CMSC 150RECURSION

CS 150: Mon 26 Mar 2012

Motivation : Bioinformatics Example A G A C T A G T T A C C G A G A C G T

Want to compare sequences for similarity

Similar sequences: common ancestors? Point mutations Insertions Deletions − − A G A C T A G T T A C

C G A G A C − − G − T − −

Global Alignment Algorithm

Think about brute force

A G A C T A G T T A C C G A G A C G T

Where should gaps go? Enumerate all possible alignments?

Global Alignment Algorithm

Think about brute force

A G A C T A G T T A C C G A G A C G T

For two sequences of length L: # of possible global alignments: ~ 22L

if L = 250, this is ~10149 alignments @ 1B alignments / second, takes 3.21 X 10132

years age of universe: ~1.4 X 1010 years

Global Alignment Algorithm

Think about brute force

A G A C T A G T T A C C G A G A C G T

For two sequences of length L: # of possible global alignments: ~ 22L

if L = 250, this is ~10149 alignments @ 1B alignments / second, takes 3.21 X 10132

years age of universe: ~1.4 X 1010 years

Needleman-Wunsch Algorithm Computes optimal global alignment

Technique: Uses dynamic programming combine optimal solutions from

subproblems number of subproblems must be

(relatively) small

Typically bottom-up: find solution using a recursive series of

simpler solutions

Recursion

Use same algorithm on smaller subproblems

Need: Base case: simplest input possible, solution

immediately available Recursive call: invoke the algorithm on a

smaller set of the input

Without base case, recursion would be infinite!

An Example

Search phone book for a name start in middle: if found, stop otherwise, repeat process in correct “half”

of book

Base case: only one name to search Recursive call: search remaining “half” of

book

Another Example : Factorial

n! = n x (n-1) x (n-2) x … x 2 x 1

5! = 5 x 4 x 3 x 2 x 1 = 120 4! = 4 x 3 x 2 x 1= 24 3! = 3 x 2 x 1 = 6 2! = 2 x 1 = 2 1! = 1 0! = 1 (multiplicative identity)

Another Example : Factorial

n! = n x (n-1) x (n-2) x … x 2 x 1

5! = 5 x 4 x 3 x 2 x 1 = 120 4! = 4 x 3 x 2 x 1= 24 3! = 3 x 2 x 1 = 6 2! = 2 x 1 = 2 1! = 1 0! = 1 (multiplicative identity)

Another Example : Factorial

n! = n x (n-1) x (n-2) x … x 2 x 1

5! = 5 x 4 x 3 x 2 x 1 = 5 x 4! = 120 4! = 4 x 3 x 2 x 1= 24

Another Example : Factorial

n! = n x (n-1) x (n-2) x … x 2 x 1 n! = n x (n-1)!

Defined recursively:

1if n = 0

n! = n(n-1)! if n > 0

Compute n! in BlueJ…

Another Example : Fibonacci Fibonacci sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …

After first two, each term is sum of previous two

Defined recursively: Let fn be the nth term, n = 0, 1, 2…0 if n = 0

fn = 1 if n = 1

fn-1 + fn-2 if n > 1

Another Example : Fibonacci Fibonacci sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …

0 if n = 0fn = 1 if n = 1

fn-1 + fn-2 if n > 1

f0 = 0 f1 = 1 f2 = f1 + f0 = 1 + 0 = 1 f3 = f2 + f1 = 1 + 1 = 2 f4 = f3 + f2 = 2 + 1 = 3

Fibonacci in Nature

Fibonacci spiral: Fibonacci tiling: squares of sizes 1, 1, 2, 3, 5,

8, 13, 21, 34 Draw circular arc connecting opposite corners

of squares

Fibonacci in Nature

Fibonacci spiral: Fibonacci tiling: squares of sizes 1, 1, 2, 3, 5, 8, 13, 21,

34 Draw circular arc connecting opposite corners of

squares More explanation: Fibonacci in nature

Compute nth Fibonacci in BlueJ…

Another Example : Towers of Hanoi 3 towers, n disks each of different size Rules:

Can move only one disk at a time No larger disk can be on top of smaller disk

Goal: move n-tower from 1st tower to 3rd tower

Think Recursively

n n - 1

Consider the n-tower as a tower of n-1 and a tower of 1…

Think Recursively

n - 1

If we can somehow move the n-1 tower to the middle…

Think Recursively

n - 1

And then the largest disk to the right…

Think Recursively

n - 1

And finally the n-1 tower to the right, we have a solution!

Think Recursively

What is the base case? a tower of n = 1 disk

Think Recursively

n n - 1

What is the recursive step? Move n-1 tower to middle Then largest disk to right Then n-1 tower from middle to right

1

2

3

Think Recursively

n n - 1

In pseudocode: moveTower( n-1, 1, 2

); moveDisk( 1, 3 ); moveTower( n-1, 2, 3

);

1

2

3

Note that we do not explicitly implement

the steps for a tower of size n-1

Solve Towers in BlueJ…