By: Lokman ChanBy: Lokman Chan

Recursive AlgorithmRecursive Algorithm

RecursionRecursion

Definition: A function that is define in terms of itself.

Goal: Reduce the solution to a problem with a particular set of input to the same problem with smaller input values. Simplify a complex problem to several sub problems

Examples: Looking up a word from a dictionary (non mathematical)

Solutions to “Towers of Hanol” problem (mathematical)

Merge Sort, reversing a list, binary search, etc…

Definition: A function that is define in terms of itself.

Goal: Reduce the solution to a problem with a particular set of input to the same problem with smaller input values. Simplify a complex problem to several sub problems

Examples: Looking up a word from a dictionary (non mathematical)

Solutions to “Towers of Hanol” problem (mathematical)

Merge Sort, reversing a list, binary search, etc…

Lookup a phone number in phone bookSince the phone numbers are listed in alphabetical,

we can actually apply the recursive strategy to find a number

Consider Search: middle page = (first page + last page)/2

Open the phone book to middle page; If (name is on middle page) then done; //this is the base case

else if (name is alphabetically before middle page) last page = middle page //redefine search area to front half

Search //recursive call with reduced number of pages

else //name must be after middle page first page = middle page //redefine search area to back half Search //recursive call with reduced number of pages

Lookup a phone number in phone bookSince the phone numbers are listed in alphabetical,

we can actually apply the recursive strategy to find a number

Consider Search: middle page = (first page + last page)/2

Open the phone book to middle page; If (name is on middle page) then done; //this is the base case

else if (name is alphabetically before middle page) last page = middle page //redefine search area to front half

Search //recursive call with reduced number of pages

else //name must be after middle page first page = middle page //redefine search area to back half Search //recursive call with reduced number of pages

Towers of HanoiTowers of Hanoi

The recursive algorithm is based on the observation that moving a

tower of height h from pole A to pole B is equivalent to moving a tower

of height h-1 to pole C, them moving the last disk from pole A to pole B,

and finally moving the the tower from pole C to B.

The recursive algorithm is based on the observation that moving a

tower of height h from pole A to pole B is equivalent to moving a tower

of height h-1 to pole C, them moving the last disk from pole A to pole B,

and finally moving the the tower from pole C to B.

So the problem of moving a tower of height h, has been reduced

to the one of moving a tower of height h-1.

So the problem of moving a tower of height h, has been reduced

to the one of moving a tower of height h-1.

The idea behind…The idea behind…

Objective:

Move tower of 3 disks from peg a to peg b, with the help of peg c.

Rules:

Only one disk can be moved at a time.

A larger disk can never be placed on top of a smaller disk.

Assume the disk1 is smallest, disk2 is larger and disk3 is largest

Objective:

Move tower of 3 disks from peg a to peg b, with the help of peg c.

Rules:

Only one disk can be moved at a time.

A larger disk can never be placed on top of a smaller disk.

Assume the disk1 is smallest, disk2 is larger and disk3 is largest

1

2

3

_______ _______ _______

A B C

1

2

3

_______ _______ _______

A B C

2

3 1

_______ _______ _______

A B C

2

3 1

_______ _______ _______

A B C

More steps …More steps …

3 1 2

_______ _______ _______

A B C

3 1 2

_______ _______ _______

A B C

1

3 2

_______ _______ _______

A B C

1

3 2

_______ _______ _______

A B C

Steps…Steps…

1

3 2

_______ _______ _______

A B C

1

3 2

_______ _______ _______

A B C

1 3 2

_______ _______ _______

A B C

1 3 2

_______ _______ _______

A B C

FinallyFinally

2

1 3

_____ _____ _____

A B C

2

1 3

_____ _____ _____

A B C

1

2

3

_______ _______ ______

A B C

1

2

3

_______ _______ ______

A B C

It turns out to be a recurrence relationship

and can be computed using recursive algorithm

It turns out to be a recurrence relationship

and can be computed using recursive algorithm

For better simulations, please go to http://www.cut-the-knot.org/recurrence/hanoi.shtmlFor better simulations, please go to http://www.cut-the-knot.org/recurrence/hanoi.shtml

Merge SortMerge Sort

Merge sort can be done nicely with recursive algorithm

but can also be implemented without using recursion. (For demonstration purpose, we will discuss only

recursive merge sort)

Merge sort is a sorting algorithm using

divide and conquer algorithm.

Merge sort can be done nicely with recursive algorithm

but can also be implemented without using recursion. (For demonstration purpose, we will discuss only

recursive merge sort)

Merge sort is a sorting algorithm using

divide and conquer algorithm.

Divide and ConquerDivide and Conquer

General Strategy:

1) Split the problem into subproblems

2) Solve subproblems

3) Combine the results

General Strategy:

1) Split the problem into subproblems

2) Solve subproblems

3) Combine the results

http://www.cs.nott.ac.uk/~nza/G5BADS01/slides4.pdfhttp://www.cs.nott.ac.uk/~nza/G5BADS01/slides4.pdf

http://www.cs.nott.ac.uk/~nza/G5BADS01/slides4.pdfhttp://www.cs.nott.ac.uk/~nza/G5BADS01/slides4.pdf

.

Recursive Merge Sort

1) Split the array into two halves

2) Call merge on each half

3) Merge sorted Halves

Recursive Merge Sort

1) Split the array into two halves

2) Call merge on each half

3) Merge sorted Halves

Time Complexity

Levels of recursion: log2N

At each level: merging N items

Time complexity: O(N log2 N)

Space complexity: O(N)

Time Complexity

Levels of recursion: log2N

At each level: merging N items

Time complexity: O(N log2 N)

Space complexity: O(N)

Basic foundation of Recursive Algo…Basic foundation of Recursive Algo…

1) Base case(s): There must be at least one base which can

be solved without recursion

2) A recursive call should always making progress, heading

towards the base case. (Simplify the problem)

3) Don’t duplicate work by solving the same instance of a

problem in separate recursive calls (don’t use recursion

instead of a simple loop)

4) Design rule: Assume that all the recursive calls work

1) Base case(s): There must be at least one base which can

be solved without recursion

2) A recursive call should always making progress, heading

towards the base case. (Simplify the problem)

3) Don’t duplicate work by solving the same instance of a

problem in separate recursive calls (don’t use recursion

instead of a simple loop)

4) Design rule: Assume that all the recursive calls work

Advantages of recursive algorithmsAdvantages of recursive algorithms

The recursive algorithm gives a more understandable,

code

The actaul code can be expressed with a few lines

of code when defined properly

The recursive algorithm gives a more understandable,

code

The actaul code can be expressed with a few lines

of code when defined properly

Disadvantages of Recursive Algo…Disadvantages of Recursive Algo…

Not all mathematically recursive functions can be

efficiently implemented by Java’s simulation of recursion

Infinite recursion can lead to stack overflow if the recursive

code run (loops) forever

Not all mathematically recursive functions can be

efficiently implemented by Java’s simulation of recursion

Infinite recursion can lead to stack overflow if the recursive

code run (loops) forever

THE ENDTHE END