1. CS 253: Algorithms Chapter 4 Divide-and-Conquer Recurrences Master Theorem Credit: Dr. George Bebis

2. Recurrences and Running Time Recurrences arise when an algorithm contains recursive calls to itself Running time is represented by an equation or inequality that describes a function in terms of its value on smaller inputs. T(n) = T(n-1) + n What is the actual running time of the algorithm? i.e. T(n) = ? Need to solve the recurrence ◦ Find an explicit formula of the expression ◦ Bound the recurrence by an expression that involves n

3. Example Recurrences T(n) = T(n-1) + nΘ(n 2 ) Recursive algorithm that loops through the input to eliminate one item T(n) = T(n/2) + cΘ(lgn) Recursive algorithm that halves the input in one step T(n) = T(n/2) + nΘ(n) Recursive algorithm that halves the input but must examine every item in the input T(n) = 2T(n/2) + 1Θ(n) Recursive algorithm that splits the input into 2 halves and does a constant amount of other work

4. BINARY-SEARCH Finds if x is in the sorted array A[lo…hi] Alg.: BINARY-SEARCH (A, lo, hi, x) if (lo > hi) return FALSE mid   (lo+hi)/2  if x = A[mid] return TRUE if ( x < A[mid] ) BINARY-SEARCH (A, lo, mid-1, x) if ( x > A[mid] ) BINARY-SEARCH (A, mid+1, hi, x) 12111097532 12345678 mid lohi

5. Example 1 A[8] = {1, 2, 3, 4, 5, 7, 9, 11} lo = 1 hi = 8 x = 7 mid = 4, lo = 5, hi = 8 mid = 6, A[mid] = x Found! 119754321 9754321 12345678 8 7 6 5

6. A[8] = {1, 2, 3, 4, 5, 7, 9, 11} lo = 1 hi = 8 x = 6 lo = 1, hi = 8, mid = 4. A[4]=4lo = 5, hi = 8, mid = 6, A[6]=7 119754321 9754321 12345678 9754321 lo = 5, hi = 5, mid = 5, A[5]=5 119754321 low high low high lo = 6, hi = 5  NOT FOUND! Example 1I

7. Analysis of BINARY-SEARCH Alg.: BINARY-SEARCH (A, lo, hi, x) if ( lo > hi ) return FALSE mid   (lo+hi)/2  if x = A[mid] return TRUE if ( x < A[mid] ) BINARY-SEARCH ( A, lo, mid-1, x ) if ( x > A[mid] ) BINARY-SEARCH ( A, mid+1, hi, x ) T(n) = c + T(n/2) constant time: c 2 same problem of size n/2 constant time: c 1 constant time: c 3

8. 8 Methods for Solving Recurrences Iteration method Recursion-tree method Master method

9. The Iteration Method Convert the recurrence into a summation and solve it using a known series Example: T(n) = c + T(n/2) T(n) = c + T(n/2) = c + c + T(n/4) = c + c + c + T(n/8) = c + c + c + c + T(n/2 4 ) Assume n=2 k then k = lg n and T(n) = c + c + c + c + c + … + T(n/2 k ) (k times) T(n) = k * c + T(1) T(n) = clg n

10. Iteration Method – Example 2 T(n) = n + 2T(n/2) Assume n=2 k  k = lg n T(n) = n + 2T(n/2) = n + 2(n/2 + 2T(n/4)) = n + n + 4T(n/4) = n + n + 4(n/4 + 2T(n/8)) = n + n + n + 8T(n/8) T(n) = 3n + 2 3 T(n/2 3 ) = kn + 2 k T(n/2 k ) = nlgn + nT(1) T(n) = O(nlgn)

11. 11 Methods for Solving Recurrences Iteration method Recursion-tree method Master method

12. The recursion-tree method Convert the recurrence into a tree: ◦ Each node represents the cost incurred at various levels of recursion ◦ Sum up the costs of all levels Used to “guess” a solution for the recurrence

13. Example 1 W(n) = 2W(n/2) + n 2 Subproblem size at level i = n/2 i At level i: Cost of each node = ( n/2 i ) 2 # of nodes = 2 i Total cost = ( n 2 /2 i ) h = Height of the tree  n/2 h =1  h = lgn Total cost at all levels:  W(n) = O(n 2 )

14. T(n) = 3T(n/4) + cn 2 Example 2 T(n) = 3T(n/4) + cn 2 Subproblem size at level i = n/4 i At level i: Cost of each node= c( n/4 i ) 2 # of nodes = 3 i Total cost =c n 2 (3/16) i h = Height of the tree  n/4 h =1  h = log 4 n Total cost at all levels: (last level has 3 log 4 n = n log 4 3 nodes)  T(n) = O(n 2 )

15. 15 Example 3 The longest path from the root to a leaf: n  (2/3)n  (2/3) 2 n  …  1 (2/3) i n =1  i = log 3/2 n Cost of the problem at level i = n Total cost: via further analysis  W(n) =Θ(nlgn) W(n) = W(n/3) + W(2n/3) + n

16. 16 Methods for Solving Recurrences Iteration method Recursion-tree method Master method

17. Master Theorem “Cookbook” for solving recurrences of the form: Idea: compare f(n) with n log b a f(n) is asymptotically smaller or larger than n log b a by a polynomial factor n  OR f(n) is asymptotically equal with n log b a

18. 18 Master Theorem Case 1: if f(n) = O(n log b a -  ) for some  > 0, then: T(n) =  (n log b a ) Case 2: if f(n) =  ( n log b a ), then: T(n) =  ( n log b a lgn) Case 3: if f(n) =  ( n log b a +  ) for some  > 0, and if af(n/b) ≤ cf(n) for some c < 1 and all sufficiently large n, then: T(n) =  (f(n)) regularity condition

19. 19 Example 1 T(n) = 2T(n/2) + n a = 2, b = 2, log 2 2 = 1 Compare n log b a =n 1 with f(n) = n f(n) =  ( n log b a =n 1 )  Case 2  T(n) =  ( n log b a lgn) =  (nlgn) Case 1: if f(n) = O(n log b a -  ) for some  > 0, then: T(n) =  (n log b a ) Case 2: if f(n) =  ( n log b a ), then: T(n) =  ( n log b a lgn) Case 3: if f(n) =  ( n log b a +  ) for some  > 0, and if af(n/b) ≤ cf(n) for some c < 1 and all sufficiently large n, then: T(n) =  (f(n))

20. T(n) = 2T(n/2) + n 2 a = 2, b = 2, log 2 2 = 1 Compare n log 2 2 =n 1 with f(n) = n 2 f(n) =  (n log 2 2 +  )  Case 3 (* need to verify regularity cond.) a f(n/b) ≤ c f(n)  2 n 2 /4 ≤ c n 2  (½ ≤ c <1)  T(n) =  (f(n)) =  (n 2 ) Example 2 Case 1: if f(n) = O(n log b a -  ) for some  > 0, then: T(n) =  (n log b a ) Case 2: if f(n) =  ( n log b a ), then: T(n) =  ( n log b a lgn) Case 3: if f(n) =  ( n log b a +  ) for some  > 0, and if af(n/b) ≤ cf(n) for some c < 1 and all sufficiently large n, then: T(n) =  (f(n))

21. 21 T(n) = 2T(n/2) + √n a = 2, b = 2, log 2 2 = 1 Compare n with f(n) = n 1/2 f(n) = O(n 1-  )  Case 1  T(n) =  (n log b a ) =  (n) Example 3 Case 1: if f(n) = O(n log b a -  ) for some  > 0, then: T(n) =  (n log b a ) Case 2: if f(n) =  ( n log b a ), then: T(n) =  ( n log b a lgn) Case 3: if f(n) =  ( n log b a +  ) for some  > 0, and if af(n/b) ≤ cf(n) for some c < 1 and all sufficiently large n, then: T(n) =  (f(n))

22. T(n) = 3T(n/4) + nlgn a = 3, b = 4, log 4 3 = 0.793 Compare n 0.793 with f(n) = nlgn f(n) =  (n log 4 3+  )  Case 3 Check regularity condition: 3  (n/4)lg(n/4) ≤ (3/4)nlgn = c  f(n), (3/4 ≤ c < 1)  T(n) =  (nlgn) Example 4 Case 1: if f(n) = O(n log b a -  ) for some  > 0, then: T(n) =  (n log b a ) Case 2: if f(n) =  ( n log b a ), then: T(n) =  ( n log b a lgn) Case 3: if f(n) =  ( n log b a +  ) for some  > 0, and if af(n/b) ≤ cf(n) for some c < 1 and all sufficiently large n, then: T(n) =  (f(n))

23. T(n) = 2T(n/2) + nlgn a = 2, b = 2, log 2 2 = 1 Compare n with f(n) = nlgn Is this a candidate for Case 3 ? Case 3: if f(n) =  (n log b a +  ) for some  > 0 In other words, If nlgn ≥ n 1.n  for some  > 0 lgn is asymptotically less than n  for any  > 0 !! Therefore, this in not Case 3! (somewhere between Case 2 & 3) Example 5 Case 1: if f(n) = O(n log b a -  ) for some  > 0, then: T(n) =  (n log b a ) Case 2: if f(n) =  ( n log b a ), then: T(n) =  ( n log b a lgn) Case 3: if f(n) =  ( n log b a +  ) for some  > 0, and if af(n/b) ≤ cf(n) for some c < 1 and all sufficiently large n, then: T(n) =  (f(n))

24. T(n) = 2T(n/2) + nlgn Use Iteration Method to show that T(n) = nlg 2 n T(n)= nlgn + 2T(n/2) = nlgn + 2(n/2*lg(n/2) + 2T(n/4)) = nlgn + nlg(n/2) + 2 2 T(n/2 2 ) = …. Exercise:

25. 25 Changing variables Try to transform the recurrence to one that you have seen before ◦ Rename: m = lgn  n = 2 m T ( 2 m ) = 2T(2 m/2 ) + m ◦ Rename: k=m and S(k) = T(2 k ) S(k) = 2S(k/2) + k  S(k) =  (k*lgk) T(n) = T(2 m ) = S(m) =  (mlgm)=  (lgn*lglgn) **See Page 86 of the Textbook. T(n) = 2T( ) + lgn

26. 26 Use Iteration Method to show that T(n) =  (lgn*lglgn) T(n)= lgn + 2T(n 1/2 ) = lgn + 2(lg(n 1/2 ) + 2T(n 1/4 )) = lgn + lgn + 2 2 T(n 1/4 ) = lgn + lgn + … + 2 k T(2) k=lglgn T(n) = 2T( ) + lgn Exercise:

27. Appendix A Summations