This article is part of the article series "MIT Introduction to Algorithms."
<- previous article next article ->
MIT Algorithms

This is the eighth post in an article series about MIT's lecture course "Introduction to Algorithms." In this post I will review lecture twelve, which introduces an efficient search structure called Skip Lists.

Skip lists are an efficient data structure that can be used in place of balanced trees. Skip lists use probabilistic balancing rather than strictly enforce balancing and as a result the algorithms for insertion and deletion in skip lists are much simpler to implement and significantly faster than the equivalent algorithms for balanced search trees (see lectures 9 and 10 for search trees).

Here is the first publication ever on skip lists by their inventor William Pugh - Skip Lists: A Probabalistic Alternative to Balanced Trees

The lecture begins with Erik Demaine (professor) telling the students that before the lecture he implemented skip lists to see how fast it could be done. It took him 10 minutes to implement linked lists (on which skip lists are based) and took another 30 to implement skip lists themselves (+30 mins debugging ;) ). He says that they are so simple that at no point did he have to think while implementing them. Compare it to red-black trees for example, where you have to think really hard to implement the operations to maintain red-black tree properties (see lecture 10 on red-black trees).

The lecture continues with derivation of skip lists from scratch. First, a single sorted linked list is considered. Searching in a sorted linked list takes O(n). Then another sorted linked list is added. The added linked list is chosen to be a sublist of the first linked list. The elements that both lists share are linked together with another link (see my scanned lecture notes below to see how it looks). The search cost for such structure is O(2·sqrt(n)). Then another and another linked list is added, the search cost goes down to O(3·n1/3), O(4·n1/4), ..., O(k·n1/k). Now, what should k be? Take it to be log(n), we get running time of O(lg(n)·n1/lg(n)), which is O(2·lg(n)) - logarithmic time search structure!

The lecture also presents Search(x) method for finding the element x in a skip list, and Insert(x) method for inserting the element x in a skip list.

The rest of the lecture is allocated to the proof that the search operation in skiplists takes O(lg(n)) time with high probability.

You're welcome to watch lecture twelve:

Topics covered in lecture twelve:

  • [00:10] A new dynamic, balanced, randomized and simple search structure - skip lists.
  • [01:20] Review of other simple search structures - treaps, red-black trees, b-trees.
  • [03:55] Skip lists are a data structure that you will never forget, so simple it is.
  • [06:10] All the skip list operations take O(lg(n)) time almost guaranteed (with high probability).
  • [07:30] Implementing skip lists from scratch.
  • [09:05] Search cost in a single sorted linked list.
  • [10:30] Adding a second linked list, search cost analysis.
  • [11:16] Trick question: what is this sequence 14, 23, 34, 42, 50, 59, 66, 72, 79, 86, 96, 103, 110, 116, 125? (I won't reveal the answer ;) , you'll have to watch the lecture).
  • [14:35] Building skip list out of this sequence.
  • [17:07] Search(x) operation algorithm for a skip list.
  • [18:40] Example of search operation.
  • [21:20] What elements do go in the second linked list?
  • [25:30] Search cost of a skip list with two sorted linked lists.
  • [27:53] Skip list with three sorted link lists.
  • [29:14] Skip list with k sorted linked lists.
  • [29:25] What should k be? (It should ideally be lg(n)).
  • [32:10] Example of an ideal skip list.
  • [38:30] Skip lists roughly maintain their structure subject to updates (insert, delete).
  • [39:40] Insert(x) operation algorithm for a skip list.
  • [47:00] Building a random skip list.
  • [54:00] Rigorous analysis of search cost of skip lists.
  • [55:00] Theorem: with high probability, search in n element skiplist costs O(lg(n)).
  • [57:10] Definition: with high probability.
  • [01:00:00] Boole's inequality.
  • [01:06:55] Joke: probability that Search(x) algorithm takes more than O(lg(n)) time is smaller than a meteor striking the earth at the same time that your computer has floating point error at the same time earth explodes :D .
  • [01:07:45] Lemma: With high probability number of levels in a skiplist is O(lg(n)).
  • [01:13:00] Cool idea - analyze search time backwards.

Lecture twelve notes:

MIT Algorithms Lecture 12 Notes Thumbnail. Page 1 of 2.
Lecture 12, page 1 of 2.

MIT Algorithms Lecture 12 Notes Thumbnail. Page 2 of 2.
Lecture 12, page 2 of 2.

Have fun building skip lists! The next post will be about two really theoretical lectures on amortized analysis, competitive analysis and self-organizing lists.

PS. This course is taught from the CLRS book (also called "Introduction to Algorithms"):

PPS. This was one of the lectures that did not have a corresponding chapter in the book.

This article is part of the article series "MIT Introduction to Algorithms."
<- previous article next article ->

Comments

eivind Permalink
October 17, 2008, 09:30

would be nice if anyone actually could do handwriting anymore

August 12, 2014, 14:30

I really love your blog.I wanted to thank you for this great read!! I definitely enjoying each and every little bit of the article and I have you bookmarked your site to check out new stuff you post.

independence day images
happy independence day images.

FIBA Permalink
August 22, 2014, 13:30

I really love your blog.I wanted to thank you for this great read!! I definitely enjoying each and every little bit of the article and I have you bookmarked your site to check out new stuff you post.

fiba 2014
fiba world cup
fiba world cup 2014
Basketball world cup 2014
fiba world cup schedule
fiba world cup 2014 teams
fiba world cup 2014 groups
fiba live streaming,

October 17, 2008, 09:49

Good old skiplists. Cyrus Imapd uses a skiplist database format as one of its storage engines. Very interesting. I got to learn more than I ever wanted to know about the format while debugging it!

The main criticism is lack of cache coherency, and the number of locations you need to update when inserting or removing a node.

July 25, 2014, 12:33

Thanks Bron for commenting here i would like to know more about it at

Well explained i would like to know more thanx for this

Happy Friendship Day 2014
Happy Friendship Day 2014 Quotes
Happy Friendship Day 2014 Wallpapers.

October 17, 2008, 12:35

Great lecture...anyone catch the reference to The Hitchhiker's Guide?

August 30, 2013, 07:11

MiniSuit Starter Kit for Nokia Lumia 520 – Hard Snap-On Rubber Case, Screen Protector, Stylus
http://www.bestcellphonecase.com/nokia/minisuit-starter-kit-for-nokia-lumia-520-hard-snap-on-rubber-case-screen-protector-stylus.html

July 25, 2014, 12:33

Hey dude i had catched every lecture of it visit more at

Well explained i would like to know more thanx for this

Happy Friendship Day 2014
Happy Friendship Day 2014 Quotes
Happy Friendship Day 2014 Wallpapers.

October 17, 2008, 17:23

The trade off, of course, is memory, which is almost n^2....

Chris Permalink
October 17, 2008, 19:44

Thanks for writing this up!

ulzha Permalink
October 18, 2008, 14:51

Uau uau uau, apsveicu ar iekļūšanu LtU :D

Ryan Ingram Permalink
October 18, 2008, 20:50

These are MIT students? They're either incredibly shy, or not as smart as we've been led to believe; lots of the questions during the lecture were no brainers yet nobody is peeping up.

He even alludes to flipping a coin during the introduction, and then when he gets to the question where the obvious answer is "flip a coin", nobody seems to remember.

On the other hand, cool data structure. The "easy to implement" advantage is a nice bonus over balanced trees; are there performance benefits as well or are you just less likely to screw it up?

Also, no love for splay trees?

Ryan Ingram Permalink
October 18, 2008, 22:08

Actually, the beginning of that last message sounds way more douche-bag than I intended; no offense was meant. But you guys should really not be afraid to speak up! You're there to participate, and you'll miss the college experience when you are gone, so make the most of it.

(From a graduate of one of your competitors!)

Bob Foster Permalink
November 06, 2008, 01:12

Cognitive dissonance. Don Knuth re-uses code?

Bob Foster Permalink
November 06, 2008, 01:15

Oh, I see. The title remains the same but the quotation rotates. This causes the quoted people to seem to endorse a platitude they would never utter.

Peteris Ledins Permalink
November 08, 2008, 06:31

Ulzha: kas diez ir LtU?
Lietuvas Universitaate?

Ryan: there is some cool kids factor, as in: "is it cool to shout what you know?"
Experienced in classes I attended, where the back rows would always know the answer, but never call out loud. Well, maybe if the prof asked for the third time.

November 08, 2008, 08:55

Peteris Ledins, LtU is Lambda the Ultimate.

<latvian>Ps. paldies par komentariem algoritmu lekcijas.</latvian>

HELP WITH ANWERS Permalink
April 07, 2009, 18:09

plis i take data str n algo course moi uni info scie

José Luis Campanello Permalink
July 16, 2009, 21:21

Oh, i've read that paper long time ago (it's from early 90s if i'm not wrong). Stunning data structure.

I had not seen the video, so perhaps what follows is on it:

It turns out that not only skip lists are simple to implement, they can can also be "easily" changed to support multiple readers/writers. The structure (with n sized next pointer buckets) and the statistical distribution of bucket size allows for "localized" pointer management, resulting in more granular changes, making many inserts local. Compare this with AVL trees, where local insertions can alter large portions of the tree.

I've read (some time ago) that many high performance databases (Oracle, for instance) have changed to use this data structure for many operations because of it's better multi-threaded properties.

If my memory is not wrong, you can search ACM digital library and many papers on skip lists will pop out in the results, including multiple writers analisys.

Cool blog, BTW.

José Luis Campanello Permalink
July 18, 2009, 05:49

Well, i went to find the original paper. It was publicated on Communications of the ACM, Volume 33 Issue 6, June 1990, pages 668 thru 676.

If you are an ACM member, it can be downloaded (free for members, paid otherwise) with the link http://portal.acm.org/citation.cfm?id=78973.78977&coll=GUIDE&dl=GUIDE&CFID=44594004&CFTOKEN=83191267

There is another paper "Lock-free linked lists and skip lists" also on ACM, the link is http://portal.acm.org/citation.cfm?id=1011767.1011776&coll=GUIDE&dl=GUIDE&CFID=44594004&CFTOKEN=83191267

Adam Permalink
August 27, 2010, 15:40

The lecture is very cool.
Tough I have one question related to a possible step of efficiency in the Search operation:
When searching X in level and comparing it against the current node Y in level :
When X < Y is not TRUE, why not trying to ask if X == Y.
If so, then we found X and we don't need to go down to level .
This will save log() operations.Regards.

dude Permalink
January 07, 2013, 22:23

Can you un-private the video? Cool people like me can't watch it in it's current state.

June 04, 2014, 07:29

Thanks for the post. A worth reading article on awk.

faheen Permalink
June 22, 2014, 04:25

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

yeshwanth Permalink
July 10, 2014, 08:06
July 20, 2014, 18:14

Actually this post is about my personal interest and the mid of the post is truly informative and I love and elaboration quality of writer. I always try to find this type of info regarding this topic now here I got this thanks Happy Friendship Day SMS Friendship Day SMS Independence Day Images Happy Independence Day Images Independence Day Images India India Independence Day independence day speech independence day sms independence day wallpaper independence day message independence day songs again thanks for this great blog.

hana Permalink
July 21, 2014, 08:45

Thank you very much., You are a good writer.
In addition to art that will give you good .I congratulate you
تور چین - تور پکن - تور شانگهای
.

mahendher Permalink
July 23, 2014, 12:09

After read a couple of the articles on your website these few days, and I truly like your style of blogging.
I tag it to my favorites internet site list and will be checking back soon.
Please check out my web site also and let me know what you think.
ramadan mubarak messages
ramadan mubarak message
ramadan quotes
ramadan mubarak images
ramadan pictures.

sanchit Permalink
July 25, 2014, 21:32

Hey its great post really enjoyed this one i found this on google ,i think there are still websmasters who provide quality contents to the reader and hope to come back soon and by the way Eid Mubarak Wishes
Eid Mubarak Sms in Hindi
Eid Mubarak sms In English
Eid Mubarak sms in Urdu
Eid Mubarak Wallpapers
Eid Mubarak Greetings
Eid Mubarak cardsto all of you also you can check Happy eid Mubarak 2014
really a great stuff

ttttttt Permalink
July 29, 2014, 11:07

Binary umbrella coding is best, do you know about that? Your post is worth my time and I sincerly thank you for spreading and sharing your valuable knowledge and information, which many less people do these days! Free Cash Formula, Free Cash Formula, Free Cash Formula, Free Cash Formula, Free Cash Formula, Free Cash Formula, Free Cash Formula Review, Free Cash Formula Review, 2K A DAY System, 2K A DAY System, 2K A DAY System, 2K A DAY System, 2K A DAY System, 2K A DAY System, 2K A DAY System Review, 2K A DAY System Review, Friendship Day Quotes, Friendship Day Quotes, Friendship Day Quotes, Friendship Day Quotes, Friendship Day Quotes, Friendship Day Quotes, Friendship Day Quotes, Friendship Day Quotes, Friendship Day Quotes, Happy Friendship Day Quotes, Happy Friendship Day Quotes, Happy Friendship Day Quotes, Daily Cash Creator, Daily Cash Creator, Daily Cash Creator, Daily Cash Creator, Daily Cash Creator, Daily Cash Creator, Daily Cash Creator Review, Daily Cash Creator Review, The $100K Club, The $100K Club, The $100K Club, The $100K Club, The $100K Club, The $100K Club, The $100K Club, The $100K Club, The $100K Club, The $100K Club Review, Binary Predictor, Binary Predictor, Binary Predictor, Binary Predictor, Binary Predictor, Binary Predictor, Binary Predictor Review, Free Cash Formula, Free Cash Formula, Free Cash Formula, Free Cash Formula, Free Cash Formula, Free Cash Formula, Free Cash Formula Review, Free Cash Formula Review, 2K A DAY System, 2K A DAY System, 2K A DAY System, 2K A DAY System, 2K A DAY System, 2K A DAY System, 2K A DAY System Review, 2K A DAY System Review, Friendship Day Quotes, Friendship Day Quotes, Friendship Day Quotes, Friendship Day Quotes, Friendship Day Quotes, Friendship Day Quotes, Friendship Day Quotes, Friendship Day Quotes, Friendship Day Quotes, Happy Friendship Day Quotes, Happy Friendship Day Quotes, Happy Friendship Day Quotes, Daily Cash Creator, Daily Cash Creator, Daily Cash Creator, Daily Cash Creator, Daily Cash Creator, Daily Cash Creator, Daily Cash Creator Review, Daily Cash Creator Review, The $100K Club, The $100K Club, The $100K Club, The $100K Club, The $100K Club, The $100K Club, The $100K Club, The $100K Club, The $100K Club, The $100K Club Review, Binary Predictor, Binary Predictor, Binary Predictor, Binary Predictor, Binary Predictor, Binary Predictor, Binary Predictor Review, Free Cash Formula, Free Cash Formula, Free Cash Formula, Free Cash Formula, Free Cash Formula, Free Cash Formula, Free Cash Formula Review, Free Cash Formula Review, 2K A DAY System, 2K A DAY System, 2K A DAY System, 2K A DAY System, 2K A DAY System, 2K A DAY System, 2K A DAY System Review, 2K A DAY System Review, Friendship Day Quotes, Friendship Day Quotes, Friendship Day Quotes, Friendship Day Quotes, Friendship Day Quotes, Friendship Day Quotes, Friendship Day Quotes, Friendship Day Quotes, Friendship Day Quotes, Happy Friendship Day Quotes, Happy Friendship Day Quotes, Happy Friendship Day Quotes, Daily Cash Creator, Daily Cash Creator, Daily Cash Creator, Daily Cash Creator, Daily Cash Creator, Daily Cash Creator, Daily Cash Creator Review, Daily Cash Creator Review, The $100K Club, The $100K Club, The $100K Club, The $100K Club, The $100K Club, The $100K Club, The $100K Club, The $100K Club, The $100K Club, The $100K Club Review, Binary Predictor, Binary Predictor, Binary Predictor, Binary Predictor, Binary Predictor, Binary Predictor, Binary Predictor Review Thanks a lot! You rock! your bolg also rock, coding you provide is nowhere to be found.

August 09, 2014, 05:59

Independence day india
india Independence day
Independence day of india
Independence Day Speech
I liked your post and I will say in a beautiful manner you have described it.

sanchit Permalink
August 11, 2014, 18:58

I am very happy to visit this site and hope to visit here again and again. I have bookmarked your site due to the interesting and relevant stuff found here. Also I have recommended the same to my other friends.
Independence day india
Independence Day Images

Independence Day Wallpapers

Independence Day Speech in English

Independence Day Quotes

Independence Day Songs

Independence Day Poems

Independence Day SMS

This is my site, you can visit it and leave your valuable feedback.

sanchit Permalink
August 13, 2014, 12:21

I am very happy to visit this site and hope to visit here again and again. I have bookmarked your site due to the interesting and relevant stuff found here. Also I have recommended the same to my other friends.
Independence day india
Happy Independence day India
Desh Bhakti songs
Patriotic songs
Whatsapp Status
Fb status
Independence Day Speech in English

Independence Day Quotes

Modi Speech on Independece Day 2014
PM Modi LIVE Speech on Independece Day 2014

This is my site, you can visit it and leave your valuable feedback.

Rosy Permalink
August 22, 2014, 14:50

This article is quite helpful and informative too. I enjoyed a lot. Thanks for sharing such a great article.

Amazon Promotional Code 2014
Amazon Promotional Code
Amazon Promotional Code for August 2014

Thanks for sharing such a great article.

hana Permalink
September 09, 2014, 06:41

Until now I did not notice these things. I'm glad you've mentioned. I understand what you mean. I must thank you for this article
دستگاه حضور و غیاب - خرید ملک در ترکیه - قیمت خودرو - سنگ کورین - خرید و فروش خودرو - otomatik kapı
.

sanchit Permalink
September 21, 2014, 11:53

Asian Games Incheon 2014 Opening Ceremony Event Place : Hello every one. So if you are here to search for the Incheon Asian Games 2014 Venue or say Incheon Asian Games 2014 Event Place or say Asian Games 2014 Incheon Opening day Place or Live News of Asian games 2014 Live streaming or Incheon Asian Games 2014 Live Streaming then you are on the right website.

Leave a new comment

(why do I need your e-mail?)

(Your twitter name, if you have one. (I'm @pkrumins, btw.))

Type the first letter of your name: (just to make sure you're a human)

Please preview the comment before submitting to make sure it's OK.

Advertisements