1 / 36

RAIK 283: Data Structures & Algorithms

RAIK 283: Data Structures & Algorithms. Recursive Algorithm Analysis Dr. Ying Lu ylu@cse.unl.edu. September 13, 2012. RAIK 283: Data Structures & Algorithms. Giving credit where credit is due: Most of the lecture notes are based on the slides from the Textbook’s companion website

mariah
Télécharger la présentation

RAIK 283: Data Structures & Algorithms

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. RAIK 283: Data Structures & Algorithms Recursive Algorithm Analysis Dr. Ying Lu ylu@cse.unl.edu September 13, 2012

  2. RAIK 283: Data Structures & Algorithms • Giving credit where credit is due: • Most of the lecture notes are based on the slides from the Textbook’s companion website http://www.aw-bc.com/info/levitin • Several slides are from Hsu Wen Jing of the National University of Singapore • I have modified them and added new slides

  3. Example: a recursive algorithm • Algorithm: if n=0 then F(n) := 1 else F(n) := F(n-1) * n return F(n)

  4. Example: another recursive algorithm Algorithm F(n) // Compute the nthFibonacci number recursively //Input: A nonnegative integer n //Output: the nthFibonacci number if n  1 return n else return F(n-1) + F(n-2)

  5. Recurrence Relation • Recurrence Relation

  6. Recurrence Relation • Recurrence Relation: an equation or inequality that describes a function in terms of its value on smaller inputs

  7. Example: a recursive algorithm • Algorithm: if n=0 then F(n) := 1 else F(n) := F(n-1) * n return F(n) • What does this algorithm compute?

  8. Example: a recursive algorithm • Algorithm: if n=0 then F(n) := 1 else F(n) := F(n-1) * n return F(n) • What does this algorithm compute? • What’s the basic operation of this algorithm?

  9. Example: recursive evaluation of n ! • Recursive definition of n!: • Algorithm: if n=0 then F(n) := 1 else F(n) := F(n-1) * n return F(n) • M(n): number of multiplications to compute n! with this recursive algorithm

  10. Example: recursive evaluation of n ! • Recursive definition of n!: • Algorithm: if n=0 then F(n) := 1 else F(n) := F(n-1) * n return F(n) • M(n): number of multiplications to compute n! with this recursive algorithm • Could we establish a recurrence relation for deriving M(n)?

  11. Example: recursive evaluation of n ! • Definition: n ! = 1*2*…*(n-1)*n • Recursive definition of n!: • Algorithm: if n=0 then F(n) := 1 else F(n) := F(n-1) * n return F(n) • M(n) = M(n-1) + 1 • Initial Condition: M(0) = ?

  12. Example: recursive evaluation of n ! • Recursive definition of n!: • Algorithm: if n=0 then F(n) := 1 else F(n) := F(n-1) * n return F(n) • M(n) = M(n-1) + 1 • Initial condition: M(0) = 0 • Explicit formula for M(n) in terms of n only?

  13. Time efficiency of recursive algorithms Steps in analysis of recursive algorithms: • Decide on parameter n indicating input size • Identify algorithm’s basic operation • Determine worst, average, and best case for inputs of size n • Set up a recurrence relation and initial condition(s) for C(n)-the number of times the basic operation will be executed for an input of size n • Solve the recurrence to obtain a closed form or determine the order of growth of the solution (see Appendix B)

  14. EXAMPLE: tower of hanoi • Problem: • Given three pegs (A, B, C) and n disks of different sizes • Initially, all the disks are on peg A in order of size, the largest on the bottom and the smallest on top • The goal is to move all the disks to peg C using peg B as an auxiliary • Only 1 disk can be moved at a time, and a larger disk cannot be placed on top of a smaller one B n disks A C

  15. EXAMPLE: tower of hanoi • Design a recursive algorithm to solve this problem: • Given three pegs (A, B, C) and n disks of different sizes • Initially, all the disks are on peg A in order of size, the largest on the bottom and the smallest on top • The goal is to move all the disks to peg C using peg B as an auxiliary • Only 1 disk can be moved at a time, and a larger disk cannot be placed on top of a smaller one B n disks A C

  16. EXAMPLE: tower of hanoi • Step 1: Solve simple case when n<=1? Just trivial B B A C A C Move(A, C)

  17. EXAMPLE: tower of hanoi • Step 2: Assume that a smaller instance can be solved, i.e. can move n-1 disks. Then? B B A C A C B A C

  18. EXAMPLE: tower of hanoi B B A C A C B A C B A C

  19. EXAMPLE: tower of hanoi B B A C A C B TOWER(n, A, B, C) A C B A C

  20. EXAMPLE: tower of hanoi B TOWER(n-1, A, C, B) B A C A C B Move(A, C) TOWER(n, A, B, C) A C TOWER(n-1, B, A, C) B A C

  21. EXAMPLE: tower of hanoi TOWER(n, A, B, C) { if n<1 return; TOWER(n-1, A, C, B); Move(A, C); TOWER(n-1, B, A, C) }

  22. EXAMPLE: tower of hanoi TOWER(n, A, B, C) { if n<1 return; TOWER(n-1, A, C, B); Move(A, C); TOWER(n-1, B, A, C) } • Algorithm analysis: • Input size? Basic operation?

  23. EXAMPLE: tower of hanoi TOWER(n, A, B, C) { if n<1 return; TOWER(n-1, A, C, B); Move(A, C); TOWER(n-1, B, A, C) } • Algorithm analysis: • Do we need to differentiate best case, worst case & average case for inputs of size n?

  24. EXAMPLE: tower of hanoi TOWER(n, A, B, C) { if n<1 return; TOWER(n-1, A, C, B); Move(A, C); TOWER(n-1, B, A, C) } • Algorithm analysis: • Set up a recurrence relation and initial condition(s) for C(n)

  25. EXAMPLE: tower of hanoi TOWER(n, A, B, C) { if n<1 return; TOWER(n-1, A, C, B); Move(A, C); TOWER(n-1, B, A, C) } • Algorithm analysis: • C(n) = 2C(n-1)+1

  26. In-Class Exercise • P. 76 Problem 2.4.1 (c): solve this recurrence relation: x(n) = x(n-1) + n for n>0, x(0)=0 • P. 77 Problem 2.4.4: consider the following recursive algorithm: • Algorithm Q(n) // Input: A positive integer n If n = 1 return 1 else return Q(n-1) + 2 * n – 1 • A. Set up a recurrence relation for this function’s values and solve it to determine what this algorithm computes • B. Set up a recurrence relation for the number of multiplications made by this algorithm and solve it. • C. Set up a recurrence relation for the number of additions/subtractions made by this algorithm and solve it.

  27. Example: BinRec(n) Algorithm BinRec(n) //Input: A positive decimal integer n //Output: The number of binary digits in n’s binary representation if n = 1 return 1 else return BinRec( n/2 ) + 1

  28. Smoothness rule If T(n)  (f(n)) for values of n that are powers of b, where b  2, then T(n)  (f(n))

  29. Example: BinRec(n) Algorithm BinRec(n) //Input: A positive decimal integer n //Output: The number of binary digits in n’s binary representation if n = 1 return 1 else return BinRec( n/2 ) + 1 If C(n)  (f(n)) for values of n that are powers of b, where b  2, then C(n)  (f(n))

  30. Fibonacci numbers • The Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, … • Fibonacci recurrence: F(n) = F(n-1) + F(n-2) F(0) = 0 F(1) = 1 • 2nd order linear homogeneous recurrence relation • with constant coefficients

  31. Solving linear homogeneous recurrence relations with constant coefficients • Easy first: 1st order LHRRCCs: C(n) = a C(n -1) C(0) = t … Solution: C(n) = t an • Extrapolate to 2nd order L(n) = aL(n-1) + bL(n-2) … A solution?: L(n) = ? • Characteristic equation (quadratic) • Solve to obtain roots r1 and r2 (quadratic formula) • General solution to RR: linear combination of r1nand r2n • Particular solution: use initial conditions

  32. Solving linear homogeneous recurrence relations with constant coefficients • Easy first: 1st order LHRRCCs: C(n) = a C(n -1) C(0) = t … Solution: C(n) = t an • Extrapolate to 2nd order L(n) = aL(n-1) + bL(n-2) … A solution?: L(n) = ? • Characteristic equation (quadratic) • Solve to obtain roots r1 and r2 (quadratic formula) • General solution to RR: linear combination of r1nand r2n • Particular solution: use initial conditions • Explicit Formula for Fibonacci Number: F(n) = F(n-1) +F(n-2)

  33. Computing Fibonacci numbers 1. Definition based recursive algorithm Algorithm F(n) // Compute the nthFibonacci number recursively //Input: A nonnegative integer n //Output: the nthFibonacci number if n  1 return n else return F(n-1) + F(n-2)

  34. Computing Fibonacci numbers 2. Nonrecursive brute-force algorithm Algorithm Fib(n) // Compute the nth Fibonacci number iteratively //Input: A nonnegative integer n //Output: the nthFibonacci number F[0]  0; F[1]  1 for i  2 to n do F[i]  F[i-1] + F[i-2] return F[n]

  35. Special care in its implementation: Intermediate results are irrational numbers Their approximations in the computer are accurate enough Final round-off yields a correct result Computing Fibonacci numbers 3. Explicit formula algorithm

  36. In-Class Exercises • What is the explicit formula for A(n)? A(n) = 3A(n-1) – 2A(n-2) A(0) = 1 A(1) = 3 • P.83 2.5.3. Climbing stairs: Find the number of different ways to climb an n-stair stair-case if each step is either one or two stairs. (For example, a 3-stair staircase can be climbed three ways: 1-1-1, 1-2, and 2-1.)

More Related