Abstract

Consider an arithmetic expression of lengthninvolving only the operations {+, ×} and non-negative constants. We prove lower bounds on the depth of any binary computation tree over the same sets of operations and constants that computes such an expression. We exhibit a family of arithmetic expressions that requires computation trees of depth at least 1.5 log₂ n−O(1), thus proving a conjecture of S. R. Kosaraju (1986,in“Proc. of the 18th ACM Symp. on Theory Computing,” pp. 231–239). In contrast, Kosaraju showed how to compute such expressions with computation trees of depth 2 log₂ n+O(1).