处理简洁矩阵和向量
Processing Succinct Matrices and Vectors
摘要：
我们研究的复杂性矩阵表示的多端决策图（MTDD）的算法问题。这些是有序决策图的一个变体，其中的终端节点被标记为半环的任意元素（而不是0和1）。一个简单的例子表明，两个MTDD表示的矩阵的乘积不能由多项式大小的MTDD表示。为了克服这个缺陷，我们通过允许规则中的变量（相同维度的）的分量符号加法将MTDD扩展到MTDD_+。它表明，访问一个条目，相等性检查，矩阵乘法，和其他基本的矩阵运算可以解决在多项式时间的MTDD_+表示的矩阵。另一方面，MTDD表示矩阵的行列式是否为零的问题是PSPACE$-完全的，而MTDD+表示对角矩阵的行列式是否为零的问题是NP-完全的。计算MTDD表示的矩阵的乘积中的特定项是#P-完全的。
英文摘要：
We study the complexity of algorithmic problems for matrices that are represented by multi-terminal decision diagrams (MTDD). These are a variant of ordered decision diagrams, where the terminal nodes are labeled with arbitrary elements of a semiring (instead of 0 and 1). A simple example shows that the product of two MTDD-represented matrices cannot be represented by an MTDD of polynomial size. To overcome this deficiency, we extended MTDDs to MTDD_+ by allowing componentwise symbolic addition of variables (of the same dimension) in rules. It is shown that accessing an entry, equality checking, matrix multiplication, and other basic matrix operations can be solved in polynomial time for MTDD_+-represented matrices. On the other hand, testing whether the determinant of a MTDD-represented matrix vanishes PSPACE$-complete, and the same problem is NP-complete for MTDD_+-represented diagonal matrices. Computing a specific entry in a product of MTDD-represented matrices is #P-complete.
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