07 dc3
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This document discusses algorithms for matrix multiplication. It first introduces the Master Theorem for solving recurrence relations. It then analyzes the complexity of standard matrix multiplication as O(n^3) using recursive calls and the Master Theorem. Next, it presents Strassen's algorithm for matrix multiplication using 7 recursive calls rather than 8, improving the complexity to O(n^2.8). Finally, it analyzes Strassen's algorithm using the Master Theorem.
07 dc3
- 1. Divide and Conquer Analysis of Algorithms
- 2. Master Theorem The solution to the equation T(n) = aT(n/b) + θ(nk ), where a >= 1 and b >1 is Ө(nlog b a ) if a > bk T(n) = Ө(nk logn) if a = bk Ө(nk ) if a < bk
- 3. Example Solve the following recurrence using Master Theorem T(n) = 8T(n/2) + O(n2 ) a = 8, b = 2, k = 2 ⇒a > bk , 8 > 22 )()()( 38lg2 nOnOnT ==⇒
- 4. Matrix Multiplication procedure stmul(n,A,B,C) if n>=1 partition A into A11,A12,A21,A22 partition B into B11,B12,B21,B22 c=AxB stmul(n/2,A11,B11,m1) stmul(n/2,A12,B21,m2) stmul(n/2,A11,B12,m3) stmul(n/2,A12,B22,m4) stmul(n/2,A21,B11,m5) stmul(n/2,A22,B21,m6) stmul(n/2,A21,B12,m7) stmul(n/2,A22,B22,m8)
- 5. c11=m1+m2, c12=m5+m6 c21=m3+m4, c22=m7+m8 Function Calls m1=A11B11, m2=A12B21, m3=A11B12 m4=A12B22, m5=A21B11, m6=A22B21 m7=A21B12, m8=A22B22
- 6. Analysis, Multiplications T(n)=8T(n/2) T(n/2)=8T(n/4) T(n/4)=8T(n/8) By backward substitution T(n)=82 T(n/4) =83 T(n/8) =8i T(n/2i ) =8 i T(n/2i ) let 2i =n T(n)=8lgn =nlg8 =n^3
- 7. Additions t(n)=8t(n/2)+4(n/2)2 a=8, b=2, k=2 by applying master theorem T(n)= n log b a = nlg8 =n3 єӨ(n3 )
- 8. Strassen’s Multiplication C11 C12 A11 A12 B11 B12 = * C21 C22 A21 A22 B21 B22 m1=A11(B12-B22) m2=(A11+A12)B22 m3=(A21+A22)B11 m4=A22(B21-B11) m5=(A11+A22)(B11+B22) m6=(A12-A22)(B21+B22) m7=(A11-A21)(B11+B12) C11=m5+m4-m2+m6, C21=m3+m4 C12=m1+m2 C22=m1+m5-m3-m7
- 9. Complexity(multiplication, addition& Subtraction) t(n)=7t(n/2)+kn 2 a=7,b=2,d=2 byapplyingmastertheorem T(n)= n log b a = nlg7 =n2.8 єӨ(n2.8 )
- 10. Complexity(multiplication, addition& Subtraction) t(n)=7t(n/2)+kn 2 a=7,b=2,d=2 byapplyingmastertheorem T(n)= n log b a = nlg7 =n2.8 єӨ(n2.8 )