Computational Complexity Analysis of Determinant Decision Diagram

Abstract-A determinant decision diagram (DDD) uses a binary decision diagram (BDD) to calculate a determinant symbolically, which is then applied for symbolic circuit analysis. The efficiency of such a technique is determined mainly by a symbol ordering scheme. Finding an optimal symbol order is an non-deterministic polynomial-time hard problem in the practice of BDD. So far, it is unknown what an optimal order is for a general sparse matrix. This brief shows that a row-wise (or column-wise) order is an optimal BDD order for full matrices in the sense that the DDD graph constructed has the minimum number of vertices (i.e., the DDD size). The optimal DDD size is proven to be (n · 2^n-1) for an n × n full matrix. This size provides a DDD complexity measure that has rarely been investigated in the literature.