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Computer Algorithms: Graph Best-First Search

Stoimen — Mon, 24 Sep 2012 10:44:53 +0000

Introduction

So far we know how to implement graph depth-first and breadth-first search. These two approaches are crucial in order to understand graph traversal algorithms. However they are just explaining how we can walk through in breadth or depth and sometimes this isn’t enough for an efficient solution of graph traversal.

In the examples so far we had an undirected, unweighted graph and we were using adjacency matrices to represent the graphs. By using adjacency matrices we store 1 in the A[i][j] if there’s an edge between vertex i and vertex j. Otherwise we put a 0. However the value of 1 gives us only the information that we have an edge between two vertices, which is not always enough when designing graphs.

Indeed graphs can be weighted. Sometimes the path between two vertices can have a value. Thinking of a road map we know that distances between cities are represented in miles or kilometers. Thus often representing a road map as a graph, we don’t put just 1 between city A and city B, to say that there is a path between them, but also we put some meaningful information – let’s say the distance in miles between A and B.

Note that this value can be the distance in miles, but it can be something else, like the time in hours we’ve to walk between those two cities. In general this value is a function of A and B. So if we keep the distance between A and B we can say this function is F(A, B) = X, or distance(A, B) = X miles.

Of course in this particular example F(A, B) = F(B, A), but this isn’t always true in practice. We can have a directed graph where F(A, B) != F(B, A).

Here I talk about distance between two cities and it is the edge that brings some additional information. However sometimes we have to store the value of the vertices. Let’s say I’m playing a game (like chess) and each move brings me some additional benefit. So each move (vertex) can be evaluated with some particular value. Thus sometimes we don’t have a function of and edge like F(A, B), but function of the vertices, like F(A) and F(B).

In breadth-first search and depth-first search we just pick up a vertex and we consecutively walk through all its successors that haven’t been visited yet.

In order to walk through an unweithed graph using DFS, we chose consecutively each successor of node i!

So in DFS in particular we started from left to right in the array above. So the first node that has to be explored is vertex “1”.

0: [0, 1, 0, 0, 1, 1]

However sometimes, as I said above, we have weighted graphs, so the question is – is there any problem, regarding to the algorithm speed, if we go consecutively through all successors. The answer in general is yes, so we must modify a bit our code in order to continue not with the first but with the best matching successor. By best-matching we mean that the successor should match some criteria like – minimal or maximal value.

Overview

In the following example we see that some of the successors of vertex 0 are very far from it, while others are closer. Thus 4 has the value of 5, while node 1’s value is 2 and 5 is 1.

Weithed graph brings us more information about the successors of a given vertex. Thus we have to chose carefully which one to get first in our path exploration!

0: [0, 2, 0, 0, 5, 1]

In this case if we’re searching for the shortest path between 1 and 3, although 1 and 4 are the first two successors in the adjacency matrix of the “start” vertex, we don’t choose them since there’s a better solution – going through node 5.

In best-first search we continue the path to the target through the best-matching successor!

Problems

The question is – are we sure that by choosing node 5, we’ll find the best path? Even more! Is there a path through node 5? As we see on the image below both cases are possible.

Somtimes best-first search doesn’t find the “best” (shortest/longest/cheapest) path to the target!

Practically best-first search is identical with depth-first search, with the main difference that we choose the best-matching successor instead of choosing the first matching successor. So we’re sure that we’re going through all the successors but in some particular order, different from DFS. Thus we know that if there’s a path we’ll find it.

However even if we find the path between A and B, we can’t be sure that there is not a better path. We only know that this path is the best so far.

Another question is – how can we find the best matching successor effectively. Well if we’re looking for the minimal or maximal value one possible solution is to sort the array of successors.

The difference between depth-first and best-first is that we change the order of chosing the next successor!

0: [0 => 0, 1 => 2, 2 => 0, 3 => 0, 4 => 5, 5 => 1]
// sorted by value
0: [5 => 1, 1 => 2, 4 => 5, 0 => 0, 2 => 0, 3 => 1]

Another good approach will be to use priority queues or heaps.

Thus on every step we’ll get the best matching successor.

Code

In general best-first search uses the ground of depth-first search, so its implementation isn’t more difficult! The following PHP code snippet shows the very small difference between these two algorithms.

class Graph 
{
    protected $_len = 0;
    protected $_g = array();
    protected $_visited = array();
    
    public function __construct()
    {
        $this->_g = array(
            array(0, 2, 0, 0, 5, 1),
            array(1, 0, 3, 0, 0, 0),
            array(0, 2, 0, 8, 0, 0),
            array(0, 0, 3, 0, 5, 0),
            array(1, 0, 0, 8, 0, 1),
            array(1, 0, 0, 0, 5, 0),
        );
        
        $this->_len = count($this->_g);
        
        $this->_initVisited();
    }
    
    protected function _initVisited()
    {
        for ($i = 0; $i < $this->_len; $i++) {
            $this->_visited[$i] = 0;
        }
    }
    
    public function bestFirst($vertex)
    {
        $this->_visited[$vertex] = 1;
    
        echo $vertex . "\n";
        
        asort($this->_g[$vertex]);
        
        foreach ($this->_g[$vertex] as $key => $v) {
            if ($v > 0 && !$this->_visited[$key]) {
                $this->bestFirst($key);
            }
        }
    }
}

$g = new Graph();
// 2 1 0 5 4 3
$g->bestFirst(2);

Application

Best-first search is a typical greedy algorithm. In its principles lies the main greedy approach of chosing the best possible solution so far. It is important to note that depth-first search and breadth-first search are the very basic graph walk through approaches, but they can be also widely extended in order to solve more complex problems.

A JavaScript Trick You Should Know

Stoimen — Tue, 12 Jul 2011 14:49:46 +0000

JavaScript Strings

You should know that in JavaScript you cannot simply write a multilined code chunk, i.e. something like this:

var str = 'hello
world';

This will result in an error, which can be a problem when you deal with large strings. Imagine a string containing some HTML that should be injected via innerHTML. Now one possible solution is to concatenate two or more strings, by splitting the initial string.

var str = 'hello'
        + 'world';

Howerver this solution result in some additional operations like concatenation. It can be very nice if there was something like the PHP multiline string format.

The PHP Example

In PHP you have the same use case with strings. You can have a string concatenated over multiple lines, but you cannot split lines like so:

// the following line is wrong
$str = 'hello
world'; 

// a possible solution to the line above is
$str = 'hello'
     . 'world';
// which is very similar to the js solution in the second example

// ... but in PHP there is the so called HEREDOC
$str = <<
The heredoc gives us the power to write multiline strings. 
The JavaScript Trick
Actually in JavaScript there is something that's exactly what Heredoc is meant to be for PHP. Take a look at the last example:
var text = <>
this 

is
my
multi-line
text
.toString();

document.body.innerHTML = text;

Thus you can write multiline strings in the JavaScript code.

Related posts:

Powerful PHP: Less Known String Manipulation 
JavaScript Flexibility – Regex match() 
Flexible JavaScript – Splitting Strings 
Flexible JavaScript – Replace in a String

jQuery.unbind()

Stoimen — Tue, 05 Apr 2011 08:51:08 +0000

Binding Problems

jQuery JavaScript Library

Typically in jQuery when you bind an HTML element with some event this event is fired every time the user (client) triggers it. Thus when you’ve a click event attached on a button you can click it as many times as you want. In some rare cases this can be tricky. Let’s imagine the following scenario.

The user clicks on a “vote” button.
Some AJAX calls are performed.
After a successful AJAX call you setup a cookie to deny further votes from this machine.

This seems to be pretty well known scenario, but as the click event is attached to the button there is enough time to click several times on the “vote” button and to vote several times. In this case before the cookie is set the user can vote more than once.

Demo

So one possible solution is to unbind the click event. In jQuery terms this is done by …

$.unbind()

With this method you can simply unbind the “click” event or whatever event there is attached on the desired HTML element. So now in the first case we had:

$(document).ready(function() {
    $('input').click(function() {
        $.ajax({ url : 'http://stoimen.com/projects/jquery.unbind/example1.php'
               , success : function() {
                    $('p').append('You\'ve voted successfully!
');
               }    
        }); 
    }); 
});

Now you can replace the code above with:

$(document).ready(function() {
    $('input').click(function() {
        $('input').unbind();
        $.ajax({ url : 'http://stoimen.com/projects/jquery.unbind/example1.php'
               , success : function() {
                    $('p').append('You\'ve voted successfully!
');
               }    
        }); 
    }); 
});

As you can see on the demo here now the button cannot be clicked more than once.

Demo

Cookies + unbind()

There is a problem however. After reloading the page the event is attached again, so the user can click on the button once more. The obvious solution is to combine both the cookie and the unbind() as shown in the following pseudo code.

$(document).ready(function() {
    if (cookie is not set) { 
        $('input').click(function() {
            $('input').unbind();
            $.ajax({ url : 'http://stoimen.com/projects/jquery.unbind/example1.php'
                   , success : function() {
                        $('p').append('You\'ve voted successfully!
');
                        set cookie
                   }    
            }); 
        });
    } 
});

Download Images with PHP

Stoimen — Tue, 18 Jan 2011 13:48:38 +0000

As it seems one possible solution while trying to download images with PHP is to write a “client” to do so. Will it be with cURL, Zend Framework or some other tool – it doesn’t matter.

However one of the most used approaches is simply with file_get_contents and file_put_contents. I’m not sure whether I wrote already about this or not, but this solution simply looks something like this.


file_put_contents('/path/to/file', 
                  file_get_contents('http://www.example.com/source.image');

In fact a client will give you more control over the process, to handle errors, etc. So maybe this is a better solution.

More on CSS Optimization

Stoimen — Fri, 07 Jan 2011 14:40:14 +0000

As CSS files are first downloaded to the client and then executed, the main optimization is to make those files smaller. But that doesn’t mean only minifing!

The Minification Process

While with minification you can strip all the symbols that only take space, but are useless when the browser parses the file, there are some other techniques, which in fact aren’t so simple, to speed up the loading process. By minification you get rid of the white spaces, tabs, new lines, etc., but the file may remain too large.

Useless Rules

Yes, sadly the browser doesn’t need all those white spaces, actually web developers need them. Just because this makes the file more readable, or readable at all. However most of the web applications have one large CSS file, typically named layout.css, main.css or whatever, that contains all the rules for the entire application. In many of the cases one page of a site doesn’t need all the rules for the site, so a possible solution is to remove all of the rules that aren’t used.

There are tools that may help you do the job. Such a tool, that I’m using is the Firefox add-on – Dust-Me Selectors. Of course there are a lot other tools doing the same job, so it’s up to you to pick up one.

After removing all those useless rules you’ll see that the size of the file can be something like 20% of the size of the source file. This is interesting to note, because most of the time the minification cannot give you such performance benefit. In fact this comes with some issues.

One Request or What?

This is the dilemma of the web programming, isn’t it? However thus you can split the main CSS file to few smaller files, but they are named differently so you cannot expect the browser to cache them when the user visits the site for the first time.

The case must be, of course, tested against various situations, so you can decide what suits you. However the typical scenario is to optimize only few of the pages, as they might be the most visited – as for example the homepage. If two of the pages are mainly visited, than you can make custom, small, CSS files for them with only the necessary rules, and let the other pages with the main CSS file. If you’ve a lot of returning visitors, you can be sure the custom files will be cached (if the client cache is turned on) and the second time somebody visits the “homepage” for example the page will load faster.

5 PHP String Functions You Need to Know

Stoimen — Fri, 17 Sep 2010 11:19:43 +0000

The Task

First of all what we’d like to achieve? The task is to convert a string, most of the cases single word, by capitalize the first letter. In my case I’ve the world countries names all lower cased, while I need them with first letter capitalized. In example “united states” must become “United States”, but not “United states” or “UNITED STATES”. So here began the journey into PHP string functions, especially those for capitalization!

1. ucwords

The first thing you find in the PHP Manual is the ucwords function. It changes the first letter to a capital letter, but it does not do the job. Why? Well let me show you an example.

$str1 = 'foo bar';
$str2 = 'Foo bar';
$str3 = 'FOO BAR';
$str4 = 'фуу бар';

echo ucwords($str1); // Foo Bar
echo ucwords($str2); // Foo Bar
echo ucwords($str3); // FOO BAR
echo ucwords($str4); // фуу бар

Here we have four strings. A lower cased, an upper cased, a mixed cased and … a lower cased Cyrillic string. First of all the main reason why ucwords doesn’t fit here is because of the Cyrillic string. Whatever non-Latin string you have you can forget about capitalization. However the other strings conversions are also interesting. Take a look at the third string! Here the string remains “FOO BAR” instead of going “Foo Bar”, which simply means that this function only looks, and hopefully changes, the first letter.

So here we have two questions. How can we overcome the Cyrillic problem and how to “normalize” the UPPER CASE string?

2. ucfirst

This is another useful function in PHP. ucfirst as you may guess from its name converts a string by only changing its first letter. So “Foo bar” will remain “Foo bar”, while with ucwords it has become “Foo Bar”. Let’s see what this function does:

$str1 = 'foo bar';
$str2 = 'Foo bar';
$str3 = 'FOO BAR';
$str4 = 'фуу бар';

echo ucfirst($str1); // Foo bar
echo ucfirst($str2); // Foo bar
echo ucfirst($str3); // FOO BAR
echo ucfirst($str4); // фуу бар

Here even the first string has only one capital letter – “foo bar” became “Foo bar”, and yet again we’ve the Cyrillic string unchanged. It simply doesn’t help us here!

3. mb_convert_case

As a PHP developer you know what the “mb_” prefix means – multibyte. This is quite useful. You can convert the string whatever the encoding is, so perhaps we can overcome the Cyrillic problem. But before proceeding to tests, let’s take a look at the parameters of this function.

The first thing to note here is that mb_convert_case doesn’t contain the case in its name – upper or lower. There’s a second parameter, after the first which is the string itself, who setups that. Note that here you don’t have the typical camel case or capitals parameter name, but MB_CASE_TITLE (as you know in English the title is always capitalized):

echo mb_convert_case($str, MB_CASE_TITLE, ...

And a third one which specifies the encoding:

echo mb_convert_case($str, MB_CASE_TITLE, 'utf-8')

Now let’s see what we can achieve with it:

$str1 = 'foo bar';
$str2 = 'Foo bar';
$str3 = 'FOO BAR';
$str4 = 'фуу бар';

echo mb_convert_case($str1, MB_CASE_TITLE, 'utf-8'); // Foo Bar
echo mb_convert_case($str2, MB_CASE_TITLE, 'utf-8'); // Foo Bar
echo mb_convert_case($str3, MB_CASE_TITLE, 'utf-8'); // Foo Bar
echo mb_convert_case($str4, MB_CASE_TITLE, 'utf-8'); // Фуу Бар

As you can see now the Cyrillic problem doesn’t exists and mb_convert_case is intelligent enough to change “FOO BAR” into “Foo Bar” – as I said this is the English style titling. That is by no means the solution when you deal with capitalization with different encoding.

However there is another approach to overcome the all UPPER CASE conversion problem. A possible solution is to convert the string first to a lower case string.

4. strtolower

strtolower is very useful PHP string function and perhaps any PHP developer has used it at least once. But yet again – it does not do the job. Again because of the encoding problem.

echo strtolower('ФУУ БАР'); // #*&$(#*%#

As you can see the Cyrillic string cannot be lower cased! Let’s search again into the “mb_” universe.

5. mb_strtolower

This is the function. Again you’ve to specify the encoding:

echo mb_strtolower('ФУУ БАР', 'utf-8')

Conclusion

It doesn’t matter whether you’re native English speaker or not. Most of the web sites are multilingual and you cannot be sure what happens when you convert strings in Alphabets different from the Latin. Thus be careful even when everything seems to be OK with Latin string tests.

Secure Forms with Zend Framework

Stoimen — Fri, 09 Apr 2010 07:16:17 +0000

Maybe the correct title is not “with Zend Framework”, but “with PHP”, because the general approach I used is purely PHP and no Zend Framework dependency is used. However let me mention that ZF allows you to build forms with Zend_Form, which gives you an abstraction over the HTML forms with many goodies like validation, filtering and protection.

Zend_Form and Zend_Form_Element_Hash

Although the technique I’m using is doing the same thing, note that in ZF there’s a Zend_Form_Element_Hash which generates and validates the form, thus protecting you from CSRF attacks. The thing is that I didn’t use it because the form I’m protecting is not generated with Zend_Form, and I cannot benefit from everything ZF is giving to me. However you can easily reproduce the basic strategy with every form and every framework till it’s written in PHP.

What’s the solution?

It’s pretty simple and it’s described many many times around the web, simply generate a random hash, a possible solution is to use uniqid in combination with mt_rand and md5, thus you’d get quite strong hash.

Step two is to pass this generated hash, also stored in the session in a hidden value of the form. Of course now the most asked question is: but that’s visible to the source and thus everybody will have a valid hash.

There’s the trick. OK everybody will have a valid hash, but on submit the hash is validated against the SESSION variable, and as you know the session is specified between the browser (client) and the web server. Although the attacker may have a valid hash he must execute the attacking script from the same domain, possibly with the same browser, which makes the task rather difficult.

An Example

Let me show a breve example, it may help make things clearer.

1. First step – start the session

2. Second step – validate the form against the $_SESSION and generate a valid token

3. Third step – make a form

Demo here.

For more to test this you may try to make the same form somewhere else on the web and to point the action to http://www.stoimen.com/projects/php.secure.forms/! Without the session validation it’s absolutely sure you can post on the attacked server.

P.S. Now I’ve to admit that this have nothing to do with Zend Framework, however it’s good practice and thus may be used with every framework.