An algorithm for the 3-Satis ability problem is presented and a probabilistic analysis is performed. The analysis is based on an instance distribution which is parameterized to simulate a variety of sample characteristics. The algorithm assigns values to variables appearing in a given instance of 3-Satis ability, one at a time, using the unit clause heuristic and a maximum occurring literal selection heuristic; at each step a variable is chosen randomly from a subset of variables which is usually large. The algorithm runs in polynomial time and it is shown that the algorithm nds a solution to a random instance of 3-Satis ability with probability bounded from below by a constant greater than zero for a range of parameter values. The heuristics studied here can be used to select variables in a Backtrack algorithm for 3-Satis ability. Experiments have shown that for about the same range of parameters as above the Backtrack algorithm using the heuristics nds a solution in polynomial average time.

Reducing branching effect and increasing boundary noise immunity are of great importance for thinning patterns. An approach based on medial axis transform (MAT) to obtain a connected 1-pixel wide skeleton with few redundant branches is presented in this paper. Though the obtained skeleton by MAT is isotropic with few redundant branches, however, the skeleton points are usually disconnected. In order to rend the merits of the MAT and avoid its disadvantages, the proposed approach is composed of distance-map generation, grouping, ridge-path linking, and refining to obtain the connected 1-pixel wide thin line. The ridge-path linking strategy can guarantee the skeletons connected, whereas the refining process can be readily performed by a conventional thinning process to obtain the 1-pixel wide thinned pattern. The performances investigated by branching effect, signal-to-noise ratio (SNR), and measurement of skeleton deviation (MSD) confirm the feasibility of the proposed MAT-based thinning for line patterns.

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It is a challenging topic to perform pattern reconstruction from a unit-width skeleton, which is obtained by a parallel thinning algorithm. The bias skeleton yielded by a fully-parallel thinning algorithm, which usually results from the so-called hidden deletable points, will result in the difficulty of pattern reconstruction. In order to make a fully-parallel thinning algorithm pattern reconstructable, a newly-defined reconstructable skeletal pixel (RSP) including a thinning flag, iteration count, as well as reconstructable structure is proposed and applied for thinning iteration to obtain a skeleton table representing the resultant thin line. Based on the iteration count and reconstructable structure associated with each skeletal pixel in the skeleton table, the pattern can be reconstructed by means of the dilating and uniting operations. Embedding a conventional fully-parallel thinning algorithm into the proposed approach, the pattern may be over-reconstructed due to the influence of a biased skeleton. A simple process of removing hidden deletable points (RHDP) in the thinning iteration is thus presented to reduce the effect of the biased skeleton. Three well-known fully-parallel thinning algorithms are used for experiments. The performances investigated by the measurement of reconstructability (MR), the number of iterations (NI), as well as the measurement of skeleton deviation (MSD) confirm the feasibility of the proposed pattern reconstruction approach with the assistance of the RHDP process.