4 research outputs found
Computing the Maximum using (min, +) Formulas
We study computation by formulas over (min,+). We consider the
computation of max{x_1,...,x_n} over N as a difference of
(min,+) formulas, and show that size n + n log n is sufficient
and necessary. Our proof also shows that any (min,+) formula
computing the minimum of all sums of n-1 out of n variables must
have n log n leaves; this too is tight. Our proofs use a
complexity measure for (min,+) functions based on minterm-like
behaviour and on the entropy of an associated graph
Complexity of Linear Operators
Let A in {0,1}^{n x n} be a matrix with z zeroes and u ones and x be an n-dimensional vector of formal variables over a semigroup (S, o). How many semigroup operations are required to compute the linear operator Ax?
As we observe in this paper, this problem contains as a special case the well-known range queries problem and has a rich variety of applications in such areas as graph algorithms, functional programming, circuit complexity, and others. It is easy to compute Ax using O(u) semigroup operations. The main question studied in this paper is: can Ax be computed using O(z) semigroup operations? We prove that in general this is not possible: there exists a matrix A in {0,1}^{n x n} with exactly two zeroes in every row (hence z=2n) whose complexity is Theta(n alpha(n)) where alpha(n) is the inverse Ackermann function. However, for the case when the semigroup is commutative, we give a constructive proof of an O(z) upper bound. This implies that in commutative settings, complements of sparse matrices can be processed as efficiently as sparse matrices (though the corresponding algorithms are more involved). Note that this covers the cases of Boolean and tropical semirings that have numerous applications, e.g., in graph theory.
As a simple application of the presented linear-size construction, we show how to multiply two n x n matrices over an arbitrary semiring in O(n^2) time if one of these matrices is a 0/1-matrix with O(n) zeroes (i.e., a complement of a sparse matrix)
Lower bounds on dynamic programming for maximum weight independent set
Publisher Copyright: © 2021 Tuukka Korhonen.We prove lower bounds on pure dynamic programming algorithms for maximum weight independent set (MWIS). We model such algorithms as tropical circuits, i.e., circuits that compute with max and + operations. For a graph G, an MWIS-circuit of G is a tropical circuit whose inputs correspond to vertices of G and which computes the weight of a maximum weight independent set of G for any assignment of weights to the inputs. We show that if G has treewidth w and maximum degree d, then any MWIS-circuit of G has 2Ω(w/d) gates and that if G is planar, or more generally H-minor-free for any fixed graph H, then any MWIS-circuit of G has 2Ω(w) gates. An MWIS-formula is an MWIScircuit where each gate has fan-out at most one. We show that if G has treedepth t and maximum degree d, then any MWIS-formula of G has 2Ω(t/d) gates. It follows that treewidth characterizes optimal MWIS-circuits up to polynomials for all bounded degree graphs and H-minor-free graphs, and treedepth characterizes optimal MWIS-formulas up to polynomials for all bounded degree graphs.Peer reviewe
Notes on Boolean Read-k and Multilinear Circuits
A monotone Boolean (OR,AND) circuit computing a monotone Boolean function f
is a read-k circuit if the polynomial produced (purely syntactically) by the
arithmetic (+,x) version of the circuit has the property that for every prime
implicant of f, the polynomial contains at least one monomial with the same set
of variables, each appearing with degree at most k. Every monotone circuit is a
read-k circuit for some k. We show that already read-1 (OR,AND) circuits are
not weaker than monotone arithmetic constant-free (+,x) circuits computing
multilinear polynomials, are not weaker than non-monotone multilinear
(OR,AND,NOT) circuits computing monotone Boolean functions, and have the same
power as tropical (min,+) circuits solving combinatorial minimization problems.
Finally, we show that read-2 (OR,AND) circuits can be exponentially smaller
than read-1 (OR,AND) circuits.Comment: A throughout revised version. To appear in Discrete Applied
Mathematic