Divide-and-Conquer Multiply two polynomials

1. Divide-and-Conquer
Multiply two polynomials
Presentation by
Hasanain ALshadoodee

2. Multiply two polynomials

3. Presentation by_Hasanain ALshadoodee
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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
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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 .
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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[].
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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
.
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8. Divide and Conquer Method
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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 .
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11. Programming and Data Structures
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 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
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Resultant polynomial

13. Consider the following polynomials
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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
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head1
head2
ptr2
ptr1
While (ptr1 != NULL)
 While (ptr2 != NULL)




15. Consider the following polynomials
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head1
head2
head3
Int res1 , res2 ;
Struct node * head3=NULL;
While (ptr1 != NULL)
 Ptr2=head2;
While (ptr2 != NULL)


 ptr1=ptr1-> link;

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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
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