Algorithms and computational complexity.ppt

Algorithms and computational complexity.ppt

• 1.
  1 CSE 417: Algorithmsand Computational Complexity
• 2.
  2 Master Divide andConquer Recurrence If T(n)=aT(n/b)+cnk for n>b then if a>bk then T(n) is if a<bk then T(n) is Q(nk) if a=bk then T(n) is Q(nk log n) Works even if it is n/b instead of n/b. ) ( log a b n Q
• 3.
  3 Multiplying Matrices n3 multiplications,n3-n2 additions                          44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11 44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11 b b b b b b b b b b b b b b b b a a a a a a a a a a a a a a a a                                                  44 44 34 43 24 42 14 41 42 44 32 43 22 42 12 41 41 44 31 43 21 42 11 41 44 34 34 33 24 32 14 31 42 34 32 33 22 32 12 31 41 34 31 33 21 32 11 31 44 24 34 23 24 22 14 21 42 24 32 23 22 22 12 21 41 24 31 23 21 22 11 21 44 14 34 13 24 12 14 11 42 14 32 13 22 12 12 11 41 14 31 13 21 12 11 11 b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a    
• 4.
  4 Multiplying Matrices                          44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11 44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11 b b b b b b b b b b b b b b b b a a a a a a a a a a a a a a a a                                                  44 44 34 43 24 42 14 41 42 44 32 43 22 42 12 41 41 44 31 43 21 42 11 41 44 34 34 33 24 32 14 31 42 34 32 33 22 32 12 31 41 34 31 33 21 32 11 31 44 24 34 23 24 22 14 21 42 24 32 23 22 22 12 21 41 24 31 23 21 22 11 21 44 14 34 13 24 12 14 11 42 14 32 13 22 12 12 11 41 14 31 13 21 12 11 11 b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a    
• 5.
  5 Multiplying Matrices                          44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11 44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11 b b b b b b b b b b b b b b b b a a a a a a a a a a a a a a a a                                                  44 44 34 43 24 42 14 41 42 44 32 43 22 42 12 41 41 44 31 43 21 42 11 41 44 34 34 33 24 32 14 31 42 34 32 33 22 32 12 31 41 34 31 33 21 32 11 31 44 24 34 23 24 22 14 21 42 24 32 23 22 22 12 21 41 24 31 23 21 22 11 21 44 14 34 13 24 12 14 11 42 14 32 13 22 12 12 11 41 14 31 13 21 12 11 11 b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a    
• 6.
  6 Multiplying Matrices                          44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11 44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11 b b b b b b b b b b b b b b b b a a a a a a a a a a a a a a a a                                                  44 44 34 43 24 42 14 41 42 44 32 43 22 42 12 41 41 44 31 43 21 42 11 41 44 34 34 33 24 32 14 31 42 34 32 33 22 32 12 31 41 34 31 33 21 32 11 31 44 24 34 23 24 22 14 21 42 24 32 23 22 22 12 21 41 24 31 23 21 22 11 21 44 14 34 13 24 12 14 11 42 14 32 13 22 12 12 11 41 14 31 13 21 12 11 11 b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a     A11 A12 A21 A11B12+A12B22 A22 A11B11+A12B21 B11B12 B21 B22 A21B12+A22B22 A21B11+A22B21
• 7.
  7 Multiplying Matrices T(n)=8T(n/2)+4(n/2)2=8T(n/2)+n2 8>22 soT(n) is A11 A12 A21 A11B12+A12B22 A22 A11B11+A12B21 B11 B12 B21 B22 A21B12+A22B22 A21B11+A22B21 = ) ( ) ( ) ( 3 log log n n n 8 a 2 b Q  Q  Q
• 8.
  8 Strassen’s algorithm Strassen’s algorithm Multiply2x2 matrices using 7 instead of 8 multiplications (and lots more than 4 additions) T(n)=7 T(n/2)+cn2 7>22 so T(n) is Q(n ) which is O(n2.81) Fastest algorithms theoretically use O(n2.376) time not practical but Strassen’s is practical provided calculations are exact and we stop recursion when matrix has size about 100 (maybe 10) log27
• 9.
  9 The algorithm P1=A12(B11+B21) P2=A21(B12+B22) P3=(A11- A12)B11 P4=(A22 - A21)B22 P5=(A22 - A12)(B21 - B22) P6=(A11 - A21)(B12 - B11) P7= (A21 - A12)(B11+B22) C11=P1+P3 C12=P2+P3+P6 - P7 C21= P1+P4+P5+P7 C22=P2+P4
• 10.
  10 Proving Master recurrence T(n)=aT(n/b)+cnk a n Problemsize n/b n/b2 b 1 # probs a2 a 1 ad T(1)=c
• 11.
  11 Proving Master recurrence T(n)=aT(n/b)+cnk a n Problemsize n/b n/b2 b 1 # probs a2 a 1 ad T(1)=c
• 12.
  12 Proving Master recurrence T(n)=aT(n/b)+cnk a n Problemsize n/b n/b2 b 1 # probs a2 a 1 ad cost cnk T(1)=c cank/bk ca2nk/b2k =cnk(a/bk)2 cnk(a/bk)d =cad
• 13.
  13 Total Cost Geometric series ratioa/bk d+1=logbn +1 terms first term cnk, last term cad If a/bk=1, all terms are equal T(n) is Q(nk log n) If a/bk<1, first term is largest T(n) is Q(nk) If a/bk>1, last term is largest T(n) is Q(ad)=Q(a ) =Q(n ) (To see this take logb of both sides) logbn logba
• 14.
  14 Quicksort Quicksort(X,left,right) if left <right split=Partition(X,left,right) Quicksort(X,left,split-1) Quicksort(X,split+1,right)
• 15.
  15 Partition - twofinger algorithm Partition(X,left,right) choose a random element to be a pivot and pull it out of the array, say at left end maintain two fingers starting at each end of the array slide them towards each other until you get a pair of elements where right finger has a smaller element and left finger has a bigger one (when compared to pivot) swap them and repeat until fingers meet put the pivot element where they meet
• 16.
  16 Partition - twofinger algorithm Partition(X,left,right) swap X[left],X[random(left,right)] pivot  X[left]; L  left; R right while L<R do while (X[L]  pivot & L  right) do L  L+1 while (X[R] > pivot & R  left) do R  R-1 if L>R then swap X[L],X[R] swap X[left],X[R] return R
• 17.
  17 In practice often choosepivot in fixed way as middle element for small arrays median of 1st, middle, and last for larger arrays median of 3 medians of 3 (9 elements in all) for largest arrays four finger algorithm is better also maintain two groups at each end of elements equal to the pivot • swap them all into middle at the end of Partition equal elements are bad cases for two fingers
• 18.
  18 Quicksort Analysis Partition doesn-1 comparisons on a list of length n pivot is compared to each other element If pivot is ith largest then two subproblems are of size i-1 and n-i Pivot is equally likely to be any one of 1st through nth largest           n 1 i i) T(n 1) T(i n 1 1 n T(n)
• 19.
  19 Quicksort analysis   2) 1)(n (n 2n 1 n T(n) 2 n 1) T(n 2n 2)T(n) (n 1) 1)T(n (n 2n T(n) 2 nT(n) - 1) 1)T(n (n T(n) 2 ... T(2) 2 T(1) 2 1)n (n 1) 1)T(n (n 1) - T(n 2 ... T(2) 2 T(1) 2 1) - n(n nT(n) n 1) - T(n 2 ... T(2) 2 T(1) 2 1 - n i) T(n 1) T(i n 1 1 n T(n) n 1 i                                            
• 20.
  20 Quicksort analysis n n 1.38 T(n) dx) 1/x n that (Recall n 38 1 n 2 2H n 1 3 1 2 1 2(1 Q(n) 1 n 2 Q(n) 1) Q(n 1 n T(n) Q(n) Let 2 n 2 n log ln ln 1 log . ) ...                    