CSCE 310: Data Structures

1
CSCE 310 Data Structures Algorithms

• Recursive Algorithm Analysis
• Dr. Ying Lu
• ylu_at_cse.unl.edu

September 8, 2010
2
CSCE 310 Data Structures Algorithms

• Giving credit where credit is due
• Most of the lecture notes are based on the slides
  from the Textbooks 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 n0 then F(n) 1
• else F(n) F(n-1) n
• return F(n)

4
Example another recursive algorithm
Algorithm F(n) // Compute the nth Fibonacci
number recursively //Input A nonnegative integer
n //Output the nth Fibonacci number if n ? 1
return n else return F(n-1) F(n-2)
5
Example recursive evaluation of n !

• Definition n ! 12(n-1)n
• Recursive definition of n!
• Algorithm
• if n0 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)?

6
Example recursive evaluation of n !

• M(n) M(n-1) 1 for n gt 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
Example recursive evaluation of n !

• Definition n ! 12(n-1)n
• Recursive definition of n!
• Algorithm
• if n0 then F(n) 1
• else F(n) F(n-1) n
• return F(n)
• M(n) M(n-1) 1
• Initial Condition M(0) ?

8
Example recursive evaluation of n !

• Recursive definition of n!
• Algorithm
• if n0 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)?

9
Time efficiency of recursive algorithms

• Steps in analysis of recursive algorithms
• Decide on parameter n indicating input size
• Identify algorithms 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
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
11
EXAMPLE tower of hanoi

• Step 1 Solve simple case when nlt1?
• Just trivial

B
B
A
C
A
C
Move(A, C)
12
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
13
EXAMPLE tower of hanoi
B
B
A
C
A
C
B
A
C
B
A
C
14
EXAMPLE tower of hanoi
B
B
A
C
A
C
B
TOWER(n, A, B, C)
A
C
B
A
C
15
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
16
EXAMPLE tower of hanoi
TOWER(n, A, B, C)
if nlt1 return
TOWER(n-1, A, C, B)
Move(A, C)
TOWER(n-1, B, A, C)
17
EXAMPLE tower of hanoi

• Algorithm analysis
• Input size? Basic operation?

18
EXAMPLE tower of hanoi

• Algorithm analysis
• Do we need to differentiate best case, worst case
  average case for inputs of size n?

19
EXAMPLE tower of hanoi

• Algorithm analysis
• Set up a recurrence relation and initial
  condition(s) for C(n)

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
  functions 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
Example BinRec(n)
Algorithm BinRec(n) //Input A positive decimal
integer n //Output The number of binary digits
in ns binary representation if n 1 return
1 else return BinRec( ?n/2? ) 1
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
Example BinRec(n)
Algorithm BinRec(n) //Input A positive decimal
integer n //Output The number of binary digits
in ns binary representation if n 1 return
1 else return BinRec( ?n/2? ) 1
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

2nd order linear homogeneous recurrence relation
with constant coefficients
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
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
Computing Fibonacci numbers

• 1. Definition based recursive algorithm

Algorithm F(n) // Compute the nth Fibonacci
number recursively //Input A nonnegative integer
n //Output the nth Fibonacci number if n ? 1
return n else return F(n-1) F(n-2)
28
Computing Fibonacci numbers

• 2. Nonrecursive brute-force algorithm

Algorithm Fib(n) // Compute the nth Fibonacci
number iteratively //Input A nonnegative integer
n //Output the nth Fibonacci number F0 ? 0
F1 ? 1 for i ? 2 to n do Fi ? Fi-1
Fi-2 return Fn
29
Computing Fibonacci numbers
3. Explicit formula algorithm
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