Lecture 7: Divide & Conquer 2. Integer Multiplication. & Matrix Multiplication. CS 341: Algorithms. Tuesday, Jan 29 th 2019
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1 Lecture 7: Divide & Conquer 2 Integer Multiplication & Matrix Multiplication CS 341: Algorithms Tuesday, Jan 29 th
2 Outline For Today 1. Integer Multiplication 2. Matrix Multiplication 2
3 Outline For Today 1. Integer Multiplication 2. Matrix Multiplication 3
4 Integer Multiplication u Input: 2 n-digit integers, X, Y u Output Z XY u E.g: X 2345 and Y 6789 then Z u Will work in base 10 b/c it is easier to think about base 10 u Same argument & analysis as in base 2 (done in handout) u Warning: There is no intuition to the DC algorithm we ll see 4
5 Grade School Algorithm x
6 Grade School Algorithm x shift by 1 digit 6
7 Grade School Algorithm x shift by 1 digit shift by 2 digits 7
8 Grade School Algorithm + x shift by 1 digit shift by 2 digits shift by 3 digits u Multiplying two digits > 1 operation u Addition of two digits > 1 operation u Shift by 1 digit > 1 operation (shift by x digits, x ops) TOTAL WORK: O(n 2 ) Observe that even the total work for shifting is ( n-1) O(n 2 )! 8
9 Question: Can we do better? u Upshot: A set of mysterious looking multiplications, additions and subtractions giving the correct answer! u X (a10 n/2 + b); Y (c10 n/2 + d) a c b d each of a, b, c, d is of size n/2 u Ex: a23, b45, so *10 n/ u XY ac10 n + ad10 n/2 + bc10 n/2 + bd u XY ac10 n + (ad + bc)10 n/2 + bd Call this expression ( ) 9
10 DC-Multiplication-1 procedure DC-Mult1(X, Y both n digit numbers): Base Case: if (X or Y is single digit): set a, b, c, d defined as before O(n) ac DC-Mult(a, c) shifts: O(n) ad DC-Mult(a, d) 3 additions of n digit numbers: O(n) bc DC-Mult(b, c) bd DC-Mult(b, d) return ac10 n + (ad + bc)10 n/2 + bd Total Work Outside of Recursive Calls: O(n) Recurrence: 4T(n/2) + O(n) Total Runtime: O(n 2 ) Not better than Grade School Algorithm 10
11 Observation Observation: We care about only 3 quantities in ( ): ac10 n + (ad + bc)10 n/2 + bd (1) ac > with 10 n padding (2) (ad + bc) > with 10 n/2 padding (3) bd > with 10 padding Question: If we care about 3 quantities, can we get these quantities with only 3 recursive calls? 11
12 Karatsuba-Ofman Algorithm (1962) ac10 n + (ad + bc)10 n/2 + bd (1) ac as before (2) bd as before (3) (a + b) (c + d) (ac + ad + bc + bd) Observation: (3) (2) (1) (ac+ad+bc+bd)-ac bd ad + bc procedure KO(X, Y both n digit numbers): Base Case: if (X or Y is single digit): set a, b, c, d defined as before ac KO(a, c) bd KO(b, d) X KO(a+b, c+d) return (ac)10 n + (X-ac-bd)10 n/2 + bd 12
13 Karatsuba-Ofman Algorithm (1962) procedure KO(X, Y both n digit numbers): Base Case: if (X or Y is single digit): set a, b, c, d defined as before O(n) ac KO(a, c) bd KO(b, d) O(n) X KO(a+b, c+d) return (ac)10 n + (X ac - bd)10 n/2 + bd O(n) Total Work Outside of Recursive Calls: O(n) Recurrence: 3T(n/2) + O(n) Total Runtime: O(n log_2(3) n 1.59 ) 13
14 Facts About Multiplication Fact 1: (By Toom & Stephen Cook): Can generalize KO to divide X and Y into k pieces instead of 2 (Math gets very messy) and get O(n log_k(2k-1) ) > O(n 1+ε ) for any ε > 1 Stephen Cook is a Canadian Computer Scientist UToronto. Fact 2: Best known alg: O(nlog(n)loglog(n)) (Schonhage & Strassen) 14
15 Outline For Today 1. Integer Multiplication 2. Matrix Multiplication 15
16 Matrix Multiplication u Input: 2 n x n matrices A, B u Output: C AxB j a 11 a 12 a 1n b 11 b 12 b 1n c 11 c 12 c 1n i a 21 a 22 a 2n b 21 b 22 b 2n X c 21 c 22 c 2n c ij a n1 a n2 a nn n b n1 b n2 b nn c n1 c n2 c nn c ij a ik b kj (a i1 b 1 j + a i2 b 2 j a in b nj ) k1 Q: By definition, how many multip. and additions needed to compute c ij? A: n multiplications, n-1 additions Then there are O(n 3 ) basic operations (by definition) to compute C. u Warning: Again, no intuition to the DC algorithm we ll see! 16
17 Standard Algorithm a 11 a 12 a 1n a 21 a 22 a 2n a n1 a n2 a nn b 11 b 12 b 1n b 21 b 22 b 2n X b n1 b n2 b nn c 11 c 12 c 1n c 21 c 22 c 2n c n1 c n2 c nn
18 Standard Algorithm a 11 a 12 a 1n a 21 a 22 a 2n a n1 a n2 a nn b 11 b 12 b 1n b 21 b 22 b 2n X b n1 b n2 b nn c 11 c 12 c 1n c 21 c 22 c 2n c n1 c n2 c nn
19 Standard Algorithm a 11 a 12 a 1n a 21 a 22 a 2n a n1 a n2 a nn b 11 b 12 b 1n b 21 b 22 b 2n X b n1 b n2 b nn c 11 c 12 c 1n c 21 c 22 c 2n c n1 c n2 c nn
20 Standard Algorithm a 11 a 12 a 1n a 21 a 22 a 2n a n1 a n2 a nn b 11 b 12 b 1n b 21 b 22 b 2n X b n1 b n2 b nn c 11 c 12 c 1n c 21 c 22 c 2n c n1 c n2 c nn
21 Standard Algorithm a 11 a 12 a 1n a 21 a 22 a 2n a n1 a n2 a nn b 11 b 12 b 1n b 21 b 22 b 2n X b n1 b n2 b nn c 11 c 12 c 1n c 21 c 22 c 2n c n1 c n2 c nn
22 Standard Algorithm a 11 a 12 a 1n a 21 a 22 a 2n a n1 a n2 a nn b 11 b 12 b 1n b 21 b 22 b 2n X b n1 b n2 b nn c 11 c 12 c 1n c 21 c 22 c 2n c n1 c n2 c nn
23 Standard Algorithm a 11 a 12 a 1n a 21 a 22 a 2n a n1 a n2 a nn b 11 b 12 b 1n b 21 b 22 b 2n X b n1 b n2 b nn c 11 c 12 c 1n c 21 c 22 c 2n c n1 c n2 c nn
24 Standard Algorithm a 11 a 12 a 1n a 21 a 22 a 2n a n1 a n2 a nn b 11 b 12 b 1n b 21 b 22 b 2n X b n1 b n2 b nn c 11 c 12 c 1n c 21 c 22 c 2n c n1 c n2 c nn
25 Standard Algorithm a 11 a 12 a 1n a 21 a 22 a 2n a n1 a n2 a nn b 11 b 12 b 1n b 21 b 22 b 2n X b n1 b n2 b nn c 11 c 12 c 1n c 21 c 22 c 2n c n1 c n2 c nn procedure StandardMM(A, B nxn matrices): C int[n][n]; for i 1 n: for j 1 n: C[i][j] 0; for k1 n: C[i][j] + a ik b kj return C Runtime: O(n 3 )
26 Question: Can we Divide & Conquer? a 11 a 12 a 1n a 21 a 22 a 2n a n1 a n2 a nn b 11 b 12 b 1n b 21 b 22 b 2n X b n1 b n2 b nn c 11 c 12 c 1n c 21 c 22 c 2n c n1 c n2 c nn 26
27 Question: Can we Divide & Conquer? A 11 A 12 B 11 B 12 C 11 C 12 X A 21 A 22 B 21 B 22 C 21 C 22 u Where A 11,, A 22, B 11,, B 22, & C 11,, C 22 are n/2 x n/2 matrices. Fact: When you split matrices into blocks & multiply, the blocks behave as they are atomic elements. n C ij A ik B kj (A i1 B 1 j + A i2 B 2 j A in B nj ) k1 where + is matrix addition operation (coordinate-wise addition) 27
28 Matrix Addition a 11 a 12 a 1n b 11 b 12 b 1n c 11 c 12 c 1n a 21 a 22 a 2n a n1 a n2 a nn b 21 b 22 b 2n + b n1 b n2 b nn c 21 c 22 c 2n c n1 c n2 c nn 28
29 Matrix Addition a 11 a 12 a 1n b 11 b 12 b 1n c 11 c 12 c 1n a 21 a 22 a 2n a n1 a n2 a nn b 21 b 22 b 2n + b n1 b n2 b nn c 21 c 22 c 2n c n1 c n2 c nn 29
30 Matrix Addition a 11 a 12 a 1n b 11 b 12 b 1n c 11 c 12 c 1n a 21 a 22 a 2n a n1 a n2 a nn b 21 b 22 b 2n + b n1 b n2 b nn c 21 c 22 c 2n c n1 c n2 c nn O(n 2 ) operation 30
31 Example of Block Multiplication X *3 + 4*2 + 3*5 + 2*
32 Example of Block Multiplication X *2 + 4*4 + 3*2 + 2*
33 Example of Block Multiplication X *3 + 5*2 + 4*5 + 2*
34 Example of Block Multiplication X *2 + 5*4 + 4*2 + 2*
35 Example of Block Multiplication X A 11 B X A 12 B X
36 DC-1 Matrix Multiplication A 11 A 12 B 11 B 12 C 11 C 12 X A 21 A 22 B 21 B 22 C 21 C 22 procedure DC1-MM(A, B nxn matrices): C 11 DC1-MM(A 11, B 11 ) + DC1-MM(A 12, B 21 ) 36
37 DC-1 Matrix Multiplication A 11 A 12 B 11 B 12 C 11 C 12 X A 21 A 22 B 21 B 22 C 21 C 22 procedure DC1-MM(A, B nxn matrices): C 11 DC1-MM(A 11, B 11 ) + DC1-MM(A 12, B 21 ) C 12 DC1-MM(A 11, B 12 ) + DC1-MM(A 12, B 22 ) 37
38 DC-1 Matrix Multiplication A 11 A 12 B 11 B 12 C 11 C 12 X A 21 A 22 B 21 B 22 C 21 C 22 procedure DC1-MM(A, B nxn matrices): C 11 DC1-MM(A 11, B 11 ) + DC1-MM(A 12, B 21 ) C 21 DC1-MM(A 11, B 12 ) + DC1-MM(A 12, B 22 ) C 21 DC1-MM(A 21, B 11 ) + DC1-MM(A 22, B 21 ) 38
39 DC-1 Matrix Multiplication A 11 A 12 B 11 B 12 C 11 C 12 X A 21 A 22 B 21 B 22 C 21 C 22 procedure DC1-MM(A, B nxn matrices): C 11 DC1-MM(A 11, B 11 ) + DC1-MM(A 12, B 21 ) C 21 DC1-MM(A 11, B 12 ) + DC1-MM(A 12, B 22 ) C 12 DC1-MM(A 21, B 11 ) + DC1-MM(A 22, B 21 ) C 22 DC1-MM(A 21, B 12 ) + DC1-MM(A 22, B 22 ) return C C 11 C 21 C 12 C 22 Outside recursion: 4 additions of n/2xn/2 matrices4n 2 /4n 2 work Runtime: T(n) 8T(n/2) + O(n 2 ) O(n 3 ) (by Master Thm) So not faster than standard method! 39
40 Strassen s (Mysterious) Algorithm (1969) A 11 A 12 B 11 B 12 C 11 Z 5 + Z 4 -Z 2 + Z 6 C 12 Z 1 + Z 2 X A 21 A 22 B 21 B 22 C 21 Z 3 + Z 4 C 22 Z 5 + Z 1 - Z 3 Z 7 Z 1 A 11 (B 12 B 22 ) Z 2 (A 11 + A 12 )B 22 Z 3 (A 21 + A 22 )B 11 Z 4 A 22 (B 21 + B 11 ) Z 5 (A 11 + A 22 )(B 11 + B 22 ) Z 6 (A 12 - A 22 )(B 21 + B 22 ) Z 7 (A 11 A 21 )(B 11 + B 12 ) Exercise: Check the rest of the quadrants to believe Strassen! Ex: C 12 Z 1 + Z 2 A 11 B 12 A 11 B 22 + A 11 B 22 + A 12 B 22 A 11 B 12 + A 12 B 22 40
41 Strassen s (Mysterious) Algorithm A 11 A 12 B 11 B 12 C 11 Z 5 + Z 4 -Z 2 + Z 6 C 12 Z 1 + Z 2 X A 21 A 22 B 21 B 22 C 21 Z 3 + Z 4 C 22 Z 5 + Z 1 - Z 3 Z 7 procedure Strassen(A, B nxn matrices): Z 1 Strassen(A 11, (B 12 B 22 )); Z 2 Strassen((A 11 + A 12 ), B 22 ) Z 3 Strassen((A 21 + A 22 ), B 11 ); Z 4 Strassen(A 22, (B 21 + B 11 )) Z 5 Strassen((A 11 + A 22 ), (B 11 + B 22 )) Z 6 Strassen((A 12 - A 22 ), (B 21 + B 22 )) Z 7 Strassen((A 11 A 21 ), (B 11 + B 12 )) C 11 Z 5 + Z 4 -Z 2 + Z 6 ; C 12 Z 1 + Z 2 ; C 21 Z 3 + Z 4 ; C 22 Z 5 + Z 1 - Z 3 Z 7 return C C 11 C 21 C 12 C 22 Runtime: T(n) 7T(n/2) + O(n 2 ) O(n log_2(7) n 2.81 ) (by Master Thm) (Asymptotically) much faster than standard algorithm! 41
42 History of Matrix Multiplication Runtime Year Authors O(n 3 ) until 1969 N/A O(n 2.81 ) 1969 Strassen O(n ) 1978 Pan O(n ) 1979 Bini et. al. O(n ) 1981 Schoenhage O(n ) 1982 Romani O(n ) 1983 Coppersmith & Winograd O(n ) 1986 Strassen O(n ) 1989 Coppersmith and Winograd O(n ) 2011 Stothers O(n ) 2012 V. Williams O(n ) 2013 Le Gall 42
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