How To Apply Classic Graph Theory To Solve Real Problems Using Ruby
How To Apply Classic Graph Theory To Solve Real Problems Using Ruby

Today we are going to talk about Graph Theory.

This is the next installment in the “Practical Computer Science” series, where you will learn how to apply classic computer science concepts to solve real problems using Ruby.

You may have heard about binary trees, they look like this:

binary-tree

The thing is that a binary tree is just a specialized version of a graph, so that should give you an idea of how widespread graphs are.

Let’s start with an overview of graph theory fundamentals, then we are going to see what are some practical uses & how to implement this in Ruby!

Graph Fundamentals

A graph is composed of two elements:

  • Nodes (or vertices)
  • Edges

One node represents one element in the graph, like a city or a street, in a graph representing a map. While the edges represent the connections between the nodes.

If you look at a computer science or math book you will see a graph defined by this formula: G(V, E).

Where G means Graph, V is the set of vertices & E is the set of edges.

Graphs can be directed or undirected. Meaning that you can only go one direction (directed graph) or both directions (undirected graph).

The most popular type of graph is the Directed Acyclic Graph (DAG). Acyclic means that there are no loops, there is no way to backtrack.

Uses for Graphs

Now that we have seen an overview of the fundamentals, let’s see some common uses for graphs.

Using graphs you can do things like:

  • Find the shortest (or longest) path between two locations
  • Check if two things are related to each other
  • Build a recommendation engine
  • Analyze dependencies

Another example includes finding the best route to a destination (think GPS devices).

How to Implement & Use Graphs

You could write your own graph implementation, but for this article we are going to stick to the RGL gem which already implements one for us.

To create a basic graph using RGL:

This code produces the following graph:

graph-1234

You can get a graphical representation of your graph like this:

Then copy the output of that method on a site that can process the dot language. Like this one.

Alternatively, you can install Graphviz on your machine to produce the image locally.

Now that we have a graph, we may want to traverse it to find out information about it.

There are two basic algorithms for searching your graph:

  • Breadth-First Search (BFS)
  • Depth-First Search (DFS)

In BFS you get the closest nodes first & in DFS you go as deep as possible for every node. These algorithms can be implemented using the stack data structure.

The RGL gem already implements those algorithms for you:

Look at the graph again & follow the path these algorithms did using just your eyes (or you can use a finger too if you want). That will help you get a sense of what’s going on.

Weighted Graphs

We can add more information to a graph in the form of weights to make it more useful.

Weights are given to edges, which are the paths between two nodes (also known as “vertices”). These weights represent the cost of going from one point to another.

For example, if we have a map of a country in the form of a graph & we want to reach a certain destination in the shortest time possible, the weights would represent the distance between two cities.

usa-graph

Or if we have a computer network, the weights may represent how many hops it takes to reach a certain network.

“In computer networking, a hop is one portion of the path between source and destination. Data packets pass through bridges, routers and gateways as they travel between source and destination. Each time packets are passed to the next network device, a hop occurs.” – Wikipedia

Here’s a code example for a weighted graph:

We can now search for the shortest path from one point to another. And that’s exactly the topic of the next section!

Finding The Shortest Path

A popular algorithm for finding the shortest path inside a graph is “Dijkstra’s Shortest Path” algorithm.

Given a weighted graph, we can use Dijkstra’s algorithm to solve this question:

“What’s the fastest way to get from point A to point B?”

Here is a code example, using the RGL gem:

This tells us the shortest path from New York to Houston, using the information available in the graph.

Summary

You have learned what a graph data structure is & how to use it with the RGL gem.
You have also learned about common algorithms for working with graphs, like DFS, BFS & Dijkstra’s.

We wish this post could have made you more informed than you were before reading it.
If you have any question, please let us know. We’ll try to solve it at the earliest possible time.
Also please share this post on social media, use the share icons just below the post.
Don’t forget to subscribe to our blog.
Thank you for reading.

251 COMMENTS

  1. Hey,
    this article was copied word-for-word from my blog (www.blackbytes.info), please remove it.

    Thank you.

  2. Lovіng the Pets, Hating their Fleas – Perhaps the reality or tthe
    philosophy of ‘the yіn and thᥱ yang’ is certainly correct in humans’
    ԁaily lives. Ꭲhey slowly Ьuiloԁ iіn our consciousness until it beϲxomes a memory that is
    beіng cherished, recalleԀ with disdaion оr peгhaps forgotten.
    Not only wiⅼl this ҝeep dirt from getting all over your
    clean ɦouse though the carpets will stay in better condition.

  3. hello there and thank you for your info – I’ve certainly picked up anything new from right here.
    I did however expertise several technical points using this site,
    since I experienced to reload the site a lot of times previous to I could
    get it to load correctly. I had been wondering if your web hosting is OK?

    Not that I am complaining, but sluggish loading instances
    times will sometimes affect your placement in google and can damage your quality score if
    advertising and marketing with Adwords. Anyway I’m adding
    this RSS to my email and can look out for a lot more
    of your respective interesting content. Make sure you
    update this again soon.

  4. whoah this blog is wonderful i love studying your articles.
    Stay up the good work! You understand, lots of people are looking round for this info, you could help them greatly.

  5. You can obtain Critical Ops at this hyperlink of official website and create
    account, however if you’re looking the right way to hack
    Crucial Ops or for Vital Ops cheats, Critical Ops hack device,
    hack Important Ops, cheats Important Ops then keep
    here as a result of only right here you’ll discover the best Critical Ops hack
    for this sport that may add Credits and Blue Credit really quick.

  6. However, annoyingly when I read my mail my i – Phone would will
    download all of my recent emails. You can keep the data and email resistant to prying eyes
    with all the world’s best web based free email service.

  7. You really make it seem really easy along with your presentation however I find this
    matter to be really something which I think I might never
    understand. It sort of feels too complicated and very broad for me.
    I’m taking a look forward in your subsequent put up, I’ll try to get the cling of
    it!

  8. Walt Disney Travel Agent Kim Oehrlein said if you do it yourself, the positive is that you
    have total control over your reservation, which will allow to make your own Advance Dining Reservations and make any needed adjustments yourself.
    “Each time of year has something that sets
    it apart from others. The White House indicated that
    the president felt the need to personally lobby the Senate against Senator Hoeven’s
    amendment because it did not provide for a means of review by appropriate agencies of the
    federal government.

  9. Hello, I think your website might be having browser
    compatibility issues. When I look at your website in Safari, it looks
    fine but when opening in Internet Explorer, it has some overlapping.
    I just wanted to give you a quick heads up! Other then that, fantastic blog!

  10. Woah! I’m really digging the template/theme of this website.
    It’s simple, yet effective. A lot of times it’s challenging to get that “perfect balance” between usability and appearance.

    I must say that you’ve done a awesome job with this. Additionally, the blog loads extremely fast for me on Firefox.
    Superb Blog!

  11. I’m truly enjoying the design and layout of your website. It’s a very easy on the eyes which makes it much more pleasant for me to come here and visit
    more often. Did you hire out a designer to create your theme?
    Great work!

  12. If you’re inquisitive about common suggestions for Pokemon Duel,
    or how the combat system works , we have got you lined in our Pokemon Duel hub We’ll
    have more on Pokemon Duel all this week, so stay tuned to Prima
    Video games!

  13. Additionally, gamers must make some in app purchases
    time to time to spice up their powers in opposition to opponents but sadly if you do not have sufficient amount of sources then you definitely cannot even replace your characters and powers with time.

  14. Somebody essentially help to make seriously posts I’d state.
    That is the first time I frequented your website page and up to now?
    I surprised with the analysis you made to make this actual put up incredible.
    Magnificent process!

  15. An outstanding share! I’ve just forwarded this onto a coworker who has been conducting a little homework on this.
    And he in fact ordered me breakfast due to the fact that I stumbled upon it for him…
    lol. So allow me to reword this…. Thanks for the meal!!
    But yeah, thanks for spending time to discuss this subject
    here on your website.

  16. Video games comparable to jacks , paper football ,
    and Jenga require solely very portable or improvised gear and could be played on any flat level surface, whereas
    other examples, akin to pinball , billiards , air hockey
    , foosball , and desk hockey require specialized tables or other self-contained modules on which the
    sport is performed.

  17. Greetings I am so delighted I found your weblog, I really
    found you by mistake, while I was researching on Google for something else,
    Regardless I am here now and would just like to say thanks for a
    remarkable post and a all round entertaining blog (I also love
    the theme/design), I don’t have time to go
    through it all at the minute but I have saved it and
    also added in your RSS feeds, so when I have time I will be back to
    read much more, Please do keep up the awesome job.

  18. It is the spiritual center of Myanmar,are and many Myanmarese go on pilgrimages of  this country.
    Here and in nearby Laem Son National Park in Thailand,
    guests can observe Burmese macaques, one of only three primate
    species on Earth to have developed a culture of stone tool use.
    The possibility of seeing new frontiers, more diverse water creature
    and sea life keeps the experience surreal.

  19. , the adventure, the dream – round-the-world voyage
    now has become a reality for many well-to-do people.
    This is just a small part of what Bangkok is all
    about. Although it has been severely damaged by war, the
    remains of the Forbidden City still portray its former
    beauty.

  20. Bom ! É seu tema personalizado feito ou você baixá-lo de algum lugar?
    A projeto como o seu com alguns simples adjustements faria mesmo meu blog
    saltar . Por favor, deixe-me saber onde você conseguiu seu projeto
    . Muito obrigado

  21. I am extremely inspired with your writing abilities and also with the format for your weblog.
    Is that this a paid topic or did you modify it your self?

    Anyway keep up the excellent high quality writing, it’s rare
    to peer a great blog like this one these days.

  22. I was recommended this blog via my cousin. I am now not sure
    whether or not this put up is written by way of him as no one else understand such
    distinct approximately my difficulty. You are wonderful!
    Thank you!cheap nfl jerseys

  23. This is the right webpage for anyone who hopes to find out
    about this topic. You know so much its almost hard to argue with you (not that I
    really would want to…HaHa). You certainly put
    a brand new spin on a topic that has been written about for years.
    Excellent stuff, just wonderful!wholesale baseball jerseys China

  24. What’s up colleagues, how is everything, and what
    you would like to say concerning this article, in my view its genuinely
    awesome in favor of me.48227

  25. Today, I went to the beachfront with my children. I found a sea shell and gave it to my 4 year
    old daughter and said “You can hear the ocean if you put this to your ear.” She put the shell to her
    ear and screamed. There was a hermit crab inside and it pinched her ear.
    She never wants to go back! LoL I know this is totally off topic but I had to tell someone!

    Life_Belonging_To_The_First_Group_Of_Jerseys

  26. Hello are using WordPress for your site platform? I’m new to the blog world but
    I’m trying to get started and create my own. Do
    you require any html coding expertise to make your own blog?
    Any help would be greatly appreciated!cheap nba jerseys

  27. Hi there! Somebody in my Myspace group shared this website with us so I came to take a look.
    I’m certainly loving the information. I’m bookmarking
    and will be tweeting this to my followers! Excellent blog and great style and design.

  28. My spouse and I stumbled over here different web address and thought I might as well check
    things out. I like what I see so now i’m following you.
    Look forward to going over your web page again.
    cheap jerseys

  29. You have made some really good points there. I looked on the
    internet to learn more about the issue and found
    most individuals will go along with your views on this site.

    Brandon

  30. hi!,I like your writing so so much! share we keep up a correspondence extra approximately your post on AOL?
    I require a specialist on this house to unravel my problem.
    May be that is you! Having a look forward to see you.

    Cheap_Nfl_Jerseys_Wholesale

  31. This new Kritika the white knights hack is available with the expectation of complimentary
    so hat you never eever require to devote any cash to utilize
    it and you will have a respectaable time even though making
    use of this new Kritika thee white knights cheat
    therefore.

  32. I am extremely inspired along with your writing skills
    and also with the format in your blog. Is this a paid theme
    or did you customize it yourself? Either way stay up the excellent high
    quality writing, it’s uncommon to see a great weblog like this one today..discount nhl jerseys China

  33. Do you mind if I quote a couple of your articles as long as I provide credit and
    sources back to your site? My blog is in the very same area of interest as yours and my visitors would certainly benefit from a lot of the information you present here.
    Please let me know if this ok with you. Regards!wholesale nfl jerseys

  34. We are a gaggle of volunteers and starting a new scheme in our community.
    Your web site provided us with useful information to work on. You
    have done a formidable task and our entire group can be
    grateful to you.cheap nba jerseys

  35. comment diminuer la libido de mon mari augmenter libido femme apres accouchement augmentation de la libido
    feminine baisse de libido chez l’homme 35 ans manque de libido homme acupuncture points increase female libido booster la
    libido de mon homme comment stimuler la libido des femmes
    manque de libido femmes rhodiola rosea female libido libido supplements gnc
    perte de libido homme 50 ans effexor xr increase libido menopause libido testosterone comment booster sa libido apres la menopause perte
    de libido apres grossesse retrouver une libido apres
    accouchement absence de libido homme 60 ans lexomil
    libido femme libido enhancing supplements baisse libido femme comment baisser la libido chez la femme

  36. Pretty section of content. I just stumbled upon your site and in accession capital to assert that I
    acquire in fact enjoyed account your blog posts. Anyway I’ll be subscribing to
    your augment and even I achievement you access consistently rapidly.discount
    basketball jerseys China

  37. you are in point of fact a just right webmaster. The web site loading pace is incredible.
    It sort of feels that you are doing any distinctive trick.

    Furthermore, The contents are masterpiece. you have performed a great activity on this topic!discount mlb jerseys China

  38. And as much as he is enjoying all of the pick-and-pops
    with sweet shooters like Ryan Anderson and Eric Gordon, or the choose-and-rolls
    with the high-flying Clint Capela, Harden is virtually giddy
    about the probability to share some of his swag” with followers.

  39. Hey There. I found your blog using msn. This
    is a really well written article. I’ll make sure to bookmark it and return to read
    more of your useful info. Thanks for the post.
    I will certainly return.

  40. As with all the best water sneakers they function a drawstring fitting to permit you to take them off without having to cut the laces.

  41. I am really impressed with your writing skills and also with the layout on your blog.
    Is this a paid theme or did you customize it yourself?
    Either way keep up the nice quality writing, it’s rare to see a great blog like
    this one today.cheap baseball jerseys free shipping

  42. Hi! I understand this is sort of off-topic but I needed to ask.
    Does managing a well-established website like yours require a large
    amount of work? I’m brand new to blogging however I do write in my
    journal daily. I’d like to start a blog so I will be able to share my experience and feelings
    online. Please let me know if you have any kind of
    recommendations or tips for new aspiring bloggers. Thankyou!discount
    mlb jerseys

  43. Good day! I know this is kinda off topic but I’d figured I’d ask.
    Would you be interested in exchanging links or maybe guest
    writing a blog post or vice-versa? My blog addresses a lot of the same
    topics as yours and I feel we could greatly benefit from each other.
    If you happen to be interested feel free to shoot
    me an e-mail. I look forward to hearing from you! Superb blog by the way!

    wholesale NFL jerseys

  44. Magnificent goods from you, man. I’ve understand your
    stuff previous to and you are just too fantastic.
    I actually like what you have acquired here, really like what you’re stating and the way in which you say it.
    You make it entertaining and you still take care of to keep it wise.
    I can not wait to read far more from you. This is actually
    a terrific web site.cheap mlb jerseys from China

  45. Have you ever thought about writing an ebook or guest
    authoring on other blogs? I have a blog based upon on the same topics you discuss and would love to have you share some stories/information. I know my subscribers
    would enjoy your work. If you are even remotely interested, feel free to send me an e-mail.

  46. Undeniably believe that which you said. Your favorite reason seemed to be at the net the easiest factor to keep in mind of.
    I say to you, I definitely get annoyed at the same time as other
    folks consider issues that they just don’t realize about.

    You controlled to hit the nail upon the highest and defined
    out the entire thing without having side effect , other people could take a signal.
    Will likely be back to get more. Thankswholesale nfl jerseys

  47. Thank you for any other magnificent post. The place else may
    anybody get that kind of information in such an ideal approach of writing?
    I have a presentation subsequent week, and I’m at the look for such information.cheap nfl jerseys

  48. Wow that was unusual. I just wrote an very long comment but after I clicked submit my comment didn’t show up.
    Grrrr… well I’m not writing all that over again. Anyhow,
    just wanted to say wonderful blog!

  49. Wonderful blog! I found it while browsing on Yahoo News.
    Do you have any tips on how to get listed in Yahoo News?
    I’ve been trying for a while but I never seem to get
    there! Many thanks

  50. With havin so much written content do you ever run into any issues of plagorism or copyright violation? My site has a lot of
    exclusive content I’ve either written myself or outsourced but it looks like a lot of
    it is popping it up all over the web without my agreement.
    Do you know any techniques to help protect against content from being ripped
    off? I’d certainly appreciate it.discount baseball jerseys

  51. Hey very cool site!! Guy .. Excellent .. Wonderful .. I’ll bookmark your web site and take the feeds also?
    I’m satisfied to search out so many helpful information right here
    within the submit, we need develop extra strategies in this regard, thank you for sharing.
    . . . . .wholesale baseball jerseys China

  52. Hi there! I could have sworn I’ve been to your blog before
    but after browsing through some of the posts I realized it’s new to me.
    Nonetheless, I’m definitely delighted I stumbled upon it and I’ll be book-marking it and checking
    back often!cheap football jerseys China

  53. Woah! I’m really loving the template/theme of this blog.
    It’s simple, yet effective. A lot of times it’s difficult to get that “perfect balance” between superb usability and appearance.
    I must say that you’ve done a superb job with this. Additionally, the
    blog loads very quick for me on Chrome. Excellent Blog!

  54. Now, if there have been a recipe for the right pair of sneakers, then certainly the two primary components could be
    style and comfort, and the Adidas Originals Superstar
    2’s seem to have each qualities.

  55. Thieves can obtain charge card information in several ways, from picking through garbage for getting credit
    statements, to high-tech skimmers which work like normal card scanners but
    store numerous card numbers that move across them.

  56. I seriously love your site.. Pleasant colors & theme. Did you build this amazing site
    yourself? Please reply back as I’m hoping to create my own personal blog and
    would love to know where you got this from or exactly what the theme
    is named. Cheers!wholesale mlb jerseys free shipping from China

  57. These people do not buy limited-run footwear as a result of they
    need to put on them; they purchase to promote for a big revenue on eBay, Flight Club , Amazon and other third-party sites.

  58. Hi there, just became alert to your blog through Google, and found that it
    is truly informative. I am gonna watch out for brussels.
    I will appreciate if you continue this in future. Numerous
    people will be benefited from your writing. Cheers!wholesale mlb jerseys free shipping

  59. You actually make it seem really easy together with your presentation however I find this matter to be actually something which I
    believe I’d never understand. It sort of feels too complicated and extremely huge for me.
    I’m having a look ahead to your next post, I will attempt
    to get the grasp of it!cheap nfl jerseys

  60. I don’t know whether it’s just me or if perhaps everybody else
    experiencing issues with your blog. It appears as if some of the text within your posts are running
    off the screen. Can somebody else please comment and let me know if this is happening to them
    too? This could be a issue with my internet browser because
    I’ve had this happen previously. Cheersauthentic nhl jerseys
    China

  61. Please let me know if you’re looking for a author for your
    site. You have some really great posts and I believe I would be
    a good asset. If you ever want to take some of the load off,
    I’d love to write some content for your blog in exchange
    for a link back to mine. Please blast me an e-mail if interested.
    Thanks!msg214

  62. Fantastic goods from you, man. I’ve understand your stuff previous
    to and you’re just too fantastic. I actually like what you have acquired here,
    certainly like what you are stating and the
    way in which you say it. You make it enjoyable and you still care for to keep it sensible.
    I cant wait to read far more from you. This is actually a great web
    site.Good_Collection_Of_Football_Kits

  63. Hello there! I could have sworn I’ve been to
    this web site before but after going through many of the articles I realized it’s new to me.
    Regardless, I’m definitely happy I came across it and I’ll be book-marking it
    and checking back frequently!wholesale nfl jerseys from China

  64. First of all I would like to say terrific blog!
    I had a quick question which I’d like to ask if you don’t mind.
    I was curious to find out how you center yourself and clear your thoughts
    prior to writing. I have had a hard time clearing my mind in getting my thoughts
    out. I do take pleasure in writing but it just seems like the first 10 to
    15 minutes are lost just trying to figure out
    how to begin. Any ideas or tips? Appreciate it!cheap jerseys China

  65. Today, while I was at work, my cousin stole my iphone
    and tested to see if it can survive a 40 foot drop,
    just so she can be a youtube sensation. My iPad is now broken and she has
    83 views. I know this is completely off topic but I had to share it with someone!cheap baseball jerseys free
    shipping

  66. Store House präsentiert u.a. Mode von Woolrich,
    bei Shoe Fashion Factory sind unter anderem Schuhbrands
    wie Adidas, Hugo Boss und Timberland zu finden und im K.I.D.S.
    Shop gibt es prime Designermode für die Kleinen von z.B.
    Chloé, DKNY Kids und Escada Kids.

  67. Please let me know if you’re looking for a author for your blog.
    You have some really good posts and I think I would be a good asset.

    If you ever want to take some of the load off,
    I’d absolutely love to write some content for your
    blog in exchange for a link back to mine. Please shoot me an email if interested.
    Thanks!

  68. Nike pants, for instance, have a 35 ½ inch inseam most, of
    their 3XL size, which is for people with a waist measurement of forty seven ½” to 52 ½”.

    Adidas pants, in accordance with Dick’s Sporting Goods, have a most inseam of 34.75” (of their 3XL measurement, which is for people with a waist measurement of 40-forty three”).

  69. Nike and its Jordan brand nonetheless command
    the lion’s share of the U.S. footwear market, however rivals Adidas and
    Under Armour are chipping away on the firm’s decades-long dominant position.

  70. What’s Taking place i am new to this, I stumbled upon this I have discovered It absolutely useful and
    it has aided me out loads. I am hoping to give a contribution &
    assist different users like its helped me. Good job.
    cheap jerseys

  71. Regardless of how the court rules in this case, I think that we are loosing sight
    of the real intention of this Philadelphia law. You may want to recolor the image and use
    Word’s washout feature to achieve the same results.

    There is absolutely no other much better area to make new colleagues and just get enjoyable.

  72. Attractive section of content. I just stumbled upon your blog and in accession capital to assert that I acquire actually enjoyed
    account your blog posts. Any way I’ll be subscribing to your augment
    and even I achievement you access consistently rapidly.

  73. You really make it seem so easy with your presentation but
    I find this matter to be really something which I think I would never understand.
    It seems too complicated and very broad for me. I am looking
    forward for your next post, I will try to get the hang of it!

  74. If you examine the creasing on the vamp of this shoe (all leather-based sneakers will crease to come
    degree) you can see it creases in a completely completely different method, and doesn’t make
    the shoe look ugly, or trigger any untimely put on points.

  75. … [Trackback]

    […] There you will find 70234 more Infos: blog.learnstartup.net/p/how-to-apply-classic-graph-theory-to-solve-real-problems-using-ruby/ […]

  76. You could definitely see your enthusiasm within the article you write.
    The arena hopes for more passionate writers such as you who aren’t afraid to mention how
    they believe. At all times follow your heart.

  77. Whilst many firming shoe manufacturers simply tweak their designs,
    repackage them, and pass them off as new footwear,
    Skechers have designed a shoe which is kind of totally
    different to earlier versions of the Shapeups range.

  78. First of all I want to say terrific blog! I had a quick
    question in which I’d like to ask if you don’t mind.
    I was interested to know how you center yourself and clear your head before writing.
    I have had a tough time clearing my thoughts in getting my thoughts out there.

    I do enjoy writing but it just seems like the first 10 to
    15 minutes are wasted just trying to figure out how to begin. Any suggestions or hints?

    Many thanks! http://www.keller.com.au/news/

  79. hello there and thank you for your info – I
    have definitely picked up something new from right here. I did however
    expertise a few technical points using this site, as I
    experienced to reload the website lots of times previous to
    I could get it to load properly. I had been wondering if your web hosting
    is OK? Not that I am complaining, but sluggish loading instances times will often affect your placement in google
    and can damage your high-quality score if advertising and marketing with Adwords.
    Well I am adding this RSS to my e-mail and can look out for much more of your respective fascinating content.
    Make sure you update this again soon.

  80. Hi, I think your website might be having browser compatibility
    issues. When I look at your blog in Safari, it looks fine but when opening in Internet Explorer, it has some overlapping.
    I just wanted to give you a quick heads up! Other then that, very good blog!

  81. Nike co-founder Philip Knight shook up the sporting items trade that Adidas has dominated till the Seventies after putting into practice his thesis paper
    arguing that sneakers from decrease-value Japan may
    compete with pricier German-made variations.

  82. They even got stamps for just a big important letter
    they’re going to have to write to some big man which has
    a white beard and also a red suit soon. A crash course in customer development while using Lean Launchpad at
    Dunn Middle School spanning a 7 day period with 1 55 minute session everyday.

  83. Click the email address contact information drop-down menu and type in the
    correct current email address for your cellphone within the input field or
    opt for a previously registered email address contact info.
    Tap the search button (the magnification device . icon) and type “Adobe Flash,”
    then press “Search.

  84. What’s Taking place i’m new to this, I stumbled upon this I’ve discovered It absolutely useful and it has helped me out
    loads. I am hoping to contribute & help other users like its helped me.
    Good job.

  85. Hi there, just became alert to your blog through Google, and found that it’s really informative.

    I’m gonna watch out for brussels. I’ll appreciate if you continue
    this in future. Many people will be benefited from your writing.
    Cheers!

  86. Howdy just wanted to give you a quick heads
    up. The words in your content seem to be running off the screen in Internet explorer.

    I’m not sure if this is a formatting issue or something to do with
    web browser compatibility but I thought I’d post to let you know.
    The style and design look great though! Hope you get the issue solved soon.
    Thanks

  87. My partner and I absolutely love your blog and find most of your
    post’s to be just what I’m looking for. Would you offer guest writers to write content for yourself?

    I wouldn’t mind producing a post or elaborating on a lot of the
    subjects you write related to here. Again, awesome web site!

  88. I do not even know how I finished up right here, however I thought this publish
    was once great. I don’t understand who you’re but definitely you are going
    to a well-known blogger in the event you are not already.
    Cheers!

  89. Excellent article, I just passed this to a colleague who was
    carrying out a bit of research on that. And he actually bought me lunch because I found it for him.

  90. I think this is one of the most vital info for me.
    And i am glad reading your article. But want to remark on some general things, The site style is great,
    the articles is really nice : D. Good job, cheers

  91. Keep up the great piece of content, I read couple of posts on this site and
    I think that your web blog is really helpful as well as includes loads of exceptional information.

  92. Does your website have a contact page? I’m having trouble locating it
    but, I’d like to send you an email. I’ve got some creative ideas for your blog you might be interested in hearing.
    Either way, great website and I look forward to seeing it develop
    over time.

  93. Just desire to say your article is as astonishing. The clarity in your post is just excellent and i
    could assume you’re an expert on this subject.
    Well with your permission let me to grab your feed to keep updated with forthcoming post.
    Thanks a million and please keep up the gratifying work.

  94. We are a gaggle of volunteers and starting a brand
    new scheme in our community. Your web site offered us with valuable information to work on. You
    have performed an impressive process and our
    whole neighborhood can be grateful to you.

  95. Hello are using WordPress for your site platform? I’m new
    to the blog world but I’m trying to get started and set up my
    own. Do you require any coding expertise to make your
    own blog? Any help would be greatly appreciated!

  96. I have noticed that your website needs some
    fresh posts. Writing manually takes a lot of time,
    but there is tool for this boring task, search for: unlimited
    content Wrastain’s tools

LEAVE A REPLY

Please enter your comment!
Please enter your name here