Wrapping%20up%20Recursion

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Wrapping up Recursion

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

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Recursion

• Recursive functions are functions that call
  themselves
• We can create recursive functions by breaking
  them up into two smaller parts

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1) Call same function with something smaller

• 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.

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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).

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Some examples of things that can be done
recursively

• Summation of numbers sum(lower,upper)
• Exponents - power(base,exp)
• Reverse a string reverse(string)
• Merge Sort mergeSort(lyst)

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Summation

• Usually with a summation, we have a lower bound
  and upper bound
• We want to sum from the lower bound to the upper
  bound
• summation(1,5) would yield
• 1234515
• Write a summation function
• summation(lower,higher)

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Summation

• def summation(lower,higher)
• if lowerhigher
• return lower
• else
• return lower summation(lower1,higher)

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Power

• Write a function that takes a base and a power
  and returns the result
• For example, we know that 24 is 16
• 2 is the base
• 4 is the power
• We break it up as 2 x 2 x 2 x 2 16
• Same as 2 x 23 16

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Power

• def power(base,exp)
• if (exp0)
• return 1
• else
• return basepower(base,exp-1)

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Reverse a String

• def reverse(string)
• if (len(string))0
• return ""
• else
• return string-1 reverse( string0-1
  )

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Merge Sort

• 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

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What if everyone hands off the work?
A
16
B
C
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What if everyone hands off the work?
A
16
B
C
8
8
D
E
F
G
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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
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Merge Sort

• At some point, the last students will only have 1
  paper.
• They can then hand those single papers back to
  the student that delegated the work in the first
  place.
• The delegating student can then sort the two
  piles (of 1) papers by performing a merge.

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Merge Sort

• Students I and J hand each of their sorted, 1
  item stacks to H.
• H performs a merge sort on the two stacks of items

H
1
1
J
I
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What is a merge?

• Start with two sorted piles (pile A and pile B)
• Take the smallest item off the top of pile A or B
  and place it in the finished pile.
• Repeat the previous step until no more items in
  either pile.
• Resulting finished pile will be sorted.

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Finishing the merge sort

• All students hand off sorted piles to delegating
  students, each of which performs a merge sort
• Finally the student that started everything (A)
  will merge sort two large piles of items and
  created the final sorted output pile

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Pseudocode for Merge Sort

• Has to be some sort of recursive algorithm
• What is the base case?
• If you have pile of size 1, it is sorted and you
  are done.
• What is the recursive case?
• This is trickier
• We can merge two sorted piles, but we want the
  two piles to be sorted first
• Each pile is of size n/2, where n is the total
  number of items we have