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Self-aligned double patterning (SADP) lithography is a promising technology which can reduce the overlay and print 2D features for sub-32nm process. Yet, how to decompose a layout to minimize the overlay and perform hot spot detection is still an open problem. In this paper, we present an algorithm that can optimally solve the SADP decomposition problem. For a decomposable layout, our algorithm guarantees to find a decomposition solution that minimizes overlay. For a non-decomposable layout our algorithm guarantees to find all hot spots. Experimental results validate our method, and decomposition results for Nangate Open Cell Library and larger testcases are also provided with competitive run-times.