Threshold Circuits Detecting Global Patterns in Two-dimensional Maps

In this paper, we consider a biologically-inspired Boolean function P D that models a simple task of detecting global spatial patterns on a twodimensional map. We prove that P D is computable by a threshold circuit of size (i.e., number of gates) O( √ n logn), which is an improvement on the previous upper bound O(n), while our circuit has larger depth O( √ n) and total wire length O(n log n). Moreover, we demonstrate that the size of our circuit is nearly optimal up to a logarithmic factor: we show that any threshold circuit computing P D has size Ω( √ n/ logn). Submitted: March 2015 Reviewed: July 2015 Revised: July 2015 Accepted: September 2015 Final: January 2016 Published: February 2016 Article type: regular paper Communicated by: M. S. Rahman and E. Tomita The early version of this paper is presented at the 9th International Workshop on Algorithms and Computation (WALCOM 2015). E-mail addresses: [email protected] (Kei Uchizawa) [email protected] (Daiki Yashima) [email protected] (Xiao Zhou) 116 Uchizawa et al. Threshold Circuits Detecting Global Patterns