Exact Minimization Algorithm Research Articles

Fast Exact Hyper-graph Matching with Dynamic Programming for Spatio-temporal Data

Graphs and hyper-graphs are frequently used to recognize complex and often non-rigid patterns in computer vision, either through graph matching or point-set matching with graphs. Most formulations resort to the minimization of a difficult energy function containing geometric or structural terms, frequently coupled with data attached terms involving appearance information. Traditional methods solve the minimization problem approximately, for instance resorting to spectral techniques. In this paper, we deal with the spatio-temporal data, for a concrete example, human actions in video sequences. In this context, we first make three realistic assumptions: (i) causality of human movements; (ii) sequential nature of human movements; and (iii) one-to-one mapping of time instants. We show that, under these assumptions, the correspondence problem can be decomposed into a set of subproblems such that each subproblem can be solved recursively in terms of the others, and hence an efficient exact minimization algorithm can be derived using dynamic programming approach. Secondly, we propose a special graphical structure which is elongated in time. We argue that, instead of approximately solving the original problem, a solution can be obtained by exactly solving an approximated problem. An exact minimization algorithm is derived for this structure and successfully applied to action recognition in two settings: video data and Kinect coordinate data.

Fast computation of the prime implicants by exact direct-cover algorithm based on the new partial ordering operation rule

In this study, a novel OFF-set based direct-cover Exact Minimization Algorithm (EMA) is proposed for single-output Boolean functions represented in a sum-of-products form. To obtain the complete set of prime implicants covering the given Target Minterm (ON-minterm), the proposed method uses OFF-cubes (OFF-minterms) expanded by this Target Minterm. The amount of temporary results produced by this method does not exceed the size of the OFF-set. In order to achieve the goal of this study, which is to make faster computations, logic operations were used instead of the standard operations. Expansion OFF-cubes, commutative absorption operations and intersection operations are realized by logic operations for fast computation. The proposed minimization method is tested on several classes of benchmarks and then compared with the ESPRESSO algorithm. The results show that the proposed algorithm obtains more accurate and faster results than ESPRESSO does.

Exact Minimization of FPRMs for Incompletely Specified Functions by Using MTBDDs

Fixed polarity Reed-Muller expressions (FPRMs) exhibit several useful properties that make them suitable for many practical applications. This paper presents an exact minimization algorithm for FPRMs for incompletely specified functions. For an n-variable function with α unspecified minterms there are 2n+α distinct FPRMs, and a minimum FPRM is one with the fewest product terms. To find a minimum FPRM the algorithm requires to determine an assignment of the incompletely specified minterms. This is accomplished by using the concept of integer-valued functions in conjunction with an extended truth vector and a weight vector. The vectors help formulate the problem as an assignment of the variables of integer-valued functions, which are then efficiently manipulated by using multi-terminal binary decision diagrams for finding an assignment of the unspecified minterms. The effectiveness of the algorithm is demonstrated through experimental results for code converters, adders, and randomly generated functions.

A simplification algorithm of and‐exor expressions guaranteeing minimality for some class of logic functions

AbstractAmong all AND‐EXOR expressions that represent a logic function f, those with a minimum number of products are called minimum AND‐EXOR expressions of f. The number of products in the minimum AND‐EXOR expressions of f is denoted by τ(f). The minimization algorithms of AND‐EXOR expressions take a great deal of time for computing (time complexity O(22n)), where n is the number of variables. Thus it is necessary to develop a simplification algorithm that does not always find the exact minimum AND‐EXOR expressions but runs fast. This paper presents a simplification algorithm of AND‐EXOR expressions, which partially guarantees minimality and has the following properties: (1) it computes a minimum AND‐EXOR expression for a logic function f with τ(f) < 3(k + 1), where k is a nonnegative integer; (2) if an AND‐EXOR expression obtained by this algorithm takes less than or equal to 3(k + 1) of products, then it is minimum; and (3) its time complexity is O(τkn2+(k+C)n). From the experimental results for k = 0, 1, 2, we can conclude that its computational time is practical for n = 20, 7, 6, respectively. Modifying this algorithm, we present an exact minimization algorithm for 5‐variable functions, which is faster than previously known ones.