Divide and Conquer: Polynomial Multiplication Version of October 1 / 7, 24201

Transcription

1 Divide and Conquer: Polynomial Multiplication Version of October 7, 2014 Divide and Conquer: Polynomial Multiplication Version of October 1 / 7, 24201

2 Outline Outline: Introduction The polynomial multiplication problem An O(n 2 ) brute force algorithm An O(n 2 ) first divide-and-conquer algorithm An improved divide-and-conquer algorithm Remarks Divide and Conquer: Polynomial Multiplication Version of October 2 / 7, 24201

3 Divide-and-Conquer Already saw some divide and conquer algorithms Divide Conquer Divide a given problem into a or more subproblems (ideally of approximately equal size n/b) Solve each subproblem (directly if small enough or recursively) a T (n/b) Combine Combine the solutions of the subproblems into a global solution f (n) Cost satisfies T(n) = at(n/b) + f(n). Divide and Conquer: Polynomial Multiplication Version of October 3 / 7, 24201

4 Divide and Conquer Examples Two major examples so far Maximum Contiguious Subarray Mergesort Both satisfied T (n) = 2T (n/2) + O(n) T (n) = O(n log n) Also saw Quicksort Divide and Conquer, but unequal size problems Divide and Conquer: Polynomial Multiplication Version of October 4 / 7, 24201

5 Divide and Conquer Analysis Main tool is the Master Theorem for soving recurrences of form where T (n) = at (n/b) + f (n) a 1 and b 1 are constants and f (n) is a (asymptotically) positive function. Note: Initial conditions T (1), T (2),... T (k) for some k. They don t contribute to asymptotic growth n/b could be either n/b or n/b Divide and Conquer: Polynomial Multiplication Version of October 5 / 7, 24201

6 The Master Theorem T(n) = at(n/b) + f(n), c = log b a If f (n) = Θ(n c ɛ ) for some ɛ > 0 then T (n) = θ(n c ) If T (n) = 4T (n/2) + n then T (n) = Θ(n 2 ) If T (n) = 3T (n/2) + n then T (n) = Θ(n log 2 3 ) = Θ(n ) If f (n) = Θ(n c ) then T (n) = Θ(n c log n) If T (n) = 2T (n/2) + n then T (n) = Θ(n log n) If f (n) = Θ(n c+ɛ ) for some ɛ > 0 and if af (n/b) df (n) for some d < 1 and large enough n then T (n) = O(f (n)) If T (n) = T (n/2) + n then T (n) = Θ(n) Divide and Conquer: Polynomial Multiplication Version of October 6 / 7, 24201

7 More Master Theorem There are many variations of the Master Theorem. Here s one... If T (n) = T (3n/4) + T (n/5) + n then T (n) = Θ(n) More generally, given constants α i > 0 with i α i < 1, if T (n) = n + k i=1 T (α in) then T (n) = Θ(n). Divide and Conquer: Polynomial Multiplication Version of October 7 / 7, 24201

8 Objective and Outline Outline: Introduction The polynomial multiplication problem An O(n 2 ) brute force algorithm An O(n 2 ) first divide-and-conquer algorithm An improved divide-and-conquer algorithm Remarks Divide and Conquer: Polynomial Multiplication Version of October 8 / 7, 24201

9 The Polynomial Multiplication Problem Definition (Polynomial Multiplication Problem) Given two polynomials Compute the product A(x)B(x) Example A(x) = a 0 + a 1 x + + a n x n B(x) = b 0 + b 1 x + + b m x m A(x) = 1 + 2x + 3x 2 B(x) = 3 + 2x + 2x 2 A(x)B(x) = 3 + 8x + 15x x 3 + 6x 4 Assume that the coefficients a i and b i are stored in arrays A[0... n] and B[0... m] Cost: number of scalar multiplications and additions Divide and Conquer: Polynomial Multiplication Version of October 9 / 7, 24201

10 What do we need to compute exactly? Define A(x) = n i=0 a ix i B(x) = m i=0 b ix i C(x) = A(x)B(x) = n+m k=0 c kx k Then c k = a i b j 0 i n, 0 j m, i+j=k for all 0 k m + n Definition The vector (c 0, c 1,..., c m+n ) is the convolution of the vectors (a 0, a 1,..., a n ) and (b 0, b 1,..., b m ) While polynomial multiplication is interesting, real goal is to calculate convolutions. Major subroutine in digital signal processing Divide and Conquer: Polynomial Multiplication Version of October 10 / 7, 24201

11 Objective and Outline Outline: Introduction The polynomial multiplication problem An O(n 2 ) brute force algorithm An O(n 2 ) first divide-and-conquer algorithm An improved divide-and-conquer algorithm Remarks Divide and Conquer: Polynomial Multiplication Version of October 11 / 7, 24201

12 Direct (Brute Force) Approach To ease analysis, assume n = m. A(x) = n i=0 a ix i and B(x) = n i=0 b ix i C(x) = A(x)B(x) = 2n k=0 c kx k with c k = a i b j, for all 0 k 2n 0 i,j n,i+j=k Direct approach: Compute all c k s using the formula above Total number of multiplications: Θ(n 2 ) Total number of additions: Θ(n 2 ) Complexity: Θ(n 2 ) Divide and Conquer: Polynomial Multiplication Version of October 12 / 7, 24201

13 Objective and Outline Outline: Introduction The polynomial multiplication problem An O(n 2 ) brute force algorithm An O(n 2 ) first divide-and-conquer algorithm An improved divide-and-conquer algorithm Remarks Divide and Conquer: Polynomial Multiplication Version of October 13 / 7, 24201

14 Divide-and-Conquer: Divide Assume n is a power of 2 Define A 0 (x) = a 0 + a 1 x + + a n 2 1 x n 2 1 A 1 (x) = a n n x + + a n x n 2 A(x) = A 0 (x) + A 1 (x)x n 2 Similarly, define B 0 (x) and B 1 (x) such that B(x) = B 0 (x) + B 1 (x)x n 2 A(x)B(x) = A 0 (x)b 0 (x)+a 0 (x)b 1 (x)x n 2 +A1 (x)b 0 (x)x n 2 +A1 (x)b 1 (x)x n The original problem (of size n) is divided into 4 problems of input size n/2 Divide and Conquer: Polynomial Multiplication Version of October 14 / 7, 24201

15 Example A(x) = 2 + 5x + 3x 2 + x 3 x 4 B(x) = 1 + 2x + 2x 2 + 3x 3 + 6x 4 A(x)B(x) = 2 + 9x + 17x x x x 5 +19x 6 + 3x 7 6x 8 A 0 (x) = 2 + 5x, A 1 (x) = 3 + x x 2, A(x) = A 0 (x) + A 1 (x)x 2 B 0 (x) = 1 + 2x, B 1 (x) = 2 + 3x + 6x 2, B(x) = B 0 (x) + B 1 (x)x 2 A 0 (x)b 0 (x) = 2 + 9x + 10x 2 A 1 (x)b 1 (x) = x + 19x 2 + 3x 3 6x 4 A 0 (x)b 1 (x) = x + 27x x 3 A 1 (x)b 0 (x) = 3 + 7x + x 2 2x 3 A 0 (x)b 1 (x) + A 1 (x)b 0 (x) = x + 28x x 3 A 0 (x)b 0 (x) + (A 0 (x)b 1 (x) + A 1 (x)b 0 (x))x 2 + A 1 (x)b 1 (x)x 4 = 2 + 9x + 17x x x x x 6 + 3x 7 6x 8 Divide and Conquer: Polynomial Multiplication Version of October 15 / 7, 24201

16 Divide-and-Conquer: Conquer Conquer: Solve the four subproblems compute A 0 (x)b 0 (x), A 0 (x)b 1 (x), A 1 (x)b 0 (x), A 1 (x)b 1 (x) by recursively calling the algorithm 4 times Combine adding the following four polynomials A 0 (x)b 0 (x)+a 0 (x)b 1 (x)x n 2 +A1 (x)b 0 (x)x n 2 +A1 (x)b 1 (x)x n takes O(n) operations (Why?) Divide and Conquer: Polynomial Multiplication Version of October 16 / 7, 24201

17 The First Divide-and-Conquer Algorithm PolyMulti1(A(x), B(x)) begin A 0 (x) = a 0 + a 1 x + + a n 2 1 x n 2 1 ; A 1 (x) = a n + a n x + + a n x n 2 ; B 0 (x) = b 0 + b 1 x + + b n 2 1 x n 2 1 ; B 1 (x) = b n + b n x + + b n x n 2 ; U(x) = PolyMulti1(A 0 (x), B 0 (x)); V (x) = PolyMulti1(A 0 (x), B 1 (x)); W (x) = PolyMulti1(A 1 (x), B 0 (x)); Z(x) = PolyMulti1(A 1 (x), B 1 (x)); return ( U(x) + [V (x) + W (x)]x n 2 + Z(x)x n) end Divide and Conquer: Polynomial Multiplication Version of October 17 / 7, 24201

18 Analysis of Running Time Assume that n is a power of 2 { 4T (n/2) + n, if n > 1, T (n) = 1, if n = 1. By the Master Theorem for recurrences T (n) = Θ(n 2 ). Same order as the brute force approach! No improvement! Divide and Conquer: Polynomial Multiplication Version of October 18 / 7, 24201

19 Objective and Outline Outline: Introduction The polynomial multiplication problem An O(n 2 ) brute force algorithm An O(n 2 ) first divide-and-conquer algorithm An improved divide-and-conquer algorithm Remarks Divide and Conquer: Polynomial Multiplication Version of October 19 / 7, 24201

20 Two Observations Observation 1: We said that we need the 4 terms: A 0 B 0, A 0 B 1, A 1 B 0, A 1 B. What we really need are the 3 terms: Observation 2: A 0 B 0, A 0 B 1 + A 1 B 0, A 1 B 1! The three terms can be obtained using only 3 multiplications: Y = (A 0 + A 1 )(B 0 + B 1 ) U = A 0 B 0 Z = A 1 B 1 We need U and Z and A 0 B 1 + A 1 B 0 = Y U Z Divide and Conquer: Polynomial Multiplication Version of October 20 / 7, 24201

21 The Second Divide-and-Conquer Algorithm PolyMulti2(A(x), B(x)) begin A 0 (x) = a 0 + a 1 x + + a n 2 1 x n 2 1 ; A 1 (x) = a n + a n x + + a n x n n 2 ; B 0 (x) = b 0 + b 1 x + + b n 2 1 x n 2 1 ; B 1 (x) = b n + b n x + + b n x n n 2 ; Y (x) = PolyMulti2(A 0 (x) + A 1 (x), B 0 (x) + B 1 (x)); U(x) = PolyMulti2(A 0 (x), B 0 (x)); Z(x) = PolyMulti2(A 1 (x), B 1 (x)); return ( ) U(x) + [Y (x) U(x) Z(x)]x n 2 + Z(x)x 2 n 2 end Divide and Conquer: Polynomial Multiplication Version of October 21 / 7, 24201

22 Running Time of the Modified Algorithm T (n) = { 3T (n/2) + n, if n > 1, 1, if n = 1. By the Master Theorem for recurrences T (n) = Θ(n log 2 3 ) = Θ(n ). Much better than previous Θ(n 2 ) algorithms! Divide and Conquer: Polynomial Multiplication Version of October 22 / 7, 24201

23 Objective and Outline Outline: Introduction The polynomial multiplication problem An O(n 2 ) brute force algorithm An O(n 2 ) first divide-and-conquer algorithm An improved divide-and-conquer algorithm Remarks Divide and Conquer: Polynomial Multiplication Version of October 23 / 7, 24201

24 Remarks This algorithm can also be used for (long) integer multiplication Really designed by Karatsuba (1960, 1962) for that purpose. Response to conjecture by Kolmogorov, founder of modern probability, that this would require Θ(n 2 ). Similar to technique deeloped by Strassen a few years later to multiply 2 n n matrices in O(n log 2 7 ) operations, instead of the Θ(n 3 ) that a straightforward algorithm would use. Takeaway from this lesson is that divide-and-conquer doesn t always give you faster algorithm. Sometimes, you need to be more clever. Coming up. An O(n log n) solution to the polynomial multiplication problem It involves strange recasting of the problem and solution using the Fast Fourier Transform algorithm as a subroutine The FFT is another classic D & C algorithm that we will learn soon. Divide and Conquer: Polynomial Multiplication Version of October 24 / 7, 24201