Algorithms: Divide and Conquer
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1 Algorithms: Divide and Conquer Amotz Bar-Noy CUNY Spring 2012 Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
2 Integer Multiplication Input: integers I and J each represented by n bits. I + J and I J can be computed with O(n) bit operations. I J can be computed with O(n 2 ) bit operations using the traditional direct algorithm. Using divide-and-conquer, I J can be computed with O(n log 2 3 ) O(n ) bit operations. Using the Fast-Transform-Fourier, I J can be computed with O(n log n) bit operations. I + J, I J, and I J need at least Ω(n) bit operations. Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
3 Ideas for Divide and Conquer Assumption: The length n of I and J is a power of 2. Input representation: I = I h 2 n/2 + I l J = J h 2 n/2 + J l Claim: Multiplying m-bit number by 2 k can be done with Θ(k + m) bit operations (shifting). Objective: Compute the product I J using multiplications on I h, I l, J h, J l whose lengths are n/2 bits. Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
4 Divide and Conquer I I = I h 2 n/2 + I l J = J h 2 n/2 + J l I J = I h J h 2 n + (I h J l + I l J h )2 n/2 + I l J l Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
5 Divide and Conquer I I = I h 2 n/2 + I l J = J h 2 n/2 + J l I J = I h J h 2 n + (I h J l + I l J h )2 n/2 + I l J l T (n) = 4T (n/2) + Θ(n) bit operations. T (n) = Θ(n 2 ) bit operations. Result: Not an improvement! Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
6 Solving the Recursion T (n) = 4T (n/2) + Θ(n) Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
7 Solving the Recursion T (n) = 4T (n/2) + Θ(n) a = 4. b = 2. log b (a) = 2. d = 1. d < log b (a). Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
8 Solving the Recursion T (n) = 4T (n/2) + Θ(n) a = 4. b = 2. log b (a) = 2. d = 1. d < log b (a). Master Theorem Case 1: T (n) = Θ ( n 2). Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
9 Divide and Conquer II (I h I l )(J l J h ) = I h J l I h J h I l J l + I l J h I h J l + I l J h = (I h I l )(J l J h ) + I h J h + I l J l A = I h J h B = I l J l C = (I h I l )(J l J h ) I J = I h J h 2 n + (I h J l + I l J h )2 n/2 + I l J l I J = A2 n + (C + A + B)2 n/2 + B Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
10 Divide and Conquer II (I h I l )(J l J h ) = I h J l I h J h I l J l + I l J h I h J l + I l J h = (I h I l )(J l J h ) + I h J h + I l J l A = I h J h B = I l J l C = (I h I l )(J l J h ) I J = I h J h 2 n + (I h J l + I l J h )2 n/2 + I l J l I J = A2 n + (C + A + B)2 n/2 + B T (n) = 3T (n/2) + Θ(n) bit operations. T (n) = Θ ( n log 2 3) bit operations. Result: Better than Θ(n 2 )! Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
11 Solving the Recursion T (n) = 3T (n/2) + Θ(n) Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
12 Solving the Recursion T (n) = 3T (n/2) + Θ(n) a = 3. b = 2. log b (a) d = 1. d < log b (a). Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
13 Solving the Recursion T (n) = 3T (n/2) + Θ(n) a = 3. b = 2. log b (a) d = 1. d < log b (a). Master Theorem Case 1: T (n) = Θ ( n log 2 3). Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
14 Divide and Conquer I and II Divide and Conquer I: I J = I h J h 2 n + (I h J l + I l J h )2 n/2 + I l J l. T (n) = 4T (n/2) + Θ(n) = Θ ( n 2) bit operations. Divide and Conquer II: I J = A2 n + (C + A + B)2 n/2 + B. T (n) = 3T (n/2) + Θ(n) = Θ ( n log 2 3) bit operations. Result: Θ ( n log 2 3) = Θ ( n 1.585) is better than Θ(n 2 )! Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
15 Arbitrary n 1 Algorithm: 2 k 1 < n 2 k 2 k < 2n. Add 2 k n zeros in front of both integers. Run the algorithm with the new integers. Omit zeros at the beginning of the product. Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
16 Arbitrary n 1 Algorithm: 2 k 1 < n 2 k 2 k < 2n. Add 2 k n zeros in front of both integers. Run the algorithm with the new integers. Omit zeros at the beginning of the product. Complexity: (2 k ) log 2 3 < (2n) log 2 3 = 3n log 2 3 The algorithm complexity Θ((2 k ) log 2 3 ) is Θ(n log 2 3 ). Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
17 Matrix Multiplication Input: 2 matrices A and B of scalars of size n n. Output: An n n matrix C = A B. Definition: c i,j = n k=1 a i,k b k,j, for 1 i, j n. Operation: An addition (subtraction) or a multiplication between two scalars. Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
18 Matrix Multiplication Algorithms Direct algorithm: n 3 multiplications and n 2 (n 1) additions. Strassen algorithm: O(n log 2 7 ) O(n 2.81 ) operations. Best known solution: O(n ) operations. Lower bound: Ω(n 2 ) operations. Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
19 Multiplying 2 2 Matrices ( r s t u ) ( a b = c d ) ( e g f h ) Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
20 Multiplying 2 2 Matrices ( r s t u ) ( a b = c d ) ( e g f h ) The values of r, s, t, u: r = ae + bf s = ag + bh t = ce + df u = cg + dh Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
21 Multiplying 2 2 Matrices ( r s t u ) ( a b = c d ) ( e g f h ) The values of r, s, t, u: r = ae + bf s = ag + bh t = ce + df u = cg + dh Complexity: Total of 8 multiplications and 4 additions. Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
22 Divide-and-Conquer I Lemma: The multiplication procedure works for sub-matrices as well. Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
23 Divide-and-Conquer I Lemma: The multiplication procedure works for sub-matrices as well. Algorithm: (for n = 2 k ) Partition A and B into 4 sub-matrices of size n 2 n 2. Recursively compute the 8 sub-matrices multiplications. Do the 4 matrices additions each with (n/2) 2 addition operations. Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
24 Divide-and-Conquer I Lemma: The multiplication procedure works for sub-matrices as well. Algorithm: (for n = 2 k ) Partition A and B into 4 sub-matrices of size n 2 n 2. Recursively compute the 8 sub-matrices multiplications. Do the 4 matrices additions each with (n/2) 2 addition operations. Complexity: T (n) = 8T (n/2) + Θ(n 2 ) = Θ(n 3 ). Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
25 Solving the Recursion T (n) = 8T (n/2) + Θ(n 2 ) Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
26 Solving the Recursion T (n) = 8T (n/2) + Θ(n 2 ) a = 8. b = 2. log b (a) = 3. d = 2. d < log b (a). Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
27 Solving the Recursion T (n) = 8T (n/2) + Θ(n 2 ) a = 8. b = 2. log b (a) = 3. d = 2. d < log b (a). Master Theorem Case 1: T (n) = Θ ( n 3). Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
28 The Magic Idea With 7 multiplications and 10 additions, compute 7 help variables: p 1 = a(g h) p 2 = (a + b)h p 3 = (c + d)e p 4 = d(f e) p 5 = (a + d)(e + h) p 6 = (b d)(f + h) p 7 = (a c)(e + g) Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
29 The Magic Idea With 7 multiplications and 10 additions, compute 7 help variables: p 1 = a(g h) p 2 = (a + b)h p 3 = (c + d)e p 4 = d(f e) p 5 = (a + d)(e + h) p 6 = (b d)(f + h) p 7 = (a c)(e + g) With 8 more additions, compute r, s, t, u: r = p 5 + p 4 p 2 + p 6 s = p 1 + p 2 t = p 3 + p 4 u = p 5 + p 1 p 3 p 7 Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
30 The Magic Idea With 7 multiplications and 10 additions, compute 7 help variables: p 1 = a(g h) p 2 = (a + b)h p 3 = (c + d)e p 4 = d(f e) p 5 = (a + d)(e + h) p 6 = (b d)(f + h) p 7 = (a c)(e + g) With 8 more additions, compute r, s, t, u: r = p 5 + p 4 p 2 + p 6 s = p 1 + p 2 t = p 3 + p 4 u = p 5 + p 1 p 3 p 7 Complexity: Total of 7 multiplications and 18 additions. Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
31 Verifying the Value of r r = p 5 + p 4 p 2 + p 6 = (a + d)(e + h) + d(f e) (a + b)h + (b d)(f + h) = ae + ah + de + dh + df de ah bh + bf + bh df dh = ae + ah / + de / + dh / + df / de / ah / bh / + bf + bh / df / dh / = ae + bf Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
32 Verifying the Value of s s = p 1 + p 2 = a(g h) + (a + b)h = ag ah + ah + bh = ag ah / + ah / + bh = ag + bh Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
33 Verifying the Value of t t = p 3 + p 4 = (c + d)e + d(f e) = ce + de + df de = ce + de / + df de / = ce + df Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
34 Verifying the Value of u u = p 5 + p 1 p 3 p 7 = (a + d)(e + h) + a(g h) (c + d)e (a c)(e + g) = ae + ah + de + dh + ag ah ce de ae ag + ce + cg = ae / + ah / + de / + dh + ag/ ah / ce / de / ae / ag/ + ce / + cg = cg + dh Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
35 Divide-and-Conquer II Lemma: The magic idea works for sub-matrices as well. Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
36 Divide-and-Conquer II Lemma: The magic idea works for sub-matrices as well. Algorithm: (for n = 2 k ) Partition A and B into 4 sub-matrices of size n 2 n 2. Recursively compute the 7 sub-matrices multiplications. Do the 18 matrices additions each with (n/2) 2 addition operations. Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
37 Divide-and-Conquer II Lemma: The magic idea works for sub-matrices as well. Algorithm: (for n = 2 k ) Partition A and B into 4 sub-matrices of size n 2 n 2. Recursively compute the 7 sub-matrices multiplications. Do the 18 matrices additions each with (n/2) 2 addition operations. Complexity: T (n) = 7T (n/2) + Θ(n 2 ) = Θ(n log 2 7 ) Θ(n 2.81 ). Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
38 Solving the Recursion T (n) = 7T (n/2) + Θ(n 2 ) Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
39 Solving the Recursion T (n) = 7T (n/2) + Θ(n 2 ) a = 7. b = 2. log b (a) d = 2. d < log b (a). Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
40 Solving the Recursion T (n) = 7T (n/2) + Θ(n 2 ) a = 7. b = 2. log b (a) d = 2. d < log b (a). Master Theorem Case 1: T (n) = Θ ( n log 2 7). Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
41 Arbitrary n 1 Algorithm: 2 k 1 < n 2 k 2 k < 2n. Add 2 k n zero columns and rows to both A and B. Run the algorithm for the new matrices. Omit the zero columns and rows from C. Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
42 Arbitrary n 1 Algorithm: 2 k 1 < n 2 k 2 k < 2n. Add 2 k n zero columns and rows to both A and B. Run the algorithm for the new matrices. Omit the zero columns and rows from C. Complexity: (2 k ) log 2 7 < (2n) log 2 7 = 7n log 2 7. The algorithm complexity Θ((2 k ) log 2 7 ) is Θ(n log 2 7 ). Amotz Bar-Noy (CUNY) Divide and Conquer Spring / 22
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