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.
Let us finally mention that straight-line programs are also used for the compact representation of other objects, e.g. polynomials @cite_34 , trees @cite_31 , graphs @cite_32 , and regular languages @cite_16 .
@cite_31 @cite_34 @cite_32 @cite_16
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We consider two formalisms for representing regular languages: constant height pushdown automata and straight line programs for regular expressions. We constructively prove that their sizes are polynomially related. Comparing them with the sizes of finite state automata and regular expressions, we obtain optimal exponential and double exponential gaps, i.e., a more concise representation of regular languages. Results on algorithmic problems on strings that are given in a compressed form via straightline programs are surveyed. A straight-line program is a context-free grammar that generates exactly one string. In this way, exponential compression rates can be achieved. Among others, we study pattern matching for compressed strings, membership problems for compressed strings in various kinds of formal languages, and the problem of querying compressed strings. Applications in combinatorial group theory and computational topology and to the solution of word equations are discussed as well. Finally, extensions to compressed trees and pictures are considered. It is shown that, over an arbitrary ring, the functions computed by polynomial-size algebraic formulas are also computed by polynomial-length algebraic straight-line programs that use only three registers. This was previously known for Boolean formulas [D. A. Barrington, J. Comput. System Sci., 38 (1989), pp. 150–164], which are equivalent to algebraic formulas over the ring @math . For formulas over arbitrary rings, the result is an improvement over previous methods that require the number of registers to be logarithmic in the size of the formulas in order to obtain polynomial-length straight-line programs. Moreover, the straight-line programs that arise in these constructions have the property that they consist of statements whose actions on the registers are linear and bijective. A consequence of this is that the problem of determining the iterated product of @math matrices is complete (under P-projections) for algebraic @math . Also, when the ring is @math , the programs that arise in the c... Straight-line programs (SLPs) offer powerful text compression by representing a text T[1,u] in terms of a restricted context-free grammar of n rules, so that T can be recovered in O(u) time. However, the problem of operating the grammar in compressed form has not been studied much. We present a grammar representation whose size is of the same order of that of a plain SLP representation, and can answer other queries apart from expanding nonterminals. This can be of independent interest. We then extend it to achieve the first grammar representation able of extracting text substrings, and of searching the text for patterns, in time o(n). We also give byproducts on representing binary relations.