Lower bound arguments with “Inaccessible” numbers

Abstract

The first result presented in this paper is a lower bound of Ω(log n) for the computation time of concurrent-write parallel random access machines (PRAMS) with operation set, multiplication by constants that recognize the “threshold set” tx̄ϵ Z n|x 1+…+ x n - ̊ 1 ⩽x n∼ for inputs from 0, 1, 2,…, 2 O( n- log n )∼ n . The same bound holds for PRAMS with arbitrary binary operations, if the size of the input numbers is not restricted. The second lower bound regards languages in R n corresponding to KNAPSACK, MINIMUM PERFECT MATCHING, SHORTEST PATH, and TRAVELING SALESPERSON on linear decision trees (LDTs) with the restriction that the number of negative coefficients ai in each test Σ1⩽ i⩽ nα i x i : α 0 is bounded by f( n). The lower bounds on the depth of such LDTs that recognize these languages are Ω( 2⌊ 2f(n)⌋ ) for KNAPSACK and Ω(2 ⌊√n 4f(n) ⌋) for the graph problems. The common new tool in the proofs of these lower bounds is the method of constructing “hard” instances ( x 1, …, x n ) of the respective problem by building up the input numbers x 1 from “mutually inaccessible” numbers, i.e., numbers of different orders of magnitude.