divide and conquer work function theorem

(rev. 2017/03/20)

The Work Function Theorem for Divide and Conquer Algorithms

Notation: ℤ is the set of integers. ℕ is the set of positive integers. ℝ is the set of real numbers. ℝ⁺ is the set of positive real numbers. Let T(n) be a work function for an algorithm A. In other words T is a function with domain ℕ and range ℝ⁺, and for each n∈ℕ, T(n) is the maximum number of "simple operations" required by algorithm A to solve a problem of size n or less. (The "or less" assures that T is non-decreasing.) Theorem: If T(n) ≤ qT(n/b) + Cn^h, ∀n∈ℕ, ∃q∈ℕ, ∃b∈ℕ, ∃C∈ℝ, ∃h∈ℝ, C≥0, h≥0, and T:ℕ→ℝ⁺ is non-decreasing, then ∃A∈ℝ, ∃B∈ℝ, A≥0, B≥0, such that case 1: if q<b^h, then T(n) ≤ An^log_b(q) + Bn^h and T(n) is O(n^h) case 2: if q=b^h, then T(n) ≤ An^hlog_b(n) + Bn^h and T(n) is O(n^hlog_b(n)) case 3: if q>b^h, then T(n) ≤ An^log_b(q) + Bn^h and T(n) is O(n^log_b(q)) c.f. page 135 of Brassard, Gilles, and Paul Bratley. Fundamentals of Algorithmics. Englewood Cliffs: Prentice Hall, 1996.