A very common problem in computer programming is finding the longest increasing (decreasing) subsequence in a sequence of numbers (usually integers). Actually this is a typical dynamic programming problem.

Dynamic programming can be described as a huge area of computer science problems that can be categorized by the way they can be solved. Unlike divide and conquer, where we were able to merge the fairly equal sub-solutions in order to receive one single solution of the problem, in dynamic programming we usually try to find an optimal sub-solution and then grow it.

Once we have an optimal sub-solution on each step we try to upgrade it in order to cover the whole problem. Thus a typical member of the dynamic programming class is finding the longest subsequence.

However this problem is interesting because it can be related to graph theory. Let’s find out how.

Overview

We already know various ways to calculate the shortest paths in a graph. Indeed finding the single-source shortest path is a typical graph problem. To model such kind of solutions we definitely need a graph represented in our solution.

However the single-source shortest path isn’t a straight-forward problem. It depends on many factors. Thus for positive edges the algorithm of Edsger Dijkstra can be a perfect solution, but when the graph contains negative edges his algorithm is no longer useful.

In the presence of negative edges the Dijkstra’s algorithm doesn’t work and we’d better use the Bellman-Ford algorithm. It is interesting to note, that it was exactly Richard Bellman who first introduced the term “dynamic programming” in the 1940s.

In fact the Bellman-Ford algorithm was able to detect negative cycles. That’s too important, because in presence of negative cycles the shortest path problem is no longer well defined.

In the other hand when we’re talking about shortest paths in a DAG (Directed Acyclic Graph) we can find a faster (linear) solution. That’s because we’re sure that there are no cycles (not even negative cycles)!

Finding the shortest paths in a DAG is closely related to the topological sorting of the DAG. This gives us a linear representation of the vertices of the DAG and we can clearly calculate the distances from the starting node to all other nodes. Note that in a DAG we have one or more nodes that can be considered as starting nodes – which means they don’t have predecessors (incoming edges).

Following the words above is pretty hard to find what all these graph algorithms have to do with finding the longest increasing (decreasing) subsequence. Actually this problem is very closely related to the toplogical sort of the DAG and the problem of finding shortest paths in a DAG.

That’s because we can represent our sequence as a DAG. The only thing we must care about is to “connect” with directed edges those elements that form an increasing (decreasing) pair.

Thus the sequence from our example [1, 8, 2, 7, 3, 4, 1, 6] is going to look like this.

Another important thing to note is that we don’t search for shortest, but for longest path, since our task is to find the longest subsequence.

Pseudo Code

Our pseudo code for finding the shortest paths in a DAG was something like that.

1. Get the toplogically sorted list L of the DAG;
2. The starting node is s;
3. The distance to s equals to 0;
4. All other distances are initialized to ∞
5. For each node (v) in L\{s} do:
5.1. If dist(v) > dist(u) + w(u, v) then
5.1.1.  dist(v) := dist(u) + w(u, v)

In the above pseudo code “u” is every predecessor of “v”!

Now we must “reverse” the solution above in order to find the longest increasing subsequence. Note that we don’t care any more about the weight of the edges, thus we can simply substitute them with 1.

1. Get the sequence (L) as a toplogically sorted DAG;
2. For each (i) in L do:
2.1. S(i) := 1 + max(S(j), where (i, j) is an edge from the DAG);

Application

Finding the longest increasing subsequence can be very useful not only at the Google/Yahoo/Facebook interview, but also in various fields of statistics.

You’ve definitely seen the “share a link” screen in Facebook. When you paste a link into the box (fig. 1) and press the “Attach” button you’ll get the prompted cite parsed with a title, description and possibly thumb (fig. 2). This functionality is well known in Facebook, but it appears to be well known also in various social services. In fact Linkedin, Reddit, Dzone‘s bookmarklet use it.

Fist thing to notice is that this information, prompted by Facebook, is the same as the meta tag information. However there is a slight difference.

Facebook prefers for the thumb the image set into the <meta property=”og:image” … />. In the case above this tag appears to be:

And the image pointed in the SRC attribute is exactly the same as the one prompted by Facebook (fig. 3).

First thing to note is that the real thumb is bigger than the thumb shown in Facebook, so Facebook resizes it and the second thing to note is that there are more meta tags of the og:… format.

Meta Tags and The Open Graph Protocol

By default meta tags contain various information about the web page. They are not visible in the webpage, but contain some info about it. The most common meta tags are the title, description and keywords tags. They of course contain the title of the page, not that this can be different from the <title> tag, a short description of the page and some keywords describing the content of the page. They are well known also because the search engines make use of them when trying to collect information about the page and the process of SEO passes through it.

However the default HTML meta tags cannot contain everything. Thus for example you cannot point the preferable thumbnail for a webpage. The solution is the Open Graph Protocol. It comes with meta tags that can contain more and more valuable info. Such a tag is the og:image meta tag. Note that all the Open Graph (og) meta tags are defined by the og: prefix before the entity name. Thus og:image comes for images, while og:longitude for geo positioning.

That’s really useful, but how you can read them?

PHP, Meta Tags and Regexps

When you try to read information from a webpage source the first possible path is by using regular expressions. However PHP is smart enough to offer you some useful functions. Such a function is get_meta_tags(). As you may guess this method reads the meta tags by given URL.

$a = get_meta_tags('http://vimeo.com/10758212');
var_dump($a);

However this method can’t read Open Graph tags. So finally you’ve to use some regexps.

preg_match('//', $source, $matches);

Now you can grab the og:image tag. And even more – grab every image (<img>) from that page.

preg_match_all('/

Related posts:

How to Sanitize User Input in PHP? 
HTML Tags: <base> 
Automatically Upload Images with PHP Directly from the URI 
Download Images with PHP 


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