Polynomial Vicinity Circuits and Nonlinear Lower Bounds

We study families of Boolean circuits with the property that the number of gates at distance t fanning into or out of any given gate in a circuit is bounded above by a polynomial in t of some degree k. We prove that such circuits require size Ω(n1+1/k / log n) to compute several natural families of functions, including sorting, finite field arithmetic, and the “rigid linear transformations” of Valiant [26]. Our proof develops a “separator theorem” in the style of Lipton and Tarjan [14] for a new class of graphs, and our methods may have independent graph-theoretic interest.

• Type: Conference
• Year: 1997
• Where: COCO
• Authors: Kenneth W. Regan
• Keywords: Algorithms, Boolean Circuits, Circuits Require Size, COCO 1997, Rigid Linear Transformations