalues of the literals are read one at a time. The numbers s(n) of steps (operations) and b(n) of working memory bits (footprint) needed to evaluate E depend on E and on the evaluation technique. A recursive evaluation performs s(n)=n–1 steps, but requires b(n)=log(n)+1 bits. Evaluating the disjunctive form of E uses only b(n)=2 bits, but may lead to an exponential growth of s(n). We propose a new Optimized Blist Form (OBF), which requires only s(n)=n steps and b(n)=⎡log2j ⎤ bits, where j=⎡log2(2n/3+2)⎤. We provide a simple and linearcost algorithm for converting positiveform expressions to their OBF. We discuss three applications: (1) Direct CSG rendering, where a candidate surfel is classified against an arbitrarily complex Boolean expression (up to 27,600,000,000,000,000,000 literals) using a footprint of only 6 stencil bits; (2) the new programmable Logic Matrix (LM), which evaluates any positiveform logical expression of n literals in a single clock cycle and uses a matrix of at most n×j wire/line connections; and (3) the new programmable Logic Pipe (LP), which uses n gates connected by a pipe of ⎡log2j⎤ lines and, when receiving a staggered stream of input vectors, produces a value of a logical expression at each clock cycle.
Approximating the Maximum Sharing Problem
"... Abstract. In the maximum sharing problem (MS), we want to compute a set of (nonsimple) paths in an undirected bipartite graph covering as many nodes as possible of the first node layer of the graph, with the constraint that all paths have both endpoints in the second node layer and no node in that ..."
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Abstract. In the maximum sharing problem (MS), we want to compute a set of (nonsimple) paths in an undirected bipartite graph covering as many nodes as possible of the first node layer of the graph, with the constraint that all paths have both endpoints in the second node layer and no node in that layer is covered more than once. MS is equivalent to the nodeduplication based crossing elimination problem (NDCE) that arises in the design of molecular quantumdot cellular automata (QCA) circuits and the physical synthesis of BDD based regular circuit structures in VLSI design. We show that MS is NPhard, present a polynomialtime 1.5approximation algorithm, and show that MS cannot be approximated with a factor better than 740 unless P = NP. 739 1
Logic Restructuring for Delay Balancing in WavePipelined Circuits: an Integer Programming Approach
"... In this paper we apply integer programming (IP) based techniques to the problem of delay balancing in wavepipelined circuits. The proposed approach considers delays, as well as fanin and fanout associated with every node in the circuit. After a weighted graph representation of the circuit is form ..."
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In this paper we apply integer programming (IP) based techniques to the problem of delay balancing in wavepipelined circuits. The proposed approach considers delays, as well as fanin and fanout associated with every node in the circuit. After a weighted graph representation of the circuit is formed a node collapsing procedure is used to preprocess (reduce the size of) the system and obtain the final formulation of the IP problem, which is solved by using a branch and bound heuristic to obtain a minimum delay in the circuit. We also compare the proposed technique with application – to the same problem – of a linear programming solver. 1.
Eliminate Wire Crossings by Node Duplication
, 2008
"... For many circuit design problem, it is imperative to carefully study the effect of physical implementation constraint. Under some condition, it is very difficult to fabricate wire crossings. In this paper, we introduce a crossing elimination model based on a node duplication method and we want to mi ..."
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For many circuit design problem, it is imperative to carefully study the effect of physical implementation constraint. Under some condition, it is very difficult to fabricate wire crossings. In this paper, we introduce a crossing elimination model based on a node duplication method and we want to minimize the number of duplication. We relate it with an artificial problem, called maximum simple sharing problem. First we prove it is NPhard, then we show a simple greedy algorithm can achieve a approximation factor of 3. We introduce maximum disjoint simple sharing problem, which is naturally a 2approximation of the maximum simple sharing problem, and show it can be solve optimally by reducing to the perfect matching problem in a series carefully constructed graph. At last, we further improve the approximation factor to 12/7 through a local search technique. 1
An Efficient Approximation Algorithm for Maximum Simple Sharing Problem
, 2008
"... For many circuit design problems, it is imperative to carefully study the effect of physical implementation constraints. Under some circumstances, it is very difficult to fabricate wire crossings. In this paper, we introduce a crossing elimination model based on a node duplication method and we wan ..."
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For many circuit design problems, it is imperative to carefully study the effect of physical implementation constraints. Under some circumstances, it is very difficult to fabricate wire crossings. In this paper, we introduce a crossing elimination model based on a node duplication method and we want to minimize the number of duplication. We relate it with an artificial problem, called the maximum simple sharing problem. First we prove it is NPhard, then we show that a simple greedy algorithm can achieve an approximation factor of 3. We then introduce the maximum disjoint simple sharing problem, which is naturally a 2approximation of the maximum simple sharing problem, and show that it can be solved optimally by reducing to the perfect matching problem in a series of carefully constructed graph. At last, we further improve the approximation factor to 12/7 with a local search technique.