Digital halftones by dot diffusion
ACM Transactions on Graphics (TOG)
Randomized rounding: a technique for provably good algorithms and algorithmic proofs
Combinatorica - Theory of Computing
Probabilistic construction of deterministic algorithms: approximating packing integer programs
Journal of Computer and System Sciences - 27th IEEE Conference on Foundations of Computer Science October 27-29, 1986
SIAM Journal on Discrete Mathematics
Improved Approximation Guarantees for Packing and Covering Integer Programs
SIAM Journal on Computing
Lattice approximation and linear discrepency of totally unimodular matrices
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Matrix rounding under the Lp-discrepancy measure and its application to digital halftoning
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Structured Randomized Rounding and Coloring
FCT '01 Proceedings of the 13th International Symposium on Fundamentals of Computation Theory
On the discrepancy of combinatorial rectangles
Random Structures & Algorithms - Special issue: Proceedings of the tenth international conference "Random structures and algorithms"
On dependent randomized rounding algorithms
Operations Research Letters
Non-independent randomized rounding
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
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We investigate the problem to round a given [0, 1]-valued matrix to a 0, 1 matrix such that the rounding error with respect to 2 脳 2 boxes is small. Such roundings yield good solutions for the digital halftoning problem as shown by Asano et al. (SODA 2002). We present a randomized algorithm computing roundings with expected error at most 0.6287 per box, improving the 0.75 non-constructive bound of Asano et al. Our algorithm is the first one solving this problem fast enough for practical application, namely in linear time.Of a broader interest might be our rounding scheme, which is a modification of randomized rounding. Instead of independently rounding the variables (expected error 0.82944 per box in the worst case), we impose a number of suitable dependencies.Experimental results show that roundings obtained by our approach look much less grainy than by independent randomized rounding, and only slightly more grainy than by error diffusion. On the other hand, the latter algorithm (like all known deterministic algorithms) tends to produce unwanted structures, a problem that randomized algorithms like ours are unlikely to encounter.