### Citations

3515 | Graph-Based Algorithms for Boolean Function Manipulation.
- Bryant
- 1986
(Show Context)
Citation Context ...cinct representation of certain objects can nowadays be found in many areas of computer science. A paradigmatic example is the use of OBDDs (ordered binary decision diagrams) in hardware verification =-=[5,26]-=-. OBDDs are a succinct representation of Boolean functions. Consider a boolean function f(x1, . . . , xn) in n input variables. One can represent f by its decision tree, which is a full binary tree of... |

1339 |
The Design and Analysis of Spatial Data Structures
- Samet
- 1990
(Show Context)
Citation Context ...e several succinct representations of sparse matrices. One of which are quad-trees, used in computer graphics for the representation of large constant areas in 2-dimensional pictures, see for example =-=[29,9]-=-. Actually, an MTDD can be seen as a quad-tree that is folded into a dag by merging identical subtrees. Two-dimensional straight-line programs. MTDDs are also a special case of 2-dimensional straight-... |

316 | Integer programming with a fixed number of variables
- Lenstra
- 1983
(Show Context)
Citation Context ...have to check, whether there exist λ1, . . . , λl ∈ N such that v = v0 + λ1v1 + · · ·λlvl. This is an instance of integer programming in the fixed dimension 2k, which can be solved in polynomial time =-=[20]-=-. For the lower bound we take elements x, y, z ∈M such that x 6= y but x+z = y+z. These elements exist since M is not cancellative. We use an encoding of 3SAT from [3]. Take a 3CNF formula C = ∧mi=1 C... |

230 |
A taxonomy of problems with fast parallel algorithms
- Cook
- 1985
(Show Context)
Citation Context ...for a function f ∈ FPSPACE we have |f(w)| ≤ 2|w|O(1) for every input. The function that maps an explicitly given integer matrix (with binary encoded entries) to its determinant belongs to uniform NC2 =-=[7]-=- and hence to FSPACE(log2(n)). We need the following simple lemma, see e.g. [24, Lemma 2.1]. Lemma 1. If f ∈ FPSPACE and L ∈ polyL then f−1(L) ∈ PSPACE. The following result can be shown in the same w... |

220 | PP is as hard as the polynomial-time hierarchy - Toda - 1991 |

183 |
Multi-terminal binary decision diagrams: An efficient datastructure for matrix representation.
- Fujita, McGeer, et al.
- 1997
(Show Context)
Citation Context ...an mappings, functions from {0, 1}n to any set S can be represented. One simply has to label the leaves of the decision tree with elements from S. This yields multi-terminal decision diagrams (MTDDs) =-=[12]-=-. Of particular interest is the case, where S is a semiring, e.g. N or Z. In the same way as an adjacency matrix (i.e., a boolean matrix) of dimension 2n can be represented by an OBDD, a matrix of dim... |

174 |
Completeness classes in algebra
- Valiant
- 1979
(Show Context)
Citation Context ...ACE-complete) if n is given unary (resp. binary). Here, #P (resp. #PSPACE) is the class of functions counting the number of accepting computations of a nondeterministic polynomial time Turing machine =-=[34]-=- (resp., a nondeterministic polynomial space Turing machine [18]). An example of a natural #PSPACE-complete counting problem is counting the number of strings not accepted by a given NFA [18]. 2 Relat... |

69 |
Succinct representations of graphs.
- Galperin, Wigderson
- 1983
(Show Context)
Citation Context ...,2, D) ) . The proof for the construction of the element-wise product is similar as for the tensorproduct. ⊓⊔ 10 4.2 Boolean circuits Another well-studied succinct representation are boolean circuits =-=[14]-=-. A boolean circuit with n inputs represents a binary string of length 2n, namely the string of output values for the 2n many input assignments (concatenated in lexicographic order). In a similar way,... |

68 |
Interleaving based variable ordering methods for ordered binary decision diagrams
- Fujii, Ootomo, et al.
- 1993
(Show Context)
Citation Context ...ed variable ordering here, where the bits of the two coordinates a and b are bitwise interleaved. This ordering turned out to be convenient in the context of OBDD-based graph representation, see e.g. =-=[11]-=-. Classical graph problems (like reachability, alternating reachability, existence of a Hamiltonian cycle) have been studied for OBDD-represented graphs in [10,35]. It turned out that these problems a... |

60 |
Testing equivalence of morphisms on context-free languages
- Plandowski
- 1994
(Show Context)
Citation Context ...sentation of the string it generates. Algorithmic problems that can be solved efficiently (in polynomial time) on SLP-represented strings are for instance equality checking (first shown by Plandowski =-=[28]-=-) and pattern matching, see [22] for a survey. In [3] a 2-dimensional extension of SLPs (2SLPs in the following) was defined. Here, every variable of the grammar generates a (not necessarily square) m... |

56 |
A very hard log-space counting class.
- Alvarez, Jenner
- 1993
(Show Context)
Citation Context ...ace nondeterministic polynomial time Turing machines by nondeterministic polynomial space Turing machines (resp. nondeterministic logspace Turing machines), we obtain the class #PSPACE [18] (resp. #L =-=[1]-=-). Note that for a mapping f ∈ #PSPACE, the number f(x) may grow doubly exponential in |x|, whereas for f ∈ #P, the number f(x) is bounded singly exponential in |x|. Ladner [18] has shown that a mappi... |

48 | Probabilistic algorithms for deciding equivalence of straight-line programs.
- Ibarra, Moran
- 1983
(Show Context)
Citation Context ... exactly four children and every root-leaf path has the same length. Let us finally mention that straight-line programs are also used for the compact representation of other objects, e.g. polynomials =-=[17]-=-, trees [23], graphs [19], and regular languages [15]. Tensor circuits. In [2,8], the authors investigated the problems of evaluating tensor formulas and tensor circuits. Let us restrict to the latter... |

42 | Counting problems computationally equivalent to the determinant. manuscript
- Toda
- 1991
(Show Context)
Citation Context ...ove that already for MTDDs over Z it is PSPACE-complete to check whether the determinant of the generated matrix is zero (Theorem 15). This result is shown by lifting a classical construction of Toda =-=[32]-=- (showing that computing the determinant of an explicitly given integer matrix is complete for the counting class GapL) to configuration graphs of polynomial space bounded Turing machines, which are o... |

40 |
Nondeterministic NC1computation.
- Caussinus, McKenzie, et al.
- 1998
(Show Context)
Citation Context ...is given explicitly and n is given unary. – #P-complete, if A is given by an MTDD and n is given unary. – #PSPACE-complete, if A is given by an MTDD and n is given binary. Let us also mention that in =-=[6,13,27]-=- the complexity of evaluating iterated matrix products and matrix powers in a fixed dimension is studied. It turns out that multiplying a sequence of (d × d)-matrices over Z in the fixed dimension d ≥... |

27 |
Polynomial space counting problems.
- Ladner
- 1989
(Show Context)
Citation Context .... #PSPACE) is the class of functions counting the number of accepting computations of a nondeterministic polynomial time Turing machine [34] (resp., a nondeterministic polynomial space Turing machine =-=[18]-=-). An example of a natural #PSPACE-complete counting problem is counting the number of strings not accepted by a given NFA [18]. 2 Related work Sparse matrices and quad-trees. To the knowledge of the ... |

26 | On the complexity of pattern matching for highly compressed two-dimensional texts
- Berman, Karpinski, et al.
- 2002
(Show Context)
Citation Context ...lems that can be solved efficiently (in polynomial time) on SLP-represented strings are for instance equality checking (first shown by Plandowski [28]) and pattern matching, see [22] for a survey. In =-=[3]-=- a 2-dimensional extension of SLPs (2SLPs in the following) was defined. Here, every variable of the grammar generates a (not necessarily square) matrix (or picture), where every position is labeled w... |

21 | Complexity of problems on graphs represented as OBDDs.
- Feigenbaum, Kannan, et al.
- 1998
(Show Context)
Citation Context ...DD-based graph representation, see e.g. [11]. Classical graph problems (like reachability, alternating reachability, existence of a Hamiltonian cycle) have been studied for OBDD-represented graphs in =-=[10,35]-=-. It turned out that these problems are exponentially harder for OBDD-represented graphs than for explicitly given graphs. In [35] an upgrading theorem for OBDD-represented graphs was shown. It roughl... |

21 |
The correlation between the complexities of nonhierarchical and hierarchical versions of graph problems
- Lengauer, Wagner
- 1992
(Show Context)
Citation Context ...d every root-leaf path has the same length. Let us finally mention that straight-line programs are also used for the compact representation of other objects, e.g. polynomials [17], trees [23], graphs =-=[19]-=-, and regular languages [15]. Tensor circuits. In [2,8], the authors investigated the problems of evaluating tensor formulas and tensor circuits. Let us restrict to the latter. A tensor circuit is a c... |

20 |
The complexity of tree automata and XPath on grammarcompressed trees
- Lohrey, Maneth
(Show Context)
Citation Context ...r children and every root-leaf path has the same length. Let us finally mention that straight-line programs are also used for the compact representation of other objects, e.g. polynomials [17], trees =-=[23]-=-, graphs [19], and regular languages [15]. Tensor circuits. In [2,8], the authors investigated the problems of evaluating tensor formulas and tensor circuits. Let us restrict to the latter. A tensor c... |

16 |
The size of reduced OBDDs and optimal read-once branching programs for almost all Boolean functions
- Wegener
- 1994
(Show Context)
Citation Context ...rule is applied that removes nodes for which the left and right child are identical. On the other hand, it is known that asymptotically the compression achieved by this elimination rule is negligible =-=[36]-=-. b1 · · · bn) is the binary representation of the index a (resp. b). Note that we use the so called interleaved variable ordering here, where the bits of the two coordinates a and b are bitwise inter... |

14 | Fast algorithms for linear algebra modulo N .
- Storjohann, Mulders
- 1998
(Show Context)
Citation Context ...re generate the full Qn. But then an+1 ∈ Un. ⊓⊔ Recall that the exponent of an abelian groupA is the smallest integer k (if it exists) such that kg = 0 for all g ∈ A. The following result is shown in =-=[30]-=-: Lemma 9. Let k ≥ 2 and let A be an abelian group of exponent k. Let ai,1x1 + · · ·+ ai,nxn = 0 for 1 ≤ i ≤ m ≤ n + 1 be equations, where ai,1, . . . , ai,n ∈ Z, and the variables x1, . . . , xn rang... |

10 | Algorithmics on SLP-compressed strings: a survey,
- Lohrey
- 2012
(Show Context)
Citation Context ...ates. Algorithmic problems that can be solved efficiently (in polynomial time) on SLP-represented strings are for instance equality checking (first shown by Plandowski [28]) and pattern matching, see =-=[22]-=- for a survey. In [3] a 2-dimensional extension of SLPs (2SLPs in the following) was defined. Here, every variable of the grammar generates a (not necessarily square) matrix (or picture), where every ... |

8 | The complexity of tensor calculus
- Damm, Holzer, et al.
- 2002
(Show Context)
Citation Context ...ally mention that straight-line programs are also used for the compact representation of other objects, e.g. polynomials [17], trees [23], graphs [19], and regular languages [15]. Tensor circuits. In =-=[2,8]-=-, the authors investigated the problems of evaluating tensor formulas and tensor circuits. Let us restrict to the latter. A tensor circuit is a circuit where the gates evaluate to matrices over a semi... |

8 |
Algorithmic problems for commutative semigroups
- Ta˘iclin
- 1968
(Show Context)
Citation Context ...ts we can compute numbers n1, . . . , nk,m1, . . . ,mk ∈ N in binary representation such that val(A1)i,j = n1a1+ · · ·+nkak and val(A2)i,j = m1a1+ · · ·+mkak. Now we can use the following result from =-=[31]-=-: There is a semilinear subset S ⊆ N2k (depending only on our fixed monoid M ) such that for all x1, . . . , xk, y1, . . . , yk ∈ N we have: x1a1 + · · · + xkak = y1a1 + · · · + ykak if and only if (x... |

7 |
How to encode a logical structure by an OBDD.
- Veith
- 1998
(Show Context)
Citation Context ...DD-based graph representation, see e.g. [11]. Classical graph problems (like reachability, alternating reachability, existence of a Hamiltonian cycle) have been studied for OBDD-represented graphs in =-=[10,35]-=-. It turned out that these problems are exponentially harder for OBDD-represented graphs than for explicitly given graphs. In [35] an upgrading theorem for OBDD-represented graphs was shown. It roughl... |

6 |
Literal shuffle of compressed words.
- Bertoni, Choffrut, et al.
- 2008
(Show Context)
Citation Context ...ed in time O(n). (2) +-circuits for the sum of all entries of val(G) and the trace of val(G) can be computed in time O(n). (3) A +-circuit for the matrix entry val(G)i,j can be computed in time O(n). =-=(4)-=- MTDD+ of size O(n ·m) for the tensor product val(G)⊗ val(H) (which includes the scalar product) and the element-wise (Hadamard) product val(G) ◦ val(H) (assuming height(G) = height(H)) can be compute... |

6 | Leaf languages and string compression.
- Lohrey
- 2011
(Show Context)
Citation Context ... size of G and H . Moreover, it is PSPACE-complete to check for two SLP-represented strings u and v and an NFA T operating on strings of pairs of symbols, whether T accepts the convolution of u and v =-=[21]-=-. MTDDs restrict 2SLPs by forbidding unbalanced derivation trees. The derivation tree of an MTDD results from unfolding the rules in (1); it is a tree, where every nonleaf node has exactly four childr... |

4 |
The complexity of tensor circuit evaluation
- Beaudry, Holzer
(Show Context)
Citation Context ...ally mention that straight-line programs are also used for the compact representation of other objects, e.g. polynomials [17], trees [23], graphs [19], and regular languages [15]. Tensor circuits. In =-=[2,8]-=-, the authors investigated the problems of evaluating tensor formulas and tensor circuits. Let us restrict to the latter. A tensor circuit is a circuit where the gates evaluate to matrices over a semi... |

4 |
Skip quadtrees: Dynamic data structures for multidimensional point sets. IJCGA 18(01n02),
- Eppstein, Goodrich, et al.
- 2008
(Show Context)
Citation Context ...e several succinct representations of sparse matrices. One of which are quad-trees, used in computer graphics for the representation of large constant areas in 2-dimensional pictures, see for example =-=[29,9]-=-. Actually, an MTDD can be seen as a quad-tree that is folded into a dag by merging identical subtrees. Two-dimensional straight-line programs. MTDDs are also a special case of 2-dimensional straight-... |

3 |
More concise representation of regular languages by automata and regular expressions
- Geffert, Mereghetti, et al.
(Show Context)
Citation Context ...he same length. Let us finally mention that straight-line programs are also used for the compact representation of other objects, e.g. polynomials [17], trees [23], graphs [19], and regular languages =-=[15]-=-. Tensor circuits. In [2,8], the authors investigated the problems of evaluating tensor formulas and tensor circuits. Let us restrict to the latter. A tensor circuit is a circuit where the gates evalu... |

3 |
Threshold circuits for iterated matrix product and powering. Informatique Théorique et Applications
- Mereghetti, Palano
(Show Context)
Citation Context ...is given explicitly and n is given unary. – #P-complete, if A is given by an MTDD and n is given unary. – #PSPACE-complete, if A is given by an MTDD and n is given binary. Let us also mention that in =-=[6,13,27]-=- the complexity of evaluating iterated matrix products and matrix powers in a fixed dimension is studied. It turns out that multiplying a sequence of (d × d)-matrices over Z in the fixed dimension d ≥... |

2 |
Functions computable in polynomial space
- Galota, Vollmer
- 2003
(Show Context)
Citation Context ...is given explicitly and n is given unary. – #P-complete, if A is given by an MTDD and n is given unary. – #PSPACE-complete, if A is given by an MTDD and n is given binary. Let us also mention that in =-=[6,13,27]-=- the complexity of evaluating iterated matrix products and matrix powers in a fixed dimension is studied. It turns out that multiplying a sequence of (d × d)-matrices over Z in the fixed dimension d ≥... |

2 | Isomorphism of regular trees and words. - Lohrey, Mathissen - 2011 |

2 |
Succinct Algebraic Branching Programs Characterizing Non-uniform Complexity Classes
- Malod
- 2011
(Show Context)
Citation Context ... checking whether for a given circuit C the determinant of the matrix MC,1 vanishes is PSPACE-complete [16]. An algebraic version of this result for the algebraic complexity class VPSPACE is shown in =-=[25]-=-. Theorem 15 from Section 6 will strengthen the result from [16] to MTDD-represented matrices. 5 Testing equality In this section, we consider the problem of testing equality of MTDD+-represented matr... |

1 | On the complexity of the multivariate resultant
- Grenet, Koiran, et al.
(Show Context)
Citation Context ...onential size. It turns out that the adjacency matrix of the configuration graph of a polynomial space bounded Turing machine can be produced by a small MTDD. Theorem 15 sharpens a recent result from =-=[16]-=- stating that it is PSPACE-complete to check whether the determinant of a matrix that is represented by a boolean circuit (see Section 4.2) vanishes. We also prove several hardness results for countin... |