197 research outputs found
Computing the Boolean product of two n\times n Boolean matrices using O(n^2) mechanical operation
We study the problem of determining the Boolean product of two n\times n
Boolean matrices in an unconventional computational model allowing for
mechanical operations. We show that O(n^2) operations are sufficient to compute
the product in this model.Comment: 11 pages, 7 figure
A Combinatorial Algorithm for All-Pairs Shortest Paths in Directed Vertex-Weighted Graphs with Applications to Disc Graphs
We consider the problem of computing all-pairs shortest paths in a directed
graph with real weights assigned to vertices.
For an 0-1 matrix let be the complete weighted graph
on the rows of where the weight of an edge between two rows is equal to
their Hamming distance. Let be the weight of a minimum weight spanning
tree of
We show that the all-pairs shortest path problem for a directed graph on
vertices with nonnegative real weights and adjacency matrix can be
solved by a combinatorial randomized algorithm in time
As a corollary, we conclude that the transitive closure of a directed graph
can be computed by a combinatorial randomized algorithm in the
aforementioned time.
We also conclude that the all-pairs shortest path problem for uniform disk
graphs, with nonnegative real vertex weights, induced by point sets of bounded
density within a unit square can be solved in time
Small Normalized Boolean Circuits for Semi-disjoint Bilinear Forms Require Logarithmic Conjunction-depth
We consider normalized Boolean circuits that use binary operations of disjunction and conjunction, and unary negation, with the restriction that negation can be only applied to input variables. We derive a lower bound trade-off between the size of normalized Boolean circuits computing Boolean semi-disjoint bilinear forms and their conjunction-depth (i.e., the maximum number of and-gates on a directed path to an output gate). In particular, we show that any normalized Boolean circuit of at most epsilon log n conjunction-depth computing the n-dimensional Boolean vector convolution has Omega(n^{2-4 epsilon}) and-gates. Analogously, any normalized Boolean circuit of at most epsilon log n conjunction-depth computing the n x n Boolean matrix product has Omega(n^{3-4 epsilon}) and-gates. We complete our lower-bound trade-offs with upper-bound trade-offs of similar form yielded by the known fast algebraic algorithms
A QPTAS for the Base of the Number of Triangulations of a Planar Point Set
The number of triangulations of a planar n point set is known to be ,
where the base lies between and The fastest known algorithm
for counting triangulations of a planar n point set runs in time.
The fastest known arbitrarily close approximation algorithm for the base of the
number of triangulations of a planar n point set runs in time subexponential in
We present the first quasi-polynomial approximation scheme for the base of
the number of triangulations of a planar point set
Consequences of APSP, triangle detection, and 3SUM hardness for separation between determinism and non-determinism
We present implications from the known conjectures like APSP, 3SUM and ETH in
a form of a negated containment of a linear-time with a non-deterministic
logarithmic-bit oracle in a respective deterministic bounded-time class They
are different for different conjectures and they exhibit in particular the
dependency on the input range parameters.Comment: The section on range reduction in the previous version contained a
flaw in a proof and therefore it has been remove
Matrix and Vector Products for Inputs Decomposable into Few Monotone Subsequences
We study the time complexity of computing the matrix
product of two integer matrices in terms of and the
number of monotone subsequences the rows of the first matrix and the
columns of the second matrix can be decomposed into. In particular,
we show that if each row of the first matrix can be decomposed into
at most monotone subsequences and each column of the second
matrix can be decomposed into at most monotone subsequences
such that all the subsequences are non-decreasing or all of them are
non-increasing then the product of the matrices can be
computed in time. On the other hand, we observe
that if all the rows of the first matrix are non-decreasing and all
columns of the second matrix are non-increasing or {\em vice versa}
then this case is as hard as the general one.
Similarly, we also study the time complexity of computing the
convolution of two -dimensional integer vectors in
terms of and the number of monotone subsequences the two vectors
can be decomposed into. We show that if the first vector can be
decomposed into at most monotone subsequences and the second
vector can be decomposed into at most subsequences such that
all the subsequences of the first vector are non-decreasing and all
the subsequences of the second vector are non-increasing or {\em
vice versa} then their convolution can be computed in
time. On the other, the case when both
vectors are non-decreasing or both of them are non-increasing is as
hard as the general case.Comment: 16 pages, accepted by COCOON 202
Lower Bounds for DeMorgan Circuits of Bounded Negation Width
We consider Boolean circuits over {or, and, neg} with negations applied only to input variables. To measure the "amount of negation" in such circuits, we introduce the concept of their "negation width". In particular, a circuit computing a monotone Boolean function f(x_1,...,x_n) has negation width w if no nonzero term produced (purely syntactically) by the circuit contains more than w distinct negated variables. Circuits of negation width w=0 are equivalent to monotone Boolean circuits, while those of negation width w=n have no restrictions. Our motivation is that already circuits of moderate negation width w=n^{epsilon} for an arbitrarily small constant epsilon>0 can be even exponentially stronger than monotone circuits.
We show that the size of any circuit of negation width w computing f is roughly at least the minimum size of a monotone circuit computing f divided by K=min{w^m,m^w}, where m is the maximum length of a prime implicant of f. We also show that the depth of any circuit of negation width w computing f is roughly at least the minimum depth of a monotone circuit computing f minus log K. Finally, we show that formulas of bounded negation width can be balanced to achieve a logarithmic (in their size) depth without increasing their negation width
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