Presentation is loading. Please wait.

Presentation is loading. Please wait.

8/2/20151 Analysis of Algorithms Lecture: Solving recurrence by recursion-tree method.

Published byMelina Farmer Modified over 9 years ago

Similar presentations

Presentation on theme: "8/2/20151 Analysis of Algorithms Lecture: Solving recurrence by recursion-tree method."— Presentation transcript:

1 8/2/20151 Analysis of Algorithms Lecture: Solving recurrence by recursion-tree method

2 8/2/20152 Problem of the day How many multiplications do you need to compute 3 16 ? 3 16 =3 8 x 3 8 3 8 =3 4 x 3 4 3 4 =3 2 x 3 2 3 2 =3 x 3 3 16 =3 x 3 x 3 …. x 3 Answer: 15 Answer: 4

3 8/2/20153 Outline Review of last lecture – analyzing recursive algorithms –Defining recurrence relation Today: analyzing recursive algorithms –Solving recurrence relation

4 8/2/20154 Solving recurrence 1.Recursion tree or iteration method - Good for guessing an answer 2.Substitution method - Generic method, rigid, but may be hard 3.Master method - Easy to learn, useful in limited cases only - Some tricks may help in other cases

5 8/2/20155 Recurrence for merge sort T(n) =  (1) if n = 1; 2T(n/2) +  (n) if n > 1. We will usually ignore the base case, assuming it is always a constant (but not 0).

6 8/2/20156 Recursion tree for merge sort Solve T(n) = 2T(n/2) + dn, where d > 0 is constant.

7 8/2/20157 Recursion tree for merge sort Solve T(n) = 2T(n/2) + dn, where d > 0 is constant. T(n)T(n)

8 8/2/20158 Recursion tree for merge sort Solve T(n) = 2T(n/2) + dn, where d > 0 is constant. T(n/2) dn

9 8/2/20159 Recursion tree for merge sort Solve T(n) = 2T(n/2) + dn, where d > 0 is constant. dn T(n/4) dn/2

10 8/2/201510 Recursion tree for merge sort Solve T(n) = 2T(n/2) + dn, where d > 0 is constant. dn dn/4 dn/2  (1) …

11 8/2/201511 Recursion tree for merge sort Solve T(n) = 2T(n/2) + dn, where d > 0 is constant. dn dn/4 dn/2  (1) … h = log n

12 8/2/201512 Recursion tree for merge sort Solve T(n) = 2T(n/2) + dn, where d > 0 is constant. dn dn/4 dn/2  (1) … h = log n dn

13 8/2/201513 Recursion tree for merge sort Solve T(n) = 2T(n/2) + dn, where d > 0 is constant. dn dn/4 dn/2  (1) … h = log n dn

14 8/2/201514 Recursion tree for merge sort Solve T(n) = 2T(n/2) + dn, where d > 0 is constant. dn dn/4 dn/2  (1) … h = log n dn …

15 8/2/201515 Recursion tree for merge sort Solve T(n) = 2T(n/2) + dn, where d > 0 is constant. dn dn/4 dn/2  (1) … h = log n dn #leaves = n (n)(n) …

16 8/2/201516 Recursion tree for merge sort Solve T(n) = 2T(n/2) + dn, where d > 0 is constant. dn dn/4 dn/2  (1) … h = log n dn #leaves = n (n)(n) Total  (n log n) … Later we will usually ignore d

17 8/2/201517 Recurrence for computing power int pow (b, n) m = n >> 1; p=pow(b,m)*pow(b,m); if (n % 2) return p * b; else return p; int pow (b, n) m = n >> 1; p = pow (b, m); p = p * p; if (n % 2) return p * b; else return p; T(n) = T(n/2)+  (1)T(n) = 2T(n/2)+  (1) Which algorithm is more efficient asymptotically?

18 8/2/201518 Time complexity for Alg1 Solve T(n) = T(n/2) + 1 T(n) = T(n/2) + 1 = T(n/4) + 1 + 1 = T(n/8) + 1 + 1 + 1 = T(1) + 1 + 1 + … + 1 = Θ (log(n)) log(n) Iteration method

19 8/2/201519 Time complexity for Alg2 Solve T(n) = 2T(n/2) + 1.

20 8/2/201520 Time complexity for Alg2 Solve T(n) = 2T(n/2) + 1. T(n)T(n)

21 8/2/201521 Time complexity for Alg2 T(n/2) 1 Solve T(n) = 2T(n/2) + 1.

22 8/2/201522 Time complexity for Alg2 1 T(n/4) 1 1 Solve T(n) = 2T(n/2) + 1.

23 8/2/201523 Time complexity for Alg2 1 11 11 1 1  (1) … Solve T(n) = 2T(n/2) + 1.

24 8/2/201524 Time complexity for Alg2 1 11 11 1 1  (1) … h = log n Solve T(n) = 2T(n/2) + 1.

25 8/2/201525 Time complexity for Alg2 1 11 11 1 1  (1) … h = log n 1 Solve T(n) = 2T(n/2) + 1.

26 8/2/201526 Time complexity for Alg2 1 11 11 1 1  (1) … h = log n 1 2 Solve T(n) = 2T(n/2) + 1.

27 8/2/201527 Time complexity for Alg2 1 11 11 1 1  (1) … h = log n 1 2 4 … Solve T(n) = 2T(n/2) + 1.

28 8/2/201528 Time complexity for Alg2 1 11 11 1 1  (1) … h = log n 1 2 4 #leaves = n (n)(n) … Solve T(n) = 2T(n/2) + 1.

29 8/2/201529 Time complexity for Alg2 1 11 11 1 1  (1) … h = log n 1 2 4 #leaves = n (n)(n) … Solve T(n) = 2T(n/2) + 1. Total  (n)

30 8/2/201530 More iteration method examples T(n) = T(n-1) + 1 = T(n-2) + 1 + 1 = T(n-3) + 1 + 1 + 1 = T(1) + 1 + 1 + … + 1 = Θ (n) n - 1

31 8/2/201531 More iteration method examples T(n) = T(n-1) + n = T(n-2) + (n-1) + n = T(n-3) + (n-2) + (n-1) + n = T(1) + 2 + 3 + … + n = Θ (n 2 )

32 8/2/201532 3-way-merge-sort 3-way-merge-sort (A[1..n]) If (n <= 1) return; 3-way-merge-sort(A[1..n/3]); 3-way-merge-sort(A[n/3+1..2n/3]); 3-way-merge-sort(A[2n/3+1.. n]); Merge A[1..n/3] and A[n/3+1..2n/3]; Merge A[1..2n/3] and A[2n/3+1..n]; Is this algorithm correct? What’s the recurrence function for the running time? What does the recurrence function solve to?

33 8/2/201533 Unbalanced-merge-sort ub-merge-sort (A[1..n]) if (n<=1) return; ub-merge-sort(A[1..n/3]); ub-merge-sort(A[n/3+1.. n]); Merge A[1.. n/3] and A[n/3+1..n]. Is this algorithm correct? What’s the recurrence function for the running time? What does the recurrence function solve to?

34 8/2/201534 More recursion tree examples T(n) = 3T(n/3) + n T(n) = T(n/3) + T(2n/3) + n T(n) = 3T(n/4) + n T(n) = 3T(n/4) + n 2

Download ppt "8/2/20151 Analysis of Algorithms Lecture: Solving recurrence by recursion-tree method."

Similar presentations

5/5/20151 Analysis of Algorithms Lecture 6&7: Master theorem and substitution method.

5/5/20151 Analysis of Algorithms Lecture 6&7: Master theorem and substitution method.
Comp 122, Spring 2004 Divide and Conquer (Merge Sort)

Comp 122, Spring 2004 Divide and Conquer (Merge Sort)
1 Divide & Conquer Algorithms. 2 Recursion Review A function that calls itself either directly or indirectly through another function Recursive solutions.

1 Divide & Conquer Algorithms. 2 Recursion Review A function that calls itself either directly or indirectly through another function Recursive solutions.
September 12, 20021 Algorithms and Data Structures Lecture III Simonas Šaltenis Nykredit Center for Database Research Aalborg University

September 12, Algorithms and Data Structures Lecture III Simonas Šaltenis Nykredit Center for Database Research Aalborg University
Divide-and-Conquer Recursive in structure –Divide the problem into several smaller sub-problems that are similar to the original but smaller in size –Conquer.

Divide-and-Conquer Recursive in structure –Divide the problem into several smaller sub-problems that are similar to the original but smaller in size –Conquer.
11 Computer Algorithms Lecture 6 Recurrence Ch. 4 (till Master Theorem) Some of these slides are courtesy of D. Plaisted et al, UNC and M. Nicolescu, UNR.

11 Computer Algorithms Lecture 6 Recurrence Ch. 4 (till Master Theorem) Some of these slides are courtesy of D. Plaisted et al, UNC and M. Nicolescu, UNR.
Algorithms Recurrences. Definition – a recurrence is an equation or inequality that describes a function in terms of its value on smaller inputs Example.

Algorithms Recurrences. Definition – a recurrence is an equation or inequality that describes a function in terms of its value on smaller inputs Example.
CS421 - Course Information Website Syllabus Schedule The Book:

CS421 - Course Information Website Syllabus Schedule The Book:
Divide-and-Conquer1 7 2  9 4  2 4 7 9 7  2  2 79  4  4 9 7  72  29  94  4.

Divide-and-Conquer1 7 2  9 4   2  2 79  4   72  29  94  4.
COMP2230 Tutorial 2. Mathematical Induction A tool for proving the truth of a statement for some integer “n”. “n” is usually a +ve integer from 1 to infinity.

COMP2230 Tutorial 2. Mathematical Induction A tool for proving the truth of a statement for some integer “n”. “n” is usually a +ve integer from 1 to infinity.
Data Structures, Spring 2004 © L. Joskowicz 1 Data Structures – LECTURE 3 Recurrence equations Formulating recurrence equations Solving recurrence equations.

Data Structures, Spring 2004 © L. Joskowicz 1 Data Structures – LECTURE 3 Recurrence equations Formulating recurrence equations Solving recurrence equations.
Lecture 2: Divide and Conquer I: Merge-Sort and Master Theorem Shang-Hua Teng.

Lecture 2: Divide and Conquer I: Merge-Sort and Master Theorem Shang-Hua Teng.
CS2420: Lecture 10 Vladimir Kulyukin Computer Science Department Utah State University.

CS2420: Lecture 10 Vladimir Kulyukin Computer Science Department Utah State University.
Data Structures, Spring 2006 © L. Joskowicz 1 Data Structures – LECTURE 3 Recurrence equations Formulating recurrence equations Solving recurrence equations.

Data Structures, Spring 2006 © L. Joskowicz 1 Data Structures – LECTURE 3 Recurrence equations Formulating recurrence equations Solving recurrence equations.
CS3381 Des & Anal of Alg (2001-2002 SemA) City Univ of HK / Dept of CS / Helena Wong 4. Recurrences - 1 Recurrences.

CS3381 Des & Anal of Alg ( SemA) City Univ of HK / Dept of CS / Helena Wong 4. Recurrences - 1 Recurrences.
Updates HW#1 has been delayed until next MONDAY. There were two errors in the assignment. 2-1. Merge sort runs in Θ(n log n). Insertion sort runs in Θ(n2).

Updates HW#1 has been delayed until next MONDAY. There were two errors in the assignment Merge sort runs in Θ(n log n). Insertion sort runs in Θ(n2).
4.Recurrences Hsu, Lih-Hsing. Computer Theory Lab. Chapter 4P.2 Recurrences -- Substitution method Recursion-tree method Master method.

4.Recurrences Hsu, Lih-Hsing. Computer Theory Lab. Chapter 4P.2 Recurrences -- Substitution method Recursion-tree method Master method.
CSC 2300 Data Structures & Algorithms January 30, 2007 Chapter 2. Algorithm Analysis.

CSC 2300 Data Structures & Algorithms January 30, 2007 Chapter 2. Algorithm Analysis.
CS Main Questions Given that the computer is the Great Symbol Manipulator, there are three main questions in the field of computer science: What kinds.

CS Main Questions Given that the computer is the Great Symbol Manipulator, there are three main questions in the field of computer science: What kinds.
Recurrences Part 3. Recursive Algorithms Recurrences are useful for analyzing recursive algorithms Recurrence – an equation or inequality that describes.

Recurrences Part 3. Recursive Algorithms Recurrences are useful for analyzing recursive algorithms Recurrence – an equation or inequality that describes.

Similar presentations

Ads by Google