The document discusses Strassen's algorithm for matrix multiplication. It begins by explaining traditional matrix multiplication that has a time complexity of O(n3). It then explains how the divide and conquer strategy can be applied by dividing the matrices into smaller square sub-matrices. Strassen improved upon this by reducing the number of multiplications from 8 to 7 terms, obtaining a time complexity of O(n2.81). His key insight was applying different equations on the sub-matrix multiplication formulas to minimize operations.
Directed Acyclic Graph Representation of basic blocks is the most important topic of compiler design.This will help for student studying in master degree in computer science.
Merge sort is a sorting technique based on divide and conquer technique. With worst-case time complexity being Ο(n log n), it is one of the most respected algorithms.
Merge sort first divides the array into equal halves and then combines them in a sorted manner.
Directed Acyclic Graph Representation of basic blocks is the most important topic of compiler design.This will help for student studying in master degree in computer science.
Merge sort is a sorting technique based on divide and conquer technique. With worst-case time complexity being Ο(n log n), it is one of the most respected algorithms.
Merge sort first divides the array into equal halves and then combines them in a sorted manner.
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펀디멘탈팀의 이재윤님이 자세한 리뷰 도와주셨습니다.
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Introduction to Matrix Chain Multiplication algorithm with an example. Matrix Chain Products algorithm comes under Dynamic Programming concept. Done for the course Advanced Data Structures and Algorithms.
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2022 ملزمة الرياضيات للصف السادس الاحيائي الفصل الثالث تطبيقات التفاضلanasKhalaf4
طبعة جديدة ومنقحة
حل تمارين الكتاب
شرح المواضيع الرياضية بالتفصيل وبأسلوب واضح ومفهوم لجميع المستويات
حلول الاسألة الوزارية
اعداد الدكتور أنس ذياب خلف
email: anasdhyiab@gmail.com
Python- Creating Dictionary,
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2. Strassen’s Matrix Multiplication
› Belongs to Divide and Conquer Paradigm
› How Matrix Multiplication done Normally
› How divide and Conquer strategy applied for matrix
multiplication
› What Strassen had done to improve matrix multiplication
using divide and conquer strategy
meghav@kannuruniv.ac.in 2
5. Divide and Conquer strategy for matrix
multiplication
A=
𝑎11 𝑎12
𝑎21 𝑎22
× B
𝑏11 𝑏12
𝑏21 𝑏22
2× 2 2× 2
= C =
𝑐11 𝑐12
𝑐21 𝑐22
2 × 2
› Multiplying a 2 × 2 matrix is a small problem
› 𝑐11 = 𝑎11 × 𝑏11+𝑎12 × 𝑏21
› 𝑐12 = 𝑎11 × 𝑏12+𝑎12 × 𝑏22
› 𝑐21 = 𝑎21 × 𝑏11+𝑎22 × 𝑏21
› 𝑐22 = 𝑎21 × 𝑏12+𝑎22 × 𝑏22
meghav@kannuruniv.ac.in 5
6. Divide and Conquer strategy for matrix
multiplication (contd.)
› If each addition term takes one unit of time, then total time is
4 unit, it’s a constant
› If there are 8 multiplications and take 8 unit of time, it is also
a constant.
› These 4 formulas take constant time.
› If we want to reduce the problem into 1x1 matrix
A = [a11], B = [b11]
A x B =C
C= [a11* b11]
meghav@kannuruniv.ac.in 6
7. Divide and Conquer strategy for matrix
multiplication (contd.)
› If the row/column size is grater than 2, we need to divide
the problem and solve each single problem
› We take the problem or assume the problem as power of
2.
› If the matrix is not power of 2 then we make it as power
of 2 by adding 0’s
meghav@kannuruniv.ac.in 7
8. Divide and Conquer strategy for matrix
multiplication (contd.)
A11 A12 B11 B12
A=
𝑎11 𝑎12 𝑎13 𝑎14
𝑎21 𝑎22 𝑎23 𝑎24
𝑎31 𝑎32 𝑎33 𝑎34
𝑎41 𝑎42 𝑎43 𝑎44
× B
𝑏11 𝑏12 𝑏13 𝑏14
𝑏21 𝑏22 𝑏23 𝑏24
𝑏31 𝑏32 𝑏33 𝑏34
𝑏41 𝑏42 𝑏43 𝑏44
A21 A22 4/2 × 4/2 B21 B22 4/2 × 4/2
A11 A12
= C =
𝑐11 𝑐12 𝑐13 𝑐14
𝑐21 𝑐22 𝑐23 𝑐24
𝑐31 𝑐32 𝑐33 𝑐34
𝑐41 𝑐42 𝑐43 𝑐44
A11 A12 4/2 × 4/2
Here A and B are divided into four 2x2 matrices
meghav@kannuruniv.ac.in 8
10. Divide and Conquer strategy for matrix
multiplication (contd.)
› Time Complexity
8 times recursive call is required.
T(n)=
1 n≤2
8𝑇
𝑛
2
+ n2 n>2
Time complexity is θ(n3)
› If we reduce the number of multiplication we can make the
algorithm faster
meghav@kannuruniv.ac.in 10
11. Strassen’s Matrix Multiplication
› Reduce 8 to 7 multiplication
› Different equations are applied on the four formulas
› No. of multiplication is reduced but addition and subtraction increased.
P= (A11+A22)(B11+B22)
Q=(A21+A22)
R= A11(B12-B22)
S=A22(B21-B11)
T=(A11+A12)B22
U=(A21–A11)(B11+B12)
V=(A12-A22)(B21+B22)
meghav@kannuruniv.ac.in
11