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CSE 421 Algorithms

Learn about recurrences and divide & conquer algorithms, including counting inversions, closest pair, multiplication, and FFT. Dive into the analysis and solution methods for recurrences. Explore the recursive matrix multiplication algorithm and Strassen's algorithm. Understand the BFPRT recurrence.

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CSE 421 Algorithms

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  1. CSE 421Algorithms Richard Anderson Lecture 11 Recurrences

  2. Divide and Conquer • Recurrences, Sections 5.1 and 5.2 • Algorithms • Counting Inversions (5.3) • Closest Pair (5.4) • Multiplication (5.5) • FFT (5.6)

  3. Divide and Conquer Array Mergesort(Array a){ n = a.Length; if (n <= 1) return a; b = Mergesort(a[0 .. n/2]); c = Mergesort(a[n/2+1 .. n-1]); return Merge(b, c); }

  4. Algorithm Analysis • Cost of Merge • Cost of Mergesort

  5. T(n) <= 2T(n/2) + cn; T(2) <= c;

  6. Recurrence Analysis • Solution methods • Unrolling recurrence • Guess and verify • Plugging in to a “Master Theorem”

  7. Unrolling the recurrence

  8. Substitution Prove T(n) <= cn log2n for n >= 2 Induction: Base Case: Induction Hypothesis: T(n/2) <= c(n/2) log2(n/2)

  9. A better mergesort (?) • Divide into 3 subarrays and recursively sort • Apply 3-way merge What is the recurrence?

  10. Unroll recurrence for T(n) = 3T(n/3) + dn

  11. T(n) = aT(n/b) + f(n)

  12. T(n) = T(n/2) + cn Where does this recurrence arise?

  13. Recurrences • Three basic behaviors • Dominated by initial case • Dominated by base case • All cases equal – we care about the depth

  14. Divide and Conquer

  15. Recurrence Examples • T(n) = 2 T(n/2) + cn • O(n log n) • T(n) = T(n/2) + cn • O(n) • More useful facts: • logkn = log2n / log2k • k log n = n log k

  16. Recursive Matrix Multiplication Multiply 2 x 2 Matrices: | r s | | a b| |e g| | t u| | c d| | f h| r = ae + bf s = ag + bh t = ce + df u = cg + dh A N x N matrix can be viewed as a 2 x 2 matrix with entries that are (N/2) x (N/2) matrices. The recursive matrix multiplication algorithm recursively multiplies the (N/2) x (N/2) matrices and combines them using the equations for multiplying 2 x 2 matrices =

  17. Recursive Matrix Multiplication • How many recursive calls are made at each level? • How much work in combining the results? • What is the recurrence?

  18. What is the run time for the recursive Matrix Multiplication Algorithm? • Recurrence:

  19. T(n) = 4T(n/2) + cn

  20. T(n) = 2T(n/2) + n2

  21. T(n) = 2T(n/2) + n1/2

  22. Recurrences • Three basic behaviors • Dominated by initial case • Dominated by base case • All cases equal – we care about the depth

  23. Solve by unrollingT(n) = n + 5T(n/2) Answer: nlog(5/2)

  24. What you really need to know about recurrences • Work per level changes geometrically with the level • Geometrically increasing (x > 1) • The bottom level wins • Geometrically decreasing (x < 1) • The top level wins • Balanced (x = 1) • Equal contribution

  25. Classify the following recurrences(Increasing, Decreasing, Balanced) • T(n) = n + 5T(n/8) • T(n) = n + 9T(n/8) • T(n) = n2 + 4T(n/2) • T(n) = n3 + 7T(n/2) • T(n) = n1/2 + 3T(n/4)

  26. Strassen’s Algorithm Multiply 2 x 2 Matrices: | r s | | a b| |e g| | t u| | c d| | f h| Where: p1 = (b + d)(f + g) p2= (c + d)e p3= a(g – h) p4= d(f – e) p5= (a – b)h p6= (c – d)(e + g) p7= (b – d)(f + h) = r = p1 + p4 – p5 + p7 s = p3 + p5 t = p2 + p5 u = p1 + p3 – p2 + p7

  27. Recurrence for Strassen’s Algorithms • T(n) = 7 T(n/2) + cn2 • What is the runtime?

  28. BFPRT Recurrence • T(n) <= T(3n/4) + T(n/5) + 20 n What bound do you expect?

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