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CSCE 310: Data Structures Algorithms

2. Giving credit where credit is due:Most of the lecture notes are based on the slides from the Textbook's companion websitehttp://www.aw-bc.com/info/levitinSeveral slides are from Hsu Wen Jing of the National University of SingaporeI have modified them and added new slides. CSCE 310: Data

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CSCE 310: Data Structures Algorithms

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    1. 1 CSCE 310: Data Structures & Algorithms

    2. 2 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 CSCE 310: Data Structures & Algorithms

    3. 3 Example: a recursive algorithm Algorithm: if n=0 then F(n) := 1 else F(n) := F(n-1) * n return F(n) Note the difference between the two recurrences. Students often confuse these! F(n) = F(n-1) n F(0) = 1 ------------ M(n) =M(n-1) + 1 M(0) = 0 (# recursive calls is the same)Note the difference between the two recurrences. Students often confuse these! F(n) = F(n-1) n F(0) = 1 ------------ M(n) =M(n-1) + 1 M(0) = 0 (# recursive calls is the same)

    4. 4 Example: another recursive algorithm

    5. 5 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): number of multiplications to compute n! Could we establish a recurrence relation for deriving M(n)? Note the difference between the two recurrences. Students often confuse these! F(n) = F(n-1) n F(0) = 1 ------------ M(n) =M(n-1) + 1 M(0) = 0 (# recursive calls is the same)Note the difference between the two recurrences. Students often confuse these! F(n) = F(n-1) n F(0) = 1 ------------ M(n) =M(n-1) + 1 M(0) = 0 (# recursive calls is the same)

    6. 6 Example: recursive evaluation of n ! M(n) = M(n-1) + 1 for n > 0 A recurrence relation is an equation or inequality that describes a function in terms of its value on smaller inputs Explicit formula for the sequence (e.g. M(n)) in terms of n only

    7. 7 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) = ? Note the difference between the two recurrences. Students often confuse these! F(n) = F(n-1) n F(0) = 1 ------------ M(n) =M(n-1) + 1 M(0) = 0 (# recursive calls is the same)Note the difference between the two recurrences. Students often confuse these! F(n) = F(n-1) n F(0) = 1 ------------ M(n) =M(n-1) + 1 M(0) = 0 (# recursive calls is the same)

    8. 8 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)? Note the difference between the two recurrences. Students often confuse these! F(n) = F(n-1) n F(0) = 1 ------------ M(n) =M(n-1) + 1 M(0) = 0 (# recursive calls is the same)Note the difference between the two recurrences. Students often confuse these! F(n) = F(n-1) n F(0) = 1 ------------ M(n) =M(n-1) + 1 M(0) = 0 (# recursive calls is the same)

    9. 9 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 estimate the order of magnitude of the solution (see Appendix B)

    10. 10 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

    11. 11 EXAMPLE: tower of hanoi Step 1: Solve simple case when n<=1? Just trivial

    12. 12 EXAMPLE: tower of hanoi Step 2: Assume that a smaller instance can be solved, i.e. can move n-1 disks. Then?

    13. 13 EXAMPLE: tower of hanoi

    14. 14 EXAMPLE: tower of hanoi

    15. 15 EXAMPLE: tower of hanoi

    16. 16 EXAMPLE: tower of hanoi

    17. 17 EXAMPLE: tower of hanoi

    18. 18 EXAMPLE: tower of hanoi

    19. 19 EXAMPLE: tower of hanoi

    20. 20 In-Class Exercise P. 76 Problem 2.4.1 (c): solve this recurrence relation: x(n) = x(n-1) + n P. 76 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.

    21. 21 Example: BinRec(n)

    22. 22 Smoothness rule If T(n) ? ?(f(n)) for values of n that are powers of b, where b ? 2, then T(n) ? ?(f(n))

    23. 23 Example: BinRec(n)

    24. 24 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

    25. 25 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) = a L(n-1) + b L(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 r1n and r2n Particular solution: use initial conditions

    26. 26 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) = a L(n-1) + b L(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 r1n and r2n Particular solution: use initial conditions Explicit Formula for Fibonacci Number: F(n) = F(n-1) +F(n-2)

    27. 27 1. Definition based recursive algorithm Computing Fibonacci numbers

    28. 28 2. Nonrecursive brute-force algorithm Computing Fibonacci numbers

    29. 29 Computing Fibonacci numbers

    30. 30 In-Class Exercises Another example: A(n) = 3A(n-1) – 2A(n-2) A(0) = 1 A(1) = 3 P.83 2.5.4. 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.)

    31. Announcement Reading List of this Week Chapter 3

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