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Chapter 3. Recursion

Lecture 6

In functions and data structures

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What is recursion?

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It is something defined in terms of itself

… but it must have a stopping condition!

… it must “bottom out”

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Recursive functions

A recursive function is one that calls itself.

E.g. to calculate N! = N (N-1) (N-2) . . . 1, do the following:

if N = 0 then 1 else N (N-1)!

We have defined the factorial function ! In terms of itself.– Note the factorial function is applied to a smaller number

every time until it is applied to 0.

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The factorial function in Java

The factorial function in Java

fact(4) =

The factorial function in Java

fact(4) = 4 * fact(3)

The factorial function in Java

fact(4) = 4 * fact(3) = 4 * 3 * fact(2)

The factorial function in Java

fact(4) = 4 * fact(3) = 4 * 3 * fact(2) = 4 * 3 * 2 * fact(1)

The factorial function in Java

fact(4) = 4 * fact(3) = 4 * 3 * fact(2) = 4 * 3 * 2 * fact(1) = 4 * 3 * 2 * 1 * fact(0)

The factorial function in Java

fact(4) = 4 * fact(3) = 4 * 3 * fact(2) = 4 * 3 * 2 * fact(1) = 4 * 3 * 2 * 1 * fact(0) = 4 * 3 * 2 * 1 * 1

The factorial function in Java

fact(4) = 4 * fact(3) = 4 * 3 * fact(2) = 4 * 3 * 2 * fact(1) = 4 * 3 * 2 * 1 * fact(0) = 4 * 3 * 2 * 1 * 1 = 4 * 3 * 2 * 1

The factorial function in Java

fact(4) = 4 * fact(3) = 4 * 3 * fact(2) = 4 * 3 * 2 * fact(1) = 4 * 3 * 2 * 1 * fact(0) = 4 * 3 * 2 * 1 * 1 = 4 * 3 * 2 * 1 = 4 * 3 * 2

The factorial function in Java

fact(4) = 4 * fact(3) = 4 * 3 * fact(2) = 4 * 3 * 2 * fact(1) = 4 * 3 * 2 * 1 * fact(0) = 4 * 3 * 2 * 1 * 1 = 4 * 3 * 2 * 1 = 4 * 3 * 2 = 4 * 6

The factorial function in Java

fact(4) = 4 * fact(3) = 4 * 3 * fact(2) = 4 * 3 * 2 * fact(1) = 4 * 3 * 2 * 1 * fact(0) = 4 * 3 * 2 * 1 * 1 = 4 * 3 * 2 * 1 = 4 * 3 * 2 = 4 * 6 = 24

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In general…

A recursive function when calling itself makes a clone and calls the clone with appropriate parameters.

Important: a workable recursive algorithm (function/procedure) must always:

• Rule 1: reduce size of data set, or the number its working on, each time

it is recursively called and

• Rule 2: provide a stopping case (terminating condition)

• Can always replace recursive algorithm with an iterative one, but often recursive solution much sleeker

• Factorial function is only simple example, but recursion is valuable tool in more complex algorithms.

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Linear recursion

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With linear recursion a method is defined so that it makes at most one recursive call each time it is invoked.Useful when we view an algorithmic problem in terms of a first and/or last element plus a remaining set with same structure as original set.

Example 1 : factorial exampleExample 2: summing the elements of an array -

Algorithm LinearSum(A,n): Input: An integer array A and integer n1, such that A has at least n elements Output:The sum of the first n integers in A if n=1 then return A[0] else return LinearSum(A,n-1)+A[n-1]

This is pseudocode. Used in GoTa a lot.

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Visualizing Recursion

• Recursion trace• A box for each recursive call• An arrow from each caller to

callee• An arrow from each callee to

caller showing return value

Example recursion trace:

RecursiveFactorial(4)

RecursiveFactorial(3)

RecursiveFactorial(2)

RecursiveFactorial(1)

RecursiveFactorial(0)

return 1

call

call

call

call

return 1*1 = 1

return 2*1 = 2

return 3*2 = 6

return 4*6 = 24 final answercall

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Actually we will remove the blue arrows on the right, to make it simpler

More examples

Raise m to the power n

...... 21 nnn mmmmmm

Raise m to the power n

1

.0

1

m

mmm nn

We now have a recursive definition

Example 3: RaisePower

Algorithm RaisePower(m,n): Input: Integers m,n Output: value of mn

if n=0 then return 1 else return (m*RaisePower(m,n-1))

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.0

1

m

mmm nn

Example 3: RaisePower

Algorithm RaisePower(m,n): Input: Integers m,n Output: value of mn

if n=0 then return 1 else return (m*RaisePower(m,n-1))

1

.0

1

m

mmm nn

Does RaisePower satisfy rules 1 and 2? Yes!

Example 3: RaisePower

Algorithm RaisePower(m,n): Input: Integers m,n Output: value of mn

if n=0 then return 1 else return (m*RaisePower(m,n-1))

1

.0

1

m

mmm nn

Does RaisePower satisfy rules 1 and 2? Yes!

Rule 1 (number function working on is decreased)

Example 3: RaisePower

Algorithm RaisePower(m,n): Input: Integers m,n Output: value of mn

if n=0 then return 1 else return (m*RaisePower(m,n-1))

1

.0

1

m

mmm nn

Does RaisePower satisfy rules 1 and 2? Yes!

Rule 1 (number function working on is decreased)

Rule 2 ( stopping case)

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Recursion trace, m=2, n=4

RaisePower(2,4)

return 1

return 2*1=2

return 2*8=16 final answercall

call

call

call

call

RaisePower(2,3)

RaisePower(2,2)

RaisePower(2,1)

RaisePower(2,0)

return 2*2=4

return 2*4=8

And in Java…

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public static int raisePower(int m, int n){ if (n==0) return 1; else return(m*raisePower(m,n-1));}

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Another example … has9

has9

Given an integer (base 10) does it contain the digit 9?

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Algorithm Has9(n): Input: Integer n Output: whether 9 appears in decimal expansion of n if (n mod 10)=9 then return true else if (n<10) then return false else return Has9(n/10)

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has9

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Does Has9 satisfy rules 1 and 2? Yes!

Algorithm Has9(n): Input: Integer n Output: whether 9 appears in decimal expansion of n if (n mod 10)=9 then return true else if (n<10) then return false else return Has9(n/10)

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has9

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Does Has9 satisfy rules 1 and 2? Yes!

Rule 1 (number function working on is decreased)

Algorithm Has9(n): Input: Integer n Output: whether 9 appears in decimal expansion of n if (n mod 10)=9 then return true else if (n<10) then return false else return Has9(n/10)

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has9

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Does Has9 satisfy rules 1 and 2? Yes!

Rule 1 (number function working on is decreased)

Rule 2 ( stopping case)

Algorithm Has9(n): Input: Integer n Output: whether 9 appears in decimal expansion of n if (n mod 10)=9 then return true else if (n<10) then return false else return Has9(n/10)

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has9

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Does Has9 satisfy rules 1 and 2? Yes!

Rule 1 (number function working on is decreased)

Rule 2 ( stopping case)

Algorithm Has9(n): Input: Integer n Output: whether 9 appears in decimal expansion of n if (n mod 10)=9 then return true else if (n<10) then return false else return Has9(n/10)

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has9

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McCreesh & ProsserarXiv :1209.45601v1

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Matula & Beck JACM 30(3) 1983

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OstergardDAM 120 (2002)

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San SegundoC&OR 38 (2011)

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Fleiner, Irving & ManloveTCS 412 (2011)

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example

reverse

Reverse an array

Reverse an array

Reverse an array

Reverse an array

Example … for you

palindrome

example

Binary search

Binary search

Binary search

Binary search

Note overloading

example

All permutations of a string

perm

perm

Ouch!

Arithmetic! (on natural numbers)

example

Similar to … Church Numerals

Recursive Arithmetic

Recursive Arithmetic

Recursive Arithmetic

Recursive Arithmetic

example

list

list

We have been defining lists in terms of nodes, where a node has

- a head (we call it “element” where we store data) - a tail (we call it “next” and it points to the next node or null)

This IS inherently recursive

list

list

list

list

list

NOTE: this is NOT destructive! It produces a new list!

Tail recursion

• Recursion is useful tool for designing algorithms with short, elegant definitions

• But comes at a cost – need to use memory to keep track of the state of each recursive call

• When memory is of primary concern, useful to be able to derive non-recursive algorithms from recursive ones.

• Can use a stack data structure to do this (see ADT chapter), but some cases we can do it more easily and efficiently

i.e. when algorithms use tail recursion• An algorithm uses tail recursion when recursion is linear

and recursive call is its very last operation

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