We suggest a new optical solution for solving the YES NO version of the Exact Cover problem by using the massive parallelism of light. The idea is to build an optical device which can generate all possible solutions of the problem and then to pick the correct one. In our case the device has a graph-like representation and the light is traversing it by following the routes given by the connections between nodes. The nodes are connected by arcs in a special way which lets us to generate all possible covers (exact or not) of the given set. For selecting the correct solution we assign to each item, from the set to be covered, a special integer number. These numbers will actually represent delays induced to light when it passes through arcs. The solution is represented as a subray arriving at a certain moment in the destination node. This will tell us if an exact cover does exist or not.
Using light instead of electric power for performing computations is not a new idea. Optical Character Recognition (OCR) machines @cite_4 were one of the first modern devices which are based on light for solving a difficult problem. Later, various researchers have shown how light can solve problems faster than modern computers. An example is the @math -point discrete Fourier transform computation which can be performed in only unit time @cite_1 @cite_8 .
@cite_1 @cite_4 @cite_8
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An all nonionic liquid shampoo which includes an amine oxide, a polyoxyethylene hexitan mono-higher fatty acid ester, and at least one of a higher alkoxy polyoxyethylene ethanol, an alkyl glycoside and a mixture of glycoside, a higher fatty acid lower alkanolamide and polyacrylamide. Optionally, the mixture of higher fatty acid lower alkanolamide and polyacrylamide may be present in the liquid shampoo containing amine oxide, polyoxyethylene hexitan mono-higher fatty acid ester and the higher alkoxy polyoxyethylene ethanol and or alkyl glycoside. Another optional constituent is a polyethylene glycol higher fatty acid ester. The shampoos are essentially free of ions and are desirably completely free of ionic materials with the pH essentially neutral. Optical-computing technology offers new challenges to algorithm designers since it can perform an n-point discrete Fourier transform (DFT) computation in only unit time. Note that the DFT is a nontrivial computation in the parallel random-access machine model, a model of computing commonly used by parallel-algorithm designers. We develop two new models, the DFT–VLSIO (very-large-scale integrated optics) and the DFT–circuit, to capture this characteristic of optical computing. We also provide two paradigms for developing parallel algorithms in these models. Efficient parallel algorithms for many problems, including polynomial and matrix computations, sorting, and string matching, are presented. The sorting and string-matching algorithms are particularly noteworthy. Almost all these algorithms are within a polylog factor of the optical-computing (VLSIO) lower bounds derived by Barakat Reif [Appl. Opt.26, 1015 (1987) and by Tyagi Reif [Proceedings of the Second IEEE Symposium on Parallel and Distributed Processing (Institute of Electrical and Electronics Engineers, New York, 1990) p. 14].