Two lower bound arguments with "inaccessible" numbers

Abstract

The lirst result presented in this paper is a lower bound of Q(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" {X E Z" 1 x1 + . . . + ,Y, <xn} for inputs from (0, 1, 2, . . . . 2 o("-i"rn)}". 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" 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 a, in each test zz , <,<,, a,xi: a,, is bounded by f(n). The lower bounds on the depth of such LDTs that recognize these languages are 51(2 L"'*/'"'J) for KNAPSACK and 52(2LA'4J(")J) for the graph problems. The common new tool in the proofs of these lower bounds is the method of constructing "hard" instances (x,, . . . . x,) of the respective problem by building up the input numbers xi from "mutually inaccessible" numbers, i.e., numbers of different orders of magnitude.

On the use of inaccessible numbers and order indiscernibles in lower bound arguments for random access machines

“…We refer to Dietzfelbinger and Maass [6], [7] for further applications of "inaccessible" numbers in lower bound arguments.…”

Some lower and upper bounds for algebraic decision trees and the separation problem

“…In the case of inputs from R", there are two main types of arguments for finding lower bounds: (i) topological arguments: Ben-Or [2] and Steele and Yao [14] use some results in algebraic geometry on the number of connected components to find an n(n1ogn) lower bound for some decision problems; (ii) Dietzfelbinger and Maass [5] use a Ramsey type argument for finding a superpolynomial lower bound for restricted linear decision trees. A similar technique was used by Moran, Snir and Manber [ll] to find an n(n1ogn) lower bound for decision trees of general type.…”