![Divide-and-Conquer
Multiply two polynomials
Problem
Given two polynomials represented by two arrays, write a function that
multiplies given two polynomials.
Input: A[] = {10, 5, 11}
B[] = {3, 2, 1}
Output: prod C[] = 10 X + 5 x1 + 11 x2
3 X + 2 x1 + 1 x2
= 30 + 10 * 2x1 + 10 * 1x1 + 5x1 * 3+ 5x1 * 2x1 + 5x1 * 1x2 + 11x2 * 3 +
11x2 * 2x1 + 11x2 * 1x2
polynomials
Presentation by_Hasanain ALshadoodee
4](https://image.slidesharecdn.com/divide-and-conquermultiplytwopolynomials-211221191010/85/Divide-and-conquer-multiply-two-polynomials-4-320.jpg)
![Algoritham
1) Create a product array prod[] of size m+n-1.
2) Initialize all entries in prod[] as 0.
3) Traverse array A[] and do following for every element A[i]
Traverse array B[] and do following for every element B[j]
so prod[i+j] = prod[i+j] + A[i] * B[j]
4) Return prod[].
Presentation by_Hasanain ALshadoodee
6](https://image.slidesharecdn.com/divide-and-conquermultiplytwopolynomials-211221191010/85/Divide-and-conquer-multiply-two-polynomials-6-320.jpg)
Divide and-conquer multiply two polynomials
- 1. Divide-and-Conquer Multiply two polynomials Presentation by Hasanain ALshadoodee
- 3. Presentation by_Hasanain ALshadoodee 3 Introduction Multiplying Polynomials To multiply two polynomials together, multiply every term of one polynomial by every term of the other polynomial.
- 4. Divide-and-Conquer Multiply two polynomials Problem Given two polynomials represented by two arrays, write a function that multiplies given two polynomials. Input: A[] = {10, 5, 11} B[] = {3, 2, 1} Output: prod C[] = 10 X + 5 x1 + 11 x2 3 X + 2 x1 + 1 x2 = 30 + 10 * 2x1 + 10 * 1x1 + 5x1 * 3+ 5x1 * 2x1 + 5x1 * 1x2 + 11x2 * 3 + 11x2 * 2x1 + 11x2 * 1x2 polynomials Presentation by_Hasanain ALshadoodee 4
- 5. Naïve Method One by one consider every term of first polynomial and multiply it with every term of second polynomial this will take O(n2) time complexity . Presentation by_Hasanain ALshadoodee 5
- 6. Algoritham 1) Create a product array prod[] of size m+n-1. 2) Initialize all entries in prod[] as 0. 3) Traverse array A[] and do following for every element A[i] Traverse array B[] and do following for every element B[j] so prod[i+j] = prod[i+j] + A[i] * B[j] 4) Return prod[]. Presentation by_Hasanain ALshadoodee 6
- 7. Analysis of Naïve Method Time complexity of the above solution is O(mn). If size of two polynomials same, then time complexity is O(n2). There are method to do multiplication faster that O(n2) time. This is method are based on Divide and conquer technique . Presentation by_Hasanain ALshadoodee 7
- 8. Divide and Conquer Method Presentation by_Hasanain ALshadoodee 8 Let the two given polynomials be A and B. For simplicity, Let us assume that the given two polynomials are of same degree and have degree in powers of 2, i.e., n = 2i * polynomial 'A' can be written as A0 + A1* xn/2 * polynomial 'B' can be written as B0 + B1* xn/2
- 9. Presentation by_Hasanain ALshadoodee Example polynomial A=2 + 3x + 6x2 - 2x3 + 5x4 polynomial B=1 - 6x + 7x2 + 2x3 + 8x4 Polynomial A=(2 + 3x) + (6 - 2x + 5x2 ) x2 A0 A1 polynomial B=(1 - 6x) + (7 + 2x + 8x2 ) x2 B0 B1 A * B = (A0 + A1*xn/2) * (B0 + B1*xn/2) = A0*B0 + A0*B1*xn/2 + A1*B0*xn/2 + A1*B1*xn = A0*B0 + (A0*B1 + A1*B0)xn/2 + A1*B1*xn
- 10. Example IN the example we can see divide and conquer method requires 4 multiplication and O(n) time to add all 4 results . T(n)=4T(n/2) + O(n) The solution of recurrence is O(n2) . But it can be reduced . T(n)=3T(n/2) + O(n) Using stressing matrix multiplication . Presentation by_Hasanain ALshadoodee 10
- 11. Programming and Data Structures Presentation by_Hasanain ALshadoodee 11 Consider the following polynomials Each term of the polynomial 1 must be multiplied with each term of the polynomial Multiplying each term means multiplying their coefficients and adding their 1 2
- 12. Consider the following polynomials Presentation by_Hasanain ALshadoodee 12 Resultant polynomial
- 13. Consider the following polynomials Presentation by_Hasanain ALshadoodee 13 head1 head2 ptr2 ptr1 We need pointers (ptr1 and otr2) for traversal , so we also need a nested loop as each term of the first polynomial must be multiplied with every term of second polynomial.
- 14. Consider the following polynomials Presentation by_Hasanain ALshadoodee 14 head1 head2 ptr2 ptr1 While (ptr1 != NULL) While (ptr2 != NULL)
- 15. Consider the following polynomials Presentation by_Hasanain ALshadoodee 15 head1 head2 head3 Int res1 , res2 ; Struct node * head3=NULL; While (ptr1 != NULL) Ptr2=head2; While (ptr2 != NULL) ptr1=ptr1-> link;
- 16. Presentation by_Hasanain ALshadoodee 16 Consider the following polynomials Resultant polynomial We will get this polynomial of term executing the cod be causes of insert function
- 17. References Section 5.6 of the text book “algorithm design” by Jon Kleinberg and Eva Tardos. The original slides were prepared by Kevin Wayne. The slides are distributed by Pearson Addison-Wesley. Wireless Algorithms, Systems, and Applications: 9th International Conference . Crack GATE & ESE with Unacademy : Divide and Conquer: Multiply 2 Polynomials. Important Links • Submit your work here: https://jovian.ai/learn/data-structures-and-algorithms-in-python/assignment/assignment-3- sorting-and-divide-conquer-practice • Ask questions and get help: https://jovian.ai/forum/c/data-structures-and-algorithms-in-python/assignment-3/89 • Lesson 3 video for review: https://jovian.ai/learn/data-structures-and-algorithms-in-python/lesson/lesson-3-sorting- algorithms-and-divide-and-conquer • Lesson 3 notebook for review: https://jovian.ai/aakashns/python-sorting-divide-and-conquer • Algebra I #5.11, Multiply two Polynomials https://www.youtube.com/watch?v=VAELGq-FViY • Application of Linked List (Multiplication of Two Polynomials) https://www.youtube.com/hashtag/linkedlist Presentation by_Hasanain ALshadoodee 17