12211028
On data structures and asymmetric communication complexity
In this paper we consider two-party communication complexity, the ``asymmetric case'', when the input sizes of the two players differ significantly. Most of previous work on communication complexity only considers the total number of bits sent, but we study trade-offs between the number of bits the first player sends and the number of bits the second sends. These types of questions are closely related to the complexity of static data structure problems in the cell probe model. We derive two generally applicable methods of proving lower bounds and obtain several applications. These applications include new lower bounds for data structures in the cell probe model. Of particular interest is our ``round elimination'' lemma, which is interesting also for the usual symmetric communication case. This lemma generalizes and abstracts in a very clean form the ``round reduction'' techniques used in many previous lower bound proofs. ] 1998 Academic Press
14452021
On Data Structures and Asymmetric Communication Complexity
In this paper we consider two-party communication complexity, the “asymmetric case”, when the input sizes of the two players differ significantly. Most of previous work on communication complexity only considers the total number of bits sent, but we study trade-offs between the number of bits the first player sends and the number of bits the second sends. These types of questions are closely related to the complexity of static data structure problems in the cell probe model. We derive two generally applicable methods of proving lower bounds and obtain several applications. These applications include new lower bounds for data structures in the cell probe model. Of particular interest is our “round elimination” lemma, which is interesting also for the usual symmetric communication case. This lemma generalizes and abstracts in a very clean form the “round reduction” techniques used in many previous lower bound proofs.
5341041
Lower bounds on arithmetic circuits via partial derivatives
In this paper we describe a new technique for obtaining lower bounds on restricted classes of non-monotone arithmetic circuits. The heart of this technique is a complexity measure for multivariate polynomials, based on the linear span of their partial derivatives. We use the technique to obtain new lower bounds for computing symmetric polynomials (that hold over fields of characteristic zero) and iterated matrix products (that hold for all fields).
45550157
Space Complexity in Propositional Calculus
We study space complexity in the framework of propositional proofs. We consider a natural model analogous to Turing machines with a read-only input tape and such popular propositional proof systems as resolution, polynomial calculus, and Frege systems. We propose two different space measures, corresponding to the maximal number of bits, and clauses/monomials that need to be kept in the memory simultaneously. We prove a number of lower and upper bounds in these models, as well as some structural results concerning the clause space for resolution and Frege systems.
7565108
Space complexity in propositional calculus
We study space complexity in the framework of propositional proofs. We consider a natural model analogous to space bounded Turing machines with a read-only input tape, and such popular propositional proof systems as Resolution, Polynomial Calculus and Frege systems. We study two different space measures. The first, introduced by [5] for Resolution and extended here to other systems, is the structured measure which counts the number of clauses/monomials kept is memory simultaneously. The other is an unstructured °measure related to the number of b i t s describing the memory content. We develop lower bound techniques that enable proving tight linear lower bounds for the first measure, and tight quadratic lower bounds for the second, for large classes of tautologies (including familiar ones like the pigeonhole principle), in both Resolution and (extensions of) Polynomial Calculus. We also prove some structural results concerning the clause space for Resolution and Frege Systems.
6164458
Rubber bands, convex embeddings and graph connectivity
We give various characterizations ofk-vertex connected graphs by geometric, algebraic, and “physical” properties. As an example, a graphG isk-connected if and only if, specifying anyk vertices ofG, the vertices ofG can be represented by points of ℝk−1 so that nok are on a hyper-plane and each vertex is in the convex hull of its neighbors, except for thek specified vertices. The proof of this theorem appeals to physics. The embedding is found by letting the edges of the graph behave like ideal springs and letting its vertices settle in equilibrium.As an algorithmic application of our results we give probabilistic (Monte-Carlo and Las Vegas) algorithms for computing the connectivity of a graph. Our algorithms are faster than the best known (deterministic) connectivity algorithms for allk≧√n, and for very dense graphs the Monte Carlo algorithm is faster by a linear factor.
20100444
Near Optimal Separation Of Tree-Like And General Resolution
We present the best known separation between tree-like and general resolution, improving on the recent exp(n∈) separation of [2]. This is done by constructing a natural family of contradictions, of size n, that have O(n)-size resolution refutations, but only exp(Ω(n/log n))- size tree-like refutations. This result implies that the most commonly used automated theorem procedures, which produce tree-like resolution refutations, will perform badly on some inputs, while other simple procedures, that produce general resolution refutations, will have polynomial run-time on these very same inputs. We show, furthermore that the gap we present is nearly optimal. Specifically, if S (ST) is the minimal size of a (tree-like) refutation, we prove that ST = exp(O(S log log S/log S)).
1334521
Rectilinear Graphs and their Embeddings
The embedding problem for a class of graphs called rectilinear graphs is discussed. These graphs have applications in many VLSI Layout Problems. An interesting topological characterization of these graphs lead to efficient algorithms for recognizing and embedding rectilinear graphs which are embeddable on the plane.