Introduction to Recursion

1
Introduction to Recursion

• Intro to Computer Science
• CS1510
• Dr. Sarah Diesburg

2
Sorting Complexity

• Three sorting algorithms
• Bubble Sort
• Insertion Sort
• Merge Sort
• All these algorithms are approximately n2
• Can we do better?

3
Sorting Programming Assignments

• Lets say I have 16 programming assignments, and
  I need to sort them in alphabetical order
• Im lazy, so I hand off half of the assignments
  to one student to sort, and the other half to
  another student to sort
• How many comparisons do I have to make to merge
  both sorted paper stacks?

4
Sorting Programming Assignments

• Lets say I have 16 programming assignments, and
  I need to sort them in alphabetical order
• Im lazy, so I hand off half of the assignments
  to one student to sort, and the other half to
  another student to sort
• How many comparisons do I have to make to merge
  both sorted paper stacks?
• Worst case 1 pass, around n comparisons

5
What if everyone hands off the work?
A
16
B
C
6
What if everyone hands off the work?
A
16
B
C
8
8
D
E
F
G
7
What if everyone hands off the work?
A
16
B
C
8
8
D
E
F
G
4
4
4
4
H
2
I
1
8
What if everyone hands off the work?
How long does this take? 5 rows of 16 units of
work each 16580 Looks better than n-squared!
A
16
B
C
8
8
D
E
F
G
4
4
4
4
H
2
I
1
9
How can we build a Merge Sort?

• We need to understand a new concept called
  recursion

10
A Function that Calls Itself

• The very basic meaning of a recursive function is
  a function that calls itself.
• However, when you first see it, it looks odd.
• Look at the recursive function that calculates
  factorial (code listing 16-2).

11
Code Listing 16-2

• def factorial(n)
• """Recursive factorial."""
• if n 1
• return 1
• else
• return n factorial(n-1)

12
It Doesnt Do Anything!

• This is a common complaint when one first sees a
  recursive function What exactly is it doing? It
  doesnt seem to do anything!
• Our goal is to understand what it means to write
  a recursive function from a programmers and
  computers point of view.

13
Defining a Recursive Function
14
1) Divide and Conquer

• Recursion is a natural outcome of a divide and
  conquer approach to problem solving.
• A recursive function defines how to break a
  problem down (divide) and how to reassemble
  (conquer) the sub-solutions into an overall
  solution.

15
2) Base Case

• Recursion is a process not unlike loop iteration.
  
• You must define how long (how many iterations)
  recursion will proceed until it stops.
• The base case defines this limit.
• Without the base case, recursion will continue
  infinitely (just like an infinite loop).

16
Simple Example

• Lao-Tzu A journey of 1000 miles begins with a
  single step.
• def journey (steps)
• the first step is easy (base case)
• the nth step is easy having completed the
  previous n-1 steps (divide and conquer)

17
Code Listing 16-1

• def takeStep(n)
• if n 1 base case
• return "Easy"
• else
• thisStep "step(" str(n) ")
• divide step, recurse
• previousSteps takeStep(n-1)
• conquer step
• return thisStep " " previousSteps

18
Factorial

• You remember factorial?
• Factorial(4) 4! 4 3 2 1
• The result of the last step, multiply by 1, is
  defined based on all the previous results that
  have been calculated.

19
Code Listing 16-2

• def factorial(n)
• """Recursive factorial."""
• if n 1
• return 1
• else
• return n factorial(n-1)

20
Trace the Calls

• 4! 4 3! start first function invocation
• 3! 3! 2 start second function invocation
• 2! 2 1! start third function invocation
• 1! 1 fourth invocation, base case
• 2! 2 1 2 third invocation finishes
• 3! 3 2 6 second invocation finishes
• 4! 4 6 24 first invocation finishes

21
Another Fibonacci

• Depends on the Fibo results for the two previous
  values in the sequence.
• The base values are Fibo(0) 1 and Fibo(1)
  1.
• fibo (x) fibo(x-1) fibo(x-2)

22
Code Listing 16-3

• def fibo(n)
• """Recursive Fibonacci sequence."""
• if n 0 or n 1 base case
• return 1
• else
• divide and conquer
• return fibonacci(n-1) fibonacci(n-2)

23
Trace

• fibo(4) fibo(3) fibo(2)
• fibo(3) fibo(2) fibo(1)
• fibo(2) fibo(1) fibo(0) 2 base case
• fibo(3) 2 fibo(1) 3 base case
• fibo(4) 3 2 5