recursion. what is recursion? solving a problem in terms of itself or repeating objects in a...

Recursion

What is recursion?

• Solving a problem in terms of itself or repeating objects in a “self-similar” way

• A recursive function is a function that calls itself!• Another way to repeat code• Takes advantage of the “activation stack”

You just moved in to a new house on a dead end street…

In celebration you shout: “Woo hoo, my new house is awesome and the house number is 1282!”

You just moved in to a new house on a dead end street…

• But you don’t know the house number!

• You do know that your house number is 2 plus the address of your neighbor to the left

• So you decide to call your neighbor…

Call your neighbor and ask them for their house number. Add two to get your house number.

You

Call your neighbor and ask them for their house number.

Howdy neighbor. What is your

house number?

Neighbor 1 You

Wait for the neighbor to give you an answer.

42.

Neighbor 1 You

Kindly thank them and hang up.

Sweet, thanks! <Hang up>

YouNeighbor 1

Add two to your neighbor’s address to get your house number.

So my address is 42 + 2 =

44!

What if your neighbor doesn’t know their address?

Call your neighbor and ask for their house number.


house number?

Neighbor 1 You

But they don’t know it!

Hm, that’s a good question. Hold

please!

Neighbor 1 You

Neighbor 1 puts you on hold and calls their neighbor.

<holding…>

Neighbor 1Neighbor 2 You

Neighbor 1 puts you on hold and calls their neighbor.


house number?

<holding…>


But their neighbor doesn’t know their house number either! (Seriously, who are these people?!)

Hm, that’s a good question. Hold

please!

<holding…>


Neighbor 2 puts neighbor 1 on hold and calls neighbor 3.

<holding…><holding…>


house number?

Neighbor 1Neighbor 2 YouNeighbor 3

Neighbor 3 tells neighbor 2 their house number (phew!)


22.


Neighbor 2 thanks neighbor 3 and hangs up.




Neighbor 2 adds two to get address.



24!


Neighbor 2 takes neighbor 3 off of hold and tells them their address.

<holding…>

24.


Neighbor 1 thanks neighbor 2 and hangs up.

<holding…>



Your neighbor adds two to get their address.

<holding…>


26!

Neighbor 1

Finally your neighbor takes you off hold and tells you their address (what took them so long?!)


Neighbor 1

You add two to your neighbors address to get your house number!


28!

Let’s analyze what just happened

• How many phone calls were made?• When did the calling chain finally stop?• From “your” perspective, what happened

when you called your neighbor?

Algorithm in writing

• If you already know your house number, say it loud!

• If you don’t know your house number…– Call neighbor to your left and ask for their house

number– When you get the number, add two to it, then say

it loud!

But…

• What if you don’t have any neighbors?

Algorithm in writing

• If you already know your house number, say it loud

• If you don’t have a neighbor to the left and don’t know your house number, say 0

• If you don’t know your house number…– Call neighbor to your left and ask for their house

number– When you get the number, add two to it, then say

it loud!

- If you already know your house number, say it loud

- If you don’t have a neighbor to the left and don’t know your house number, say 0

- If you don’t know your house number…

-Call neighbor to your left and ask for their house number

-When you get the number, add two to it, then say it loud!

22 You






22 You <calling neighbor, waiting for answer>












N1<answers call, decides what to do>












N1<calling neighbor, waiting for answer>
























N3<knows answer! Tells N2 and hangs up>

























22


















N2<adds two to neighbor’s answer>

22 + 2 = 24



















22 + 2 = 24

24












N1<adds two to neighbor’s answer>

24 + 2 = 26












N1<knows answer! Tells you and hangs up>

24 + 2 = 26

26






22 You <add two to neighbor’s answer>

26 + 2 = 28

22 You <done! Your address is 28>

But…

• What if no-one on your street knows their house number?

But…

• What if no-one on your street knows their house number?– Think about it….don’t we cover this case by adding

a check to see if you have a neighbor to the left and returning 0 if you don’t?

Three “Pillars” of Recursion

• A recursive function…– Calls clone of itself– Has terminating condition(s)– Moves toward terminating condition(s)

House number problem components

• Calling clone of itself?• Terminating condition(s)?• Moving toward the terminating conditions?

• If you already know your house number, say it loud• If you don’t have a neighbor to the left and don’t know

your house number, say 0• If you don’t know your house number…

– Call neighbor to your left and ask for their house number

– When you get the number, add two to it, then say it loud!

House number problem components

• Calling clone of itself?• Terminating condition(s)?• Moving toward the terminating conditions?

• If you already know your house number, say it loud• If you don’t have a neighbor to the left and don’t know

your house number, say 0• If you don’t know your house number…

– Call neighbor to your left and ask for their house number

– When you get the number, add two to it, then say it loud!

Terminating conditions

Call clone andmove towardterm cond

Another problem: Factorial

• How do you calculate 5!5*4*3*2*1

Another problem: Factorial

• How do you calculate 5!5! = 5*4! 4! = 4*3!3! = 3*2! 2! = 2*1! 1! = 1 (what about 0!)

In general…

• n! = n * (n-1)!• If we had a factorial function instead of !, it

would be fact(n) = n * fact(n-1)

Factorial Recursive Algorithm

• What are are terminating (aka base) conditions? What is the simplest factorial to calculate?

• What inputs does the function need? How do we move toward the terminating condition?

Factorial Recursive Algorithm

• If number is 0 or 1– Answer is 1

• Else– Answer is number * factorial of number-1

Call fact(3) from command window…

>> myans = fact(3)

fact(3)if 3 == 1 | 3==0

ans = 1else

ans = 3 * fact(3-1)

>> myans = fact(3)

fact(3)if 3 == 1 | 3==0

ans = 1else

ans = 3 * fact(3-1)

Cal fact(3) from command window…

>> myans = fact(3)

fact(3)if 3 == 1 | 3==0

ans = 1else

ans = 3 * fact(3-1)


>> myans = fact(3) fact(2)if 2 == 1 | 2==0

ans = 1else

ans = 2 * fact(2-1)

fact(3)if 3 == 1 | 3==0

ans = 1else

ans = 3 * fact(3-1)



ans = 1else

ans = 2 * fact(2-1)

fact(1)if 1 == 1 | 1==0

ans = 1else

ans = 1 * fact(1-1)

fact(3)if 3 == 1 | 3==0

ans = 1else

ans = 3 * fact(3-1)



ans = 1else

ans = 2 * fact(2-1)

fact(1)if 1 == 1 | 1==0

ans = 1else

ans = 1 * fact(1-1)1

fact(3)if 3 == 1 | 3==0

ans = 1else

ans = 3 * fact(3-1)



ans = 1else

ans = 2 * fact(2-1)

1

fact(3)if 3 == 1 | 3==0

ans = 1else

ans = 3 * fact(3-1)



ans = 1else

ans = 2 * fact(2-1)

12

fact(3)if 3 == 1 | 3==0

ans = 1else

ans = 3 * fact(3-1)


>> myans = fact(3)

2

fact(3)if 3 == 1 | 3==0

ans = 1else

ans = 3 * fact(3-1)


>> myans = fact(3)

26


>> myans = fact(3)6


>> myans = fact(3)myans =

6>>

6

Answer is returned to place that called it, which in this case was the command window!

What about bad inputs?

• What happens if input is -1?• If input is -5?

What about bad inputs?

• What happens if input is -1?• If input is -5?• We theoretically recurse forever! (though our

activation stack keeping track of all the temporary workspaces will eventually run out of memory and we will get an error)

Wrapper Functions• Use a “wrapper function” to preprocess your data!• A wrapper function checks input(s), and if they are

value, makes the first call to the recursive function• Captures result from recursive function and then

returns it back to where called the wrapper function originally

• Note, the wrapper function should usually only run once per call (unlike the recursive helper function)

Why recursion?

• In general, any recursive function could be written as a loop (likely a while loop)

• Loops are usually faster (because Matlab doesn’t have to make all those temporary workspaces and copy the parameter values in)

• But some problems are solved more simply using recursion– Printing out structures (e.g. nested structures)– Listing out directories and files– Crawling the web

Recursively List Contents of Directory

Recursively List Contents of Directory

• Given a directory, we want to list all items in that folder, including items in

• Don’t even worry about matlab yet…let’s think about the steps involved in general

• Given a directory (you, yourself), what would you do to start listing out the filenames?

• What would you do when you encountered another folder?

Recursively Listing Directories

• Open folder to see list of items• For each item listed in folder– Check to see if it is a directory• If it is a directory, we need to search it (start whole

process over)• Else, if not a directory (meaning it’s just a file name),

print it out to command window and continue on

Translating to Matlabfunction mydir(dirname)% recursive dir traversal lst = dir(dirname); % returns a structure array for pos = 3:length(lst) %skip over first two items, as they are always '.' and '..' if lst(pos).isdir % check if current is a directory mydir([dirname '/' lst(pos).name]) else fprintf('%s\n', lst(pos).name); % print out file name end endend

When you shouldn’t use recursion

• Consider Fibonacci numbers

• fib(n) = fib(n-1) + fib(n-2)• fib(0) = 0• fib(1) = 1

Redundant Calculations

• To compute fib(n), we recursively compute fib(n-1). When that recursive call returns, we compute fib(n-2) using another recursive call– We have already computed fib(n-2) in the process

of computing fib(n-1)– We make two calls to fib(n-2)

Redundant Calculations

• Making two method calls would double the running time

• Compounding effect: each recursive call does more and more redundant work– Each call to fib(n-1) and each call to fib(n-2) makes

a call to fib(n-3); there are 3 calls to fib(n-3)– Each call to fib(n-2) or fib(n-3) results in a call to

fib(n-4), so 5 calls to fib(n-4)

Redundant Calculations III

• The recursive routine fib is exponential– Meaning the amount of work done grows

exponentially based on the input value n

recursion. what is recursion? solving a problem in terms of itself or repeating objects in a...

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