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Grade School Revisited: How To Multiply Two Numbers

2 X 2 = 5. Grade School Revisited: How To Multiply Two Numbers. The best way is often far from obvious!. (a+bi)(c+di). Gauss. Gauss’ Complex Puzzle. Remember how to multiply two complex numbers a + bi and c + di? (a+bi)(c+di) = [ac –bd] + [ad + bc] i Input: a,b,c,d

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Grade School Revisited: How To Multiply Two Numbers

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  1. 2 X 2 = 5 Grade School Revisited:How To Multiply Two Numbers

  2. The best way isoften far from obvious!

  3. (a+bi)(c+di) Gauss

  4. Gauss’ Complex Puzzle • Remember how to multiply two complex numbers a + bi and c + di? • (a+bi)(c+di) = [ac –bd] + [ad + bc] i • Input: a,b,c,d • Output: ac-bd, ad+bc • If multiplying two real numbers costs $1 and adding them costs a penny, what is the cheapest way to obtain the output from the input? Can you do better than $4.02?

  5. Gauss’ $3.05 Method (a+bi)(c+di) = [ac –bd] + [ad + bc] i Input: a,b,c,d Output: ac-bd, ad+bc • X1 = a + b • X2 = c + d • X3 = X1 X2 = ac + ad + bc + bd • X4 = ac • X5 = bd • X6 = X4 – X5 = ac - bd • X7 = X3 – X4 – X5 = bc + ad c c $ $ $ c cc

  6. The Gauss optimization saves one multiplication out of four. It requires 25% less work.

  7. * * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * * ** * ** * + We saw that T(n) was linear. Time complexity of grade school addition T(n) = The amount of time grade school addition uses to add two n-bit numbers T(n) = Θ(n)

  8. X * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * n2 * * * * * * * * We saw that T(n) was quadratic. * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Time complexity of grade school multiplication T(n) = The amount of time grade school multiplication uses to add two n-bit numbers T(n) = Θ(n2)

  9. Grade School Addition: Linear timeGrade School Multiplication: Quadratic time • No matter how dramatic the difference in the constants the quadratic curve will eventually dominate the linear curve time # of bits in numbers

  10. Grade school addition is linear time. Is there a sub-linear time method for addition?

  11. Any addition algorithm takes Ω(n) time Claim: Any algorithm for addition must read all of the input bits Proof: Suppose there is a mystery algorithm A that does not examine each bit Give A a pair of numbers. There must be some unexamined bit position i in one of the numbers

  12. Any addition algorithm takes Ω(n) time • If A is not correct on the inputs, we found a bug • If A is correct, flip the bit at position i and give A the new pair of numbers. A gives the same answer as before, which is now wrong. * * * * * * * * * A did not read this bit at position i * * * * * * * * * * * * * * * * * * *

  13. So any algorithm for addition must use time at least linear in the size of the numbers. Grade school addition can’t be improved upon by more than a constant factor.

  14. Grade School Addition: Θ(n) timeFurthermore, it is optimalGrade School Multiplication: Θ(n2) time Is there a clever algorithm to multiply two numbers in linear time?

  15. Despite years of research, no one knows! If you resolve this question, Carnegie Mellon will give you a PhD!

  16. Can we even break the quadratic time barrier? In other words, can we do something very different than grade school multiplication?

  17. Grade School Multiplication:The Kissing Intuition Intuition: Let’s say that each time an algorithm has to multiply a digit from one number with a digit from the other number, we call that a “kiss”. It seems as if any correct algorithm must kiss at least n2 times.

  18. Divide And Conquer • An approach to faster algorithms: • DIVIDE a problem into smaller subproblems • CONQUER them recursively • GLUE the answers together so as to obtain the answer to the larger problem

  19. Multiplication of 2 n-bit numbers n bits • X = • Y = X a b Y c d n/2 bits n/2 bits X = a 2n/2 + b Y = c 2n/2 + d X × Y = ac 2n + (ad + bc) 2n/2 + bd

  20. Multiplication of 2 n-bit numbers a b • X = • Y = c d n/2 bits n/2 bits X × Y = ac 2n + (ad + bc) 2n/2 + bd MULT(X,Y): If |X| = |Y| = 1 then return XY break X into a;b and Y into c;d return MULT(a,c) 2n + (MULT(a,d) + MULT(b,c)) 2n/2 + MULT(b,d)

  21. Same thing for numbers in decimal! n digits • X = • Y = a b c d n/2 digits n/2 digits X = a 10n/2 + b Y = c 10n/2 + d X × Y = ac 10n + (ad + bc) 10n/2 + bd

  22. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • *4276 5678*2139 5678*4276 • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  23. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 1234*2139 1234*4276 5678*2139 5678*4276 • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  24. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 1234*2139 1234*42765678*2139 5678*4276 • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  25. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 1234*2139 1234*4276 5678*21395678*4276 • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  26. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 1234*2139 1234*4276 5678*2139 5678*4276 • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  27. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 1234*2139 1234*4276 5678*2139 5678*4276 • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  28. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 1234*2139 1234*4276 5678*2139 5678*4276 • 12*2112*39 34*21 34*39 xxxxxxxxxxxxxxxxxxxxxxxxx • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  29. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 1234*2139 1234*4276 5678*2139 5678*4276 • 12*21 12*39 34*21 34*39 xxxxxxxxxxxxxxxxxxxxxxxxx • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  30. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 1234*2139 1234*4276 5678*2139 5678*4276 • 12*21 12*39 34*21 34*39 xxxxxxxxxxxxxxxxxxxxxxxxx • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  31. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 1234*2139 1234*4276 5678*2139 5678*4276 • 12*21 12*39 34*21 34*39 xxxxxxxxxxxxxxxxxxxxxxxxx • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  32. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 1234*2139 1234*4276 5678*2139 5678*4276 • 12*21 12*39 34*21 34*39 xxxxxxxxxxxxxxxxxxxxxxxxx • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  33. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 1234*2139 1234*4276 5678*2139 5678*4276 • 12*21 12*39 34*21 34*39 xxxxxxxxxxxxxxxxxxxxxxxxx • 1*2 1*1 2*2 2*1xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  34. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 1234*2139 1234*4276 5678*2139 5678*4276 • 12*2112*39 34*21 34*39 xxxxxxxxxxxxxxxxxxxxxxxxx • 1*2 1*1 2*2 2*1 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx • Hence: 12*21 = 2*102 + (1 + 4)101 + 2 = 252 2 1 4 2 • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  35. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 1234*2139 1234*4276 5678*2139 5678*4276 • 25212*39 34*21 34*39 xxxxxxxxxxxxxxxxxxxxxxxxx • 1*2 1*1 2*2 2*1 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx • Hence: 12*21 = 2*102 + (1 + 4)101 + 2 = 252 2 1 4 2 • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  36. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 1234*2139 1234*4276 5678*2139 5678*4276 • 25212*39 34*21 34*39 xxxxxxxxxxxxxxxxxxxxxxxxx • 1*2 1*1 2*2 2*1 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx • Hence: 12*21 = 2*102 + (1 + 4)101 + 2 = 252 • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  37. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 1234*2139 1234*4276 5678*2139 5678*4276 • 252 12*39 34*21 34*39 xxxxxxxxxxxxxxxxxxxxxxxxx • xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 1*3 1*9 2*3 2*9 • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  38. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 1234*2139 1234*4276 5678*2139 5678*4276 • 242 12*39 34*21 34*39 xxxxxxxxxxxxxxxxxxxxxxxxx • xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 3 9 6 18 *102 + *101 + *101 + *1 • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  39. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 1234*2139 1234*4276 5678*2139 5678*4276 • 242 468 34*21 34*39 xxxxxxxxxxxxxxxxxxxxxxxxx • xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 3 9 6 18 *102 + *101 + *101 + *1 • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  40. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 1234*2139 1234*4276 5678*2139 5678*4276 • 12*21 12*39 34*21 34*39 xxxxxxxxxxxxxxxxxxxxxxxxx 252 468 714 1326 • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  41. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 1234*2139 1234*4276 5678*2139 5678*4276 • 12*21 12*39 34*21 34*39 xxxxxxxxxxxxxxxxxxxxxxxxx 252 468 714 1326 *104 + *102 + *102 + *1 • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  42. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 1234*2139 1234*4276 5678*2139 5678*4276 • 12*21 12*39 34*21 34*39 xxxxxxxxxxxxxxxxxxxxxxxxx • = 2639526 252 468 714 1326 *104 + *102 + *102 + *1 • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  43. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 2639526 1234*4276 5678*2139 5678*4276 • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  44. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 2639526 5276584 12145242 24279128 • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  45. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 2639526 5276584 12145242 24279128 *108 + *104 + *104 + *1 • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  46. a b c d Multiplying (Divide & Conquer style) • 12345678 * 21394276 • 2639526 5276584 12145242 24279128 *108 + *104 + *104 + *1 = 264126842539128 • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  47. a b c d Multiplying (Divide & Conquer style) 264126842539128 • X = • Y = • X × Y = ac 10n + (ad + bc) 10n/2 + bd

  48. Divide, Conquer, and Glue • MULT(X,Y)

  49. Divide, Conquer, and Glue if |X| = |Y| = 1 then return XY, else… • MULT(X,Y):

  50. Divide, Conquer, and Glue X=a;b Y=c;d • MULT(X,Y):

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