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CSC 282 – Algorithms

CSC 282 – Algorithms. Daniel Stefankovic – CSB 620 stefanko@cs.rochester.edu TA: Girts Folkmanis – CSB 614 gfolkman@cs.rochester.edu www.cs.rochester.edu/~stefanko/Teaching/06CS282. Grading formula. 25% - homework 30% - quizzes 25% - midterm (Tu, Oct. 24)

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CSC 282 – Algorithms

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  1. CSC 282 – Algorithms Daniel Stefankovic – CSB 620 stefanko@cs.rochester.edu TA: Girts Folkmanis – CSB 614 gfolkman@cs.rochester.edu www.cs.rochester.edu/~stefanko/Teaching/06CS282

  2. Grading formula 25% - homework 30% - quizzes 25% - midterm (Tu, Oct. 24) 30% - final (Th, Dec. 21)

  3. 25% - homework • turn in before class on the due date • no late homework accepted • two lowest homework scores dropped 30% - quizzes • 1 each week • 10 min • closed book • no make-up quizzes • two lowest quiz grades dropped

  4. What is an algorithm? algorithm = problem-solving procedure Algoritmi de numero Indorum (Al-Khwarizmi Concerning the Hindu Art of Reckoning) CORRECTNESS EFFICIENCY

  5. Problem: is n a prime? PRIMALITY: INSTANCE: a natural number n QUESTION: is n a prime? Is 12345678987654321 a prime?

  6. Problem: is n a prime? Algorithm 1: 1 for k from 2 to n-1 do 2 if k divides n then RETURN “composite” 3 RETURN “prime” Is 12345678987654321 a prime?

  7. Problem: is n a prime? Algorithm 1: 1 for k from 2 to n-1 do 2 if k divides n then RETURN “composite” 3 RETURN “prime” Algorithm 2: 1 for k from 2 to √n do 2 if k divides n then RETURN “composite” 3 RETURN “prime”

  8. Problem: is n a prime? Algorithm 2: 1 for k from 2 to √n do 2 if k divides n then RETURN “composite” 3 RETURN “prime” CORRECT?

  9. Problem: is n a prime? RSA cryptosystem needs primes with 1024-4096 bits. Running time of our algorithms: Algorithm 1: 21024 – 24096 Algorithm 2:

  10. Problem: is n a prime? RSA cryptosystem needs primes with 1024-4096 bits. Running time of our algorithms: Algorithm 1: 21024 – 24096 Algorithm 2: 2512 – 22048 NOT EFFICIENT

  11. What means efficient? “efficient program using other efficient as subroutines is efficient” running time is bounded by a polynomial in the input size

  12. Input size How many bits needed to represent n?

  13. Input size How many bits needed to represent n? log n Algorithm 2: 1 for k from 2 to √n do 2 if k divides n then RETURN “composite” 3 RETURN “prime” √n  polynomial(log n) ???

  14. What means efficient? running time is bounded by a polynomial in the input size More refined classification asymptotic notation

  15. Asymptotic notation DEF: Let f,g: N  R+. We say f(n) = O(g(n)) if ( C) ( n) f(n)  C . g(n)

  16. Asymptotic notation DEF: Let f,g: N  R+. We say f(n) = O(g(n)) if ( C) ( n) f(n)  C . g(n) 1 n2 + n3 = O(n4) 2 n2 / log(n) = O(n . log n) 3 5n + log(n) = O(n) 4 nlog n = O(n100) 5 3n = O(2n . n100)

  17. Asymptotic notation 1 n2 + n3 = O(n4) 2 n2 / log(n)  O(n . log n) 3 5n + log(n) = O(n) 4 nlog n O(n100) 5 3n O(2n . n100)

  18. Asymptotic notation 1 n! = O(3n) 2 n +1 = O(n) 3 2n+1 = O(2n) 4 (n+1)! = O(n!) 5 1+c+c2+…+cn = O(cn) 6 1+c+c2+…+cn = O(1) n!= n.(n-1).(n-2) … 3.2.1

  19. Asymptotic notation 1 n!  O(3n) 2 n+1 = O(n) 3 2n+1 = O(2n) 4 (n+1)!  O(n!) 5 1+c+c2+…+cn = O(cn) for c>1 6 1+c+c2+…+cn = O(1) for c<1

  20. Asymptotic notation DEF: Let f,g: N  R+. We say f(n) = (g(n)) if f(n)=O(g(n)) and g(n)=O(f(n))

  21. What means efficient? polynomial-time = running time is bounded by a polynomial in the input size, i.e., ( k) T(n) = O(nk) More refined analysis = asymptotics for the running time (as a function of input-size) ideally we would like f(n) such that T(n) = (f(n))

  22. An applied problem INSTANCE: n points in the plane SOLUTION: a tour connecting the points MEASURE: the length of the tour

  23. An applied problem INSTANCE: n points in the plane SOLUTION: a tour connecting the points MEASURE: the length of the tour

  24. An applied problem INSTANCE: n points in the plane SOLUTION: a tour connecting the points MEASURE: the length of the tour

  25. An efficient algorithm?

  26. A correct algorithm • best  (1 2 3 … n) • for each permutation  • if cost()<cost(best) then • best EFFICIENCY?

  27. Sorting SORTING: INSTANCE: a sequence of n numbers a1,a2, … , an SOLUTION: reordering bi of the input such that b1 b2 …  bn

  28. Insertion sort i  1 while i  n do j  i while j  2 and a[ j-1] > a[ j ] do swap a[ j ],a[ j-1] j  j - 1 i  i + 1

  29. Insertion sort – correctness? i  1 while i  n do j  i while j  2 and a[ j-1] > a[ j ] do swap a[ j ],a[ j-1] j  j - 1 i  i + 1

  30. Insertion sort – running time?

  31. Insertion sort – running time? The worst-case running time of insertion sort is (n2).

  32. Merge 2 sorted lists MERGE INSTANCE: 2 lists xi, yi such that x1  x2  …  xn y1  y2  …  ym SOLUTION: ordered merge

  33. Merge 2 sorted lists 1 i  1, j  1 2 while i  n and j  n do 3 if xi  yj then 4 output xi, i  i + 15 else 6 output yj, j  j + 1 7 output remaining elements

  34. Mergesort MERGE-SORT(a,l,r) if l < r then m  (l+r)/2  MERGE-SORT(a,I,m) MERGE-SORT(a,m+1,r) MERGE(a,l,m,r)

  35. Mergesort Running time?

  36. Mergesort Running time? [ ... n … ] [ … n/2 … ] [ … n/2 … ] [ … n/4 … ] [ … n/4 … ] [ … n/4 … ] [ … n/4 … ] Depth ?

  37. Mergesort Running time? [ ... n … ] [ … n/2 … ] [ … n/2 … ] [ … n/4 … ] [ … n/4 … ] [ … n/4 … ] [ … n/4 … ] Depth = log n

  38. Mergesort Time spent on merge? [ ... n … ] [ … n/2 … ] [ … n/2 … ] [ … n/4 … ] [ … n/4 … ] [ … n/4 … ] [ … n/4 … ] Depth = log n

  39. Mergesort [ ... n … ] [ … n/2 … ] [ … n/2 … ] [ … n/4 … ] [ … n/4 … ] [ … n/4 … ] [ … n/4 … ] Time spent on merge? O(n) O(n) O(n) Depth = log n O(n.logn)

  40. Mergesort recurrence T(n)= T(n/2) + (n) if n>1 T(1)= (1)

  41. RAM model r0 r1 Program r2 r3 r4 r5 . • Each register holds an integer • Operations: • simple arithmetic • if-then, goto, etc. . . memory

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