Follow me on Twitter for my latest adventures!

**55**Comments July 07, 2009

# On the Linear Time Algorithm For Finding Fibonacci Numbers

In this article I'd like to show how the theory does not always match the practice. I am sure you all know the linear time algorithm for finding Fibonacci numbers. The analysis says that the running time of this algorithm is O(n). But is it still O(n) if we actually run it? If not, what is wrong?

Let's start with the simplest linear time implementation of the Fibonacci number generating algorithm in Python:

def LinearFibonacci(n): fn = f1 = f2 = 1 for x in xrange(2, n): fn = f1 + f2 f2, f1 = f1, fn return fn

The theory says that this algorithm should run in O(n) - given the n-th Fibonacci number to find, the algorithm does a single loop up to n.

Now let's verify if this algorithm is really linear in practice. If it's linear then the plot of *n* vs. running time of LinearFibonacci(n) should be a line. I plotted these values for n up to 200,000 and here is the plot that I got:

**Note**: Each data point was averaged over 10 calculcations.

Oh no! This does not look linear at all! It looks quadratic! I fitted the data with a quadratic function and it fit nearly perfectly. Do you know why the seemingly linear algorithm went quadratic?

The answer is that the theoretical analysis assumed that all the operations in the algorithm executed in constant time. But this is not the case when we run the algorithm on a real machine! As the Fibonacci numbers get larger, each addition operation for calculating the next Fibonacci number "fn = f1 + f2 " runs in time proportional to the length of the previous Fibonacci number. It's because these huge numbers no longer fit in the basic units of computation in the CPU; so a big integer library is required. The addition of two numbers of length O(n) in a big integer library takes time of O(n).

I'll show you that the running time of the real-life linear Fibonacci algorithm really is O(n^2) by taking into account this hidden cost of bigint library.

So at each iteration *i* we have a hidden cost of O(number of digits of f_{i}) = O(digits(f_{i})). Let's sum these hidden cost for the whole loop up to n:

Now let's find the number of digits in the n-th Fibonacci number. To do that let's use the well-known Binet's formula, which tells us that the n-th Fibonacci number f_{n} can be expressed as:

It is also well-known that the number of digits in a number is integer part of log_{10}(number) + 1. Thus the number of digits in the n-th Fibonacci number is:

Thus if we now sum all the hidden costs for finding the n-th Fibonacci number we get:

There we have it. The running time of this "linear" algorithm is actually quadratic if we take into consideration that each addition operation runs proportionally to the length of addends.

Next time I'll show you that if the addition operation runs in constant time, then the algorithm is truly linear; and later I will do a similar analysis of the logarithmic time algorithm for finding Fibonnaci numbers that uses this awesome matrix identity:

Don't forget to subscribe if you are interested! It's well worth every byte!

## Comments

Hi all!

Just a note about the Python code:

fn is not needed here, and if one opts to use it anyway, need not be initialised before the loop. Furthermore, the variable names "f1" and "f2" are confusing here, because I expect f2 to be bigger than f1. I would write this code as:

Tested and works.

Without reading your analysis it's clear that LnearFibonacci computes all the fib numbers up to "n", and then throws all but the last number away. Next time it recomputes the same numbers again, so of course you get complexity O(n^2).

If the function filled an array then computation would be linear and only one call would be required.

You can always use the O(logN) algorithm to compute nth fibonacci number (or O(NlogN) using your analysis). Your article's point is that big numbers (ie. not implemented in the hardware) don't have constant time operations. :|

excuse me sir,

I m ram sandesh.I m doing my B.tech degree.I m interested in knowing this algorithm with time complexity of O(log n).can u please mail me back graph & algorithm.i guess u have done with matrix multiplication method...

Actually, the «O(log n)» algorithm is not O(n log n) but O(n^a) where a single call is also O(n^a). If you use naïve long multiplication, a is 2; if you use Karatsuba multiplication, a is log(3)/log(2), which is better.

yes, LinearFibonacci() is O(n), but you run it m number of times which means the total computation is O(mn) time. And on each run of LinearFibonacci() m = n so that's O(n^2). Why should that be a surprise?

Also, Richie is correct: if you cached your previous results, it would be O(n). You've only come up with an O(n^2) algorithm for a problem that could be solved in O(n) time.

Richie: your point about throwing the results away is valid when multiple calls to LinearFibonacci are made. However, it seems that Peteris considers only the performance of single call to LinearFib; by modifying LinearFib according to your suggestion the running time will still be quadratic.

However your point is noted when the application requires multiple Fibonacci numbers - dropping previously computed results is a bit foolish then.

It seems beside the point to bring up caching the results. This is a very good analysis of the given algorithm

Corey, Richie: I believe you are incorrect. The following is a naive O(n^2) algorithm for calculating the nth fibonacci number:

def fib(n):

if n == 0 or n == 1:

return n

else: return fib(n-1) + fib(n-2)

However, the OP's algorithm simply performs n additions to find the nth fibonacci number -- there is no recursion. To work out [LinearFibonacci(x) for x in xrange(1,n)] would be O(n^2), but LinearFibonacci(n) itself is just O(n) (counting additions as constant time).

As Madars says, I consider only the performance of a single call to LinearFibonacci.

For example, if I did a call to LinearFib(1000) and LinearFib(2000), from the analysis algorithm I would expect that LinearFib(2000) would take twice as much time as LinearFib(1000). But in practice it takes four times as much time. The caching would not help here, as you would still have to iterate from 1000 to 2000 without caching!

Corey & Richie - you are wrong. It's only the test where the code iterates through 1 to 16 for the parameters. The points on the graph are the time take to complete the operation for a single value of N.

Alexandru hits the nail on the head - "big numbers don't have constant time operations"

Common sense: theory does not (necessarily) match practice when the theoretical assumptions don't hold.

http://gmplib.org/manual/Fibonacci-Numbers-Algorithm.html

Not sure if this is still quadratic or not, but either way it's going to be an effective speed-up.

How about a closed form evaluation?

F(n) = (Phi^n - (1-Phi)^n) / sqrt(5)

Where Phi is the Golden Ratio.

Phi = (1 + sqrt(5)) / 2

In that case, you'll need phi and sqrt(5) to have enough precision to round correctly. For the 1000th Fibonacci number, you'd need both to have over 200 digits of accuracy. You then have to take your large (data-wise) decimal number and take it to a large power. Computationally, it would be much worse than repeated addition.

To David: It's not a naive

O(N^2)algorithm, it's actually O(Fib(n)) (assuming constant time addition).To Peteris: Nice analysis :) It was well worth every byte :)

Thanks for the analysis, sure surprised me.

If possible, I'd like to hear how you generated that graph and those formulas. I generally use gnuplot and latex/lyx, but I'm sure there are numerous tips I could learn from you :)

Why have you compared the running time of Fib(n) to Fib(n)? There's no reason why this should give you any information about the complexity of the function. You must have meant to compare the running time of Fib(n) to n, which should be linear. It's no surprise that you found a quadratic relationship since the running time is linear in n and Fib(n) is quadratic in n, so Fib(n) is quadratic with respect to the running time.

All you've done is point out that you have to know what you're doing and not blindly follow your textbook and it's assumptions.

its*

John Tantalo, no, I compared n to running_time(Fib(n)).

Dror Levin, here is how I did it:

First I created a Python program to time execution for an individual n:

Next I created a loop in shell to run the program 10 times for each individual value on n:

Next, I used an Awk program to calculate the average of 10 runs:

Finally I used the following script for GnuPlot to create the graph and fit the data with a quadratic function:

For the formulas I used this online latex generator.

Have fun!

Since you've introduced the matrix form for the fibonacci recurrence, I think it would interest your readers to decompose to PDP^-1 form and thereby derive the closed form formula one reader mentions.

Then why does the plot have "n-th fibonacci number" on the x axis? This should be "n".

And why do you say "the plot of LinearFibonacci(n) vs. running time of LinearFibonacci(n) should be a line"? This should be "the plot of n vs. running time of LinearFibonacci(n) should be a line".

John, that's a mistake right there. Will fix it now! Thanks for spotting.

I see by your sample code that you were indeed comparing n to the running time of fib(n). Please correct your graphs and analysis!

Learn all about memoization =]

http://en.wikipedia.org/wiki/Memoization

John Tantalo, I have now corrected the graph and analysis.

Brandon Pelfrey, sorry, but this time it has nothing to do with memoization! See one of my previous comments above!

But if we used a 128-bit processor (or created instructions to do a single-step 128-bit add) and updated our language to support these 128-bit integers, we'd return to O(n) time, since adding is once again an O(1) operation. I think that most modern processors support special case 128-bit-adds through SSE - so it's likely your Python Big-Number library that's the bottleneck?

Zach, you're missing the point, which is that in the real world, you need to take the finite capabilities of your computational unit into consideration when considering the order of your algorithm. Great analysis Peteris!

Zach, you're also just plain wrong. 128-bit integers only get you up to around f(200) or so (too lazy to calculate the exact number of bits, but it's close). Fibonacci numbers are *big*.

Anyway, any decent implementation will have a short linear part at the beginning, and then an n^2 part after that.

Unless, of course, you can play tricks to get your computation time down to constant again. No existing processor can do this, but a processor that had a configurable number of bits could stay linear for as long as it had computation units. Again, difference between theory and practice.

No technically is it exponential. This is a common problem used to trip up CS grad students during their qualifying exam. When finding the complexity of an algorithm, you must consider the number of steps with respect to the input. The input is log(n) and the number of steps in the for loop is n, hence the algorithm is exponential with respect to the input.

This is why the naive primality test:

fails to be O(n). Although long suspected to be in P, primality was not classified as such until the recent discovery of the AKS primality test.

See: http://en.wikipedia.org/wiki/Pseudo-polynomial_time for more info

Yes, in my comment I assumed addition time is constant. With long precision numbers it is not. So my comment was not entirely correct. :)

Another practical issue theoretical computer scientist often ignore (apart from machine code word sizes) is the memory hierarchy. I.e. the evaluation of an algorithm is ignorant of the locality of memory accesses. But in practice this makes a HUGE difference.

@ZachPruckowski: the 187th fibonacci number cannot be expressed in one 128-bit register, which renders the point rather moot.

The point is that physical limitation of the machine can and will increase computational time, and can make things inaccurate. Not realizing that fibonacci sequence can quickly exceed your register size can lead to unexpected increase in computational time, and not taking care while calculating your floating point calculation can cause erroneous results.

Just like the good old days, when int multiplications ran in time proportional to the number of bits set in the second operand ... and divisions ran *eventually*.

Interesting analysis & shows the (occasional) perils of abstraction. Thanks.

-----N

>In this article I’d like to show how the theory does not always match the practice

Just be careful to pick right theory. Theory saying that f1+f2 takes a constant time is useful in some situation, not in this one. You picked the right theory and it matches the practice.

Hi Cantonmat,

I looked at this a while ago because I was annoyed about stupid implementations of Fibonacci. I quickly noticed that the value of Fn quickly overflows fixed precision datatypes so as you point out you can't use fixed precision arithmetic and have to take account the non-trivial complexity of your operations.

I looks at the real complexity for Binet's formula and implement the better identity used by GMP

F2k = (Fk+1)^2 − (Fk−1)^2

Glimpses of Lisp: Fibonacci and the Netflix prize

As ZachPruckowski and other pointed out, the analysis is misleading. The skew is due to the big integer computation and has no bearing on the theory of asymptotic analysis. The syntactical sugar of python disguising the underlying big number implementation has mislead your analysis.

It is like expecting length(string) is O(1) operation, because you could express it as an say # operator.

The fact that several commentators are still confused show how deep rooted this misconception is.

Possibly one the most "simple yet insightful" posts I've read. Great work!

Happy Mothers Day Messages

Mothers Day Poems 2014

Mother's day Poems

Mothers day 2014 messages

Mother's day message

Mothers Day SMS

Mothers day Sayings

Mothers day Quotes

nice

bihar board result 2014

good

nice

bihar board result 2014

good

dkjahsadhlsd

Solidworks Tutorials

SolidWorks Tutorials for beginners

solidworks 2014

learning solidworks

solidworks simulation

3d cad tutorials

Hil

This is a very good analysis of the given algorithm

Thank you for clean and simply article. your article is wonderful. nice and informative.!

no Website Millionaire no Website Millionaire no Website Millionaire no Website Millionaire Review no Website Millionaire Review no Website Millionaire no Website Millionaire no Website Millionaire no Website Millionaire Review no Website Millionaire Review no Website Millionaire no Website Millionaire no Website Millionaire no Website Millionaire Review no Website Millionaire Review no Website Millionaire no Website Millionaire no Website Millionaire no Website Millionaire Review no Website Millionaire Review no Website Millionaire no Website Millionaire no Website Millionaire no Website Millionaire Review no Website Millionaire Review no Website Millionaire no Website Millionaire no Website Millionaire no Website Millionaire Review no Website Millionaire Review no Website Millionaire no Website Millionaire no Website Millionaire no Website Millionaire Review no Website Millionaire Review no Website Millionaire no Website Millionaire no Website Millionaire no Website Millionaire Review no Website Millionaire Review no Website Millionaire no Website Millionaire.

Thank you for clean and simply article. your article is wonderful. nice and informative.!

no Website Millionaire no Website Millionaire no Website Millionaire no Website Millionaire Review no Website Millionaire Review no Website Millionaire no Website Millionaire no Website Millionaire no Website Millionaire Review no Website Millionaire Review no Website Millionaire no Website Millionaire no Website Millionaire no Website Millionaire Review no Website Millionaire Review no Website Millionaire no Website Millionaire no Website Millionaire no Website Millionaire Review no Website Millionaire Review no Website Millionaire no Website Millionaire no Website Millionaire no Website Millionaire Review no Website Millionaire Review no Website Millionaire no Website Millionaire no Website Millionaire no Website Millionaire Review no Website Millionaire Review no Website Millionaire no Website Millionaire no Website Millionaire no Website Millionaire Review no Website Millionaire Review no Website Millionaire no Website Millionaire no Website Millionaire no Website Millionaire Review no Website Millionaire Review no Website Millionaire no Website Millionaire.

fathers day quotes

father's day quotes

happy fathers day quotes

fathers day quotes from daughter

fathers day messages

fathers day 2014 Quotes

Fathers Day Quotes

fathers day quotes

father's day quotes

happy fathers day quotes

fathers day quotes from daughter

fathers day messages

fathers day 2014 Quotes

Fathers Day Quotes

happy fathers day messages

fathers day greetings messages

Fathers day Messages from Daughter

fathers day sms

fathers day messages in hindi

happy fathers day Quotes from daughter

Happy Father Day Messages

Thanks for wonderful post. Your writing skill is very nice and informative.

12 Minute PayDay 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay Review 12 Minute PayDay Review 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay Review 12 Minute PayDay Review 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay Review 12 Minute PayDay Review 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay Review 12 Minute PayDay Review 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay Review 12 Minute PayDay Review 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay Review 12 Minute PayDay Review 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay Review 12 Minute PayDay Review 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay Review 12 Minute PayDay Review 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay Review 12 Minute PayDay Review 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay 12 Minute PayDay Review 12 Minute PayDay Review.

Ramadan Wallpaper

Ramadan Wallpaper HD Download

Ramadan Wallpaper 2014

Ramadan Wallpaper 2014 Download

Ramadan Wallpaper 2014 HD Images

Ramadan Wallpaper 2014 Images

Ramadan Wallpaper quotes

Ramadan wallpaper greetings

Ramadan wallpaper 2014 greetings

Ramadan Wallpaper 2014 quotes

Ramadan Kareem 2014 images

Ramadan Kareem Mubarak Images

Ramadan Kareem Mubarak 2014 images

Ramadan Screen Saver 2014 download

Ramadan Mubarak images

Ramadan images 2014

Ramadan Mubarak greetings

Wish you happy Ramadan 2014

Happy Ramadan Wallpaper 2014

Ramadan wallpaper 2014

Wish you happy Ramadan 2014

ramadan Hd wallpaper 2014

Wish you happy Ramadan wallpaper 2014

Ramadan 2014 images

Ramadan 2014 pictures

wish you happy Ramadan 2014 images

Wish you happy Ramadan 2014 pictures

Happy ramadan 2014 wallpaper

Ramadan Kareem 2014 Wallpaper

Ramadan Mubarak 2014 wallpaper

Wallpaper Ramadan 2014

Ramadan HD wallpaper

Ramadan 2014 HD wallpaper

wish you Ramadan 2014 HD wallpaper

Happy Ramadan HD wallpaper

Happy Ramadan 2014 HD wallpaper

Wish you Ramadan HD wallpaper Download 2014

Ramadan HD wallpaper Download 2014

Ramadan HD wallpaper 2014 Download

Ramadan HD wallpaper 1080p

Ramadan 2014 HD wallpaper 1080p

Best Ramadan 2014 HD wallpaper

Top 10 Ramadan 2014 HD wallpaper

Best Collections Of Ramadan 2014 HD wallpaper

Ramadan 2014 HD wallpaper For Android

Ramadan 2014 best Screensaver download

Ramadan 2014 Screensaver download

screensaver download for mobile Ramadan 2014

Ramadan 2014 HD Screensaver Download

Ramadan 2014 download best screen saver

Screensaver for 2014 ramadan

Ramadan mubarak screensaver download

Ramadan mubarak 2014 best screen saver

Ramadan Kareem Mubarak 2014 best screen saver

Ramadan mubarak 2014 best screen saver

screen saver 2014 download free for android

Ramadan Kareem Mubarak Greetings 2014

Best Collections of Ramadan 2014 Greetings

Best Collections of Ramadan 2014 Greetings

Top Collections of Ramadan 2014 Greetings

Ramadan 2014 Greetings

Ramadan 2014 Greetings Free

Ramadan 2014 Greetings Kareem best

Best Collections of Ramadan 2014 Kareem Greetings

Best Collections of Ramadan 2014 Kareem Greetings

Best Collections of Ramadan 2014 kareem greetings

Top Ramadan 2014 wallpapers

Top Ramadan 2014 pictures

Top ramadan 2014 pictures for pc

Top ramadan 2014 Wallpaper for Android

Ramadan 2014 best softwares

Ramadan 2014 Best Images of Desktop

Ramadan 2014 best pictures for desktop

ramadan 2014 collection of Hd wallpaper

Ramadan 2014 Best collections

Ramadan 2014 top Hd wallpaper

Ramadan 2014 in uk

When Ramadan 2014 in Uk

ramadan 2014 in United Kingdom

When Ramadan 2014 in United States

When ramadan 2014 in America

Ramadan date in uk

Ramadan date in uk

Ramadan facts in united kingdom

Ramadan date in France

Ramadan 2014 in france

Ramadan 2014 date in france

Ramadan 2014 date in Frace country

Happy Ramadan 2014 best wallpaper

Ramadan Wallpaper best collections

Ramadan Kareem Wallpaper collections

wish you happy Ramadan wallpaper 2014

happy Ramadan wallpaper 2014

wish you happy Ramadan Hd wallpaper

Happy Ramadan 2014 HD wallpaper

Hd Ramadan wallpaper 2014

Wallpaper 2014 Hd in Ramadan

Ramadan wallpaper 2014 for Desktop Background

Best Islamic Ramadan wallpaper 2014

Wallpaper 2014 for facebook Cover

Best Wallpaper 2014 Download

Ramadan HD wallpaper 2014 Download

When ramadan in India 2014

Ramadan in India 2014

Ramadan in India 2014

Ramadan Kareem Greetings

Ramadan kareem Mubarak Greetings

Ramadan kareeem 2014 Greetings

Ramadan 2014 kareem Greetings

Ramadan 2014 kareem greetings

Ramadan Kareem Mubarak 2014

Ramadan kareem Mubarak 2014

Ramadan kareem Mubarak 2014

Ramadan Kareem greetings 2014

Ramadan Kareem greetings 2014

Ramadan Kareem greetings 2014

Ramadan Kareem greetings 2014

Ramadan Kareem greetings 2014

Ramadan Mubarak Greetings 2014

Ramadan Mubarak Greetings 2014

Ramadan Mubarak Greetings 2014

Ramadan Mubarak Greetings 2014

Ramadan Mubarak Greetings 2014

Ramadan Mubarak Greetings 2014

Ramadan Mubarak Greetings 2014

Ramadan Mubarak Greetings 2014

Ramadan Mubarak Greetings 2014

Ramadan Mubarak Greetings 2014

Ramadan Mubarak Greetings 2014

Inspirational pictures of ramadan 2014

Inspirational pictures of ramadan 2014

Inspirational pictures of ramadan 2014

Inspirational pictures of ramadan 2014

Inspirational pictures of ramadan 2014

Inspirational pictures of ramadan 2014

Inspirational pictures of ramadan 2014

Inspirational pictures of ramadan 2014

Inspirational pictures of ramadan 2014

Inspirational pictures of ramadan 2014

Ramadan 2014 Wallpaper and Mubarak pics

Ramadan 2014 Wallpaper and Mubarak pics

Ramadan 2014 Wallpaper and Mubarak pics

Ramadan 2014 Wallpaper and Mubarak pics

Ramadan 2014 Wallpaper and Mubarak pics

Ramadan 2014 Wallpaper and Mubarak pics

Ramadan 2014 Wallpaper and Mubarak pics

Ramadan 2014 Wallpaper and Mubarak pics

Ramadan 2014 Wallpaper and Mubarak pics

Ramadan 2014 Wallpaper and Mubarak pics

Ramadan 2014 Wallpaper and Mubarak pics

Awesome Post Ramadan Wallpaper Ramadan Wallpaper HD Download Ramadan Wallpaper 2014 Ramadan Wallpaper 2014 Download Ramadan Wallpaper 2014 HD Images Ramadan Wallpaper 2014 Images Ramadan Wallpaper quotes Ramadan wallpaper greetings Ramadan wallpaper 2014 greetings Ramadan Wallpaper 2014 quotes Ramadan Kareem 2014 images Ramadan Kareem Mubarak Images Ramadan Kareem Mubarak 2014 images Ramadan Screen Saver 2014 download Ramadan Mubarak images Ramadan images 2014 Ramadan Mubarak greetings Wish you happy Ramadan 2014 Happy Ramadan Wallpaper 2014 Ramadan wallpaper 2014 Wish you happy Ramadan 2014 ramadan Hd wallpaper 2014 Wish you happy Ramadan wallpaper 2014 Ramadan 2014 images Ramadan 2014 pictures wish you happy Ramadan 2014 images Wish you happy Ramadan 2014 pictures Happy ramadan 2014 wallpaper Ramadan Kareem 2014 Wallpaper Ramadan Mubarak 2014 wallpaper Wallpaper Ramadan 2014 Ramadan HD wallpaper Ramadan 2014 HD wallpaper wish you Ramadan 2014 HD wallpaper Happy Ramadan HD wallpaper Happy Ramadan 2014 HD wallpaper Wish you Ramadan HD wallpaper Download 2014 Ramadan HD wallpaper Download 2014 Ramadan HD wallpaper 2014 Download Ramadan HD wallpaper 1080p Ramadan 2014 HD wallpaper 1080p Best Ramadan 2014 HD wallpaper Top 10 Ramadan 2014 HD wallpaper Best Collections Of Ramadan 2014 HD wallpaper Ramadan 2014 HD wallpaper For Android Ramadan 2014 best Screensaver download Ramadan 2014 Screensaver download screensaver download for mobile Ramadan 2014 Ramadan 2014 HD Screensaver Download Ramadan 2014 download best screen saver Screensaver for 2014 ramadan Ramadan mubarak screensaver download Ramadan mubarak 2014 best screen saver Ramadan Kareem Mubarak 2014 best screen saver Ramadan mubarak 2014 best screen saver screen saver 2014 download free for android Ramadan Kareem Mubarak Greetings 2014 Best Collections of Ramadan 2014 Greetings Best Collections of Ramadan 2014 Greetings Top Collections of Ramadan 2014 Greetings Ramadan 2014 Greetings Ramadan 2014 Greetings Free Ramadan 2014 Greetings Kareem best Best Collections of Ramadan 2014 Kareem Greetings Best Collections of Ramadan 2014 Kareem Greetings Best Collections of Ramadan 2014 kareem greetings Top Ramadan 2014 wallpapers Top Ramadan 2014 pictures Top ramadan 2014 pictures for pc Top ramadan 2014 Wallpaper for Android Ramadan 2014 best softwares Ramadan 2014 Best Images of Desktop Ramadan 2014 best pictures for desktop ramadan 2014 collection of Hd wallpaper Ramadan 2014 Best collections Ramadan 2014 top Hd wallpaper Ramadan 2014 in uk When Ramadan 2014 in Uk ramadan 2014 in United Kingdom When Ramadan 2014 in United States When ramadan 2014 in America Ramadan date in uk Ramadan date in uk Ramadan facts in united kingdom Ramadan date in France Ramadan 2014 in france Ramadan 2014 date in france Ramadan 2014 date in Frace country Happy Ramadan 2014 best wallpaper Ramadan Wallpaper best collections Ramadan Kareem Wallpaper collections wish you happy Ramadan wallpaper 2014 happy Ramadan wallpaper 2014 wish you happy Ramadan Hd wallpaper Happy Ramadan 2014 HD wallpaper Hd Ramadan wallpaper 2014 Wallpaper 2014 Hd in Ramadan Ramadan wallpaper 2014 for Desktop Background Best Islamic Ramadan wallpaper 2014 Wallpaper 2014 for facebook Cover Best Wallpaper 2014 Download Ramadan HD wallpaper 2014 Download When ramadan in India 2014 Ramadan in India 2014 Ramadan in India 2014 Ramadan Kareem Greetings Ramadan kareem Mubarak Greetings Ramadan kareeem 2014 Greetings Ramadan 2014 kareem Greetings Ramadan 2014 kareem greetings Ramadan Kareem Mubarak 2014 Ramadan kareem Mubarak 2014 Ramadan kareem Mubarak 2014 Ramadan Kareem greetings 2014 Ramadan Kareem greetings 2014 Ramadan Kareem greetings 2014 Ramadan Kareem greetings 2014 Ramadan Kareem greetings 2014 Ramadan Mubarak Greetings 2014 Ramadan Mubarak Greetings 2014 Ramadan Mubarak Greetings 2014 Ramadan Mubarak Greetings 2014 Ramadan Mubarak Greetings 2014 Ramadan Mubarak Greetings 2014 Ramadan Mubarak Greetings 2014 Ramadan Mubarak Greetings 2014 Ramadan Mubarak Greetings 2014 Ramadan Mubarak Greetings 2014 Ramadan Mubarak Greetings 2014 Inspirational pictures of ramadan 2014 Inspirational pictures of ramadan 2014 Inspirational pictures of ramadan 2014 Inspirational pictures of ramadan 2014 Inspirational pictures of ramadan 2014 Inspirational pictures of ramadan 2014 Inspirational pictures of ramadan 2014 Inspirational pictures of ramadan 2014 Inspirational pictures of ramadan 2014 Inspirational pictures of ramadan 2014 Ramadan 2014 Wallpaper and Mubarak pics Ramadan 2014 Wallpaper and Mubarak pics Ramadan 2014 Wallpaper and Mubarak pics Ramadan 2014 Wallpaper and Mubarak pics Ramadan 2014 Wallpaper and Mubarak pics Ramadan 2014 Wallpaper and Mubarak pics Ramadan 2014 Wallpaper and Mubarak pics Ramadan 2014 Wallpaper and Mubarak pics Ramadan 2014 Wallpaper and Mubarak pics Ramadan 2014 Wallpaper and Mubarak pics Ramadan 2014 Wallpaper and Mubarak pics oHack Anna University Paper Valuation Anna University result 2014 Thanks for sharing

clear and informative. I just feel your blog is my worth. thanks a lot!

Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Review Fast Income Club.

clear and informative. I just feel your blog is my worth. thanks a lot!

Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Fast Income Club Review Fast Income Club Fast Income Club Review Fast Income Club.

Nokia Lumia PC Suite free Downloaddrooling

Thank You For Sharing This Excellent Post,

Friendship Day 2014 Quotes

Happy Friendship Day Images

Friendship Day Images 2014

Happy Friendship Day 2014 SMS

Thank You For Sharing.

## Leave a new comment