Theory of Computing Systems

, Volume 61, Issue 2, pp 322–351 | Cite as

Processing Succinct Matrices and Vectors

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Abstract

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 + MTDDs 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 is 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.

Keywords

Multi-terminal decision diagrams Succinct matrix representations Matrix computation 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department für Elektrotechnik und InformatikUniversität SiegenSiegenGermany
  2. 2.Institut für InformatikGoethe-UniversitätFrankfurtGermany

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