Abstract. We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis:• The Blum-Shub-Smale model of computation over the reals.• A problem we call the "Generic Task of Numerical Computation," which captures an aspect of doing numerical computation in floating point, similar to the "long exponent model" that has been studied in the numerical computing community. We show that both of these approaches hinge on the question of understanding the complexity of the following problem, which we call PosSLP: Given a division-free straight-line program producing an integer N , decide whether N > 0.• In the Blum-Shub-Smale model, polynomial time computation over the reals (on discrete inputs) is polynomial-time equivalent to PosSLP, when there are only algebraic constants. We conjecture that using transcendental constants provides no additional power, beyond nonuniform reductions to PosSLP, and we present some preliminary results supporting this conjecture.• The Generic Task of Numerical Computation is also polynomial-time equivalent to PosSLP. We prove that PosSLP lies in the counting hierarchy. Combining this with work of Tiwari, we obtain that the Euclidean Traveling Salesman Problem lies in the counting hierarchy -the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE.In the course of developing the context for our results on arithmetic circuits, we present some new observations on the complexity of ACIT: the Arithmetic Circuit Identity Testing problem. In particular, we show that if n! is not ultimately easy, then ACIT has subexponential complexity.
We study the static membership problem: Given a set S of at most n keys drawn from a universe of size m, store it so that queries of the form "Is x in S?" can be answered quickly. We study schemes for this problem that use space close to the information theoretic lower bound of 12(nlog(~)) bits and yet answer queries by reading a small number of bits of the memory. We show that there is a randomized scheme with error e that stores O(;~-logm) bits and answers queries using a single bitprobe. It is based on a family of sets with small intersections. If the error is required to be restricted to queries not in S, then we have a scheme that stores o((n) 2 logm) bits, answers queries with one bitprobe and works with probability of error less than e. We also show that better schemes with one-sided error can be obtained if more probes are allowed. We show lower bounds that come close to our upper bounds (for a large range of n and e): Schemes that answer queries with just one bitprobe and error probability e must use f~(~ log m) bits of storage; if the error is restricted to r~ 2 queries not in S, then the scheme must use ~(~ log m) bits of storage. We also consider deterministic schemes for the static membership problem and show upper and lower bounds.
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. ]
In this paper we consider two party communication complexity when the input sizes of the two players differ significantly, the "asymmetric" case. Most of previous work on communication complexity only considers the total number of bits sent, but we study tradeoffs 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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.