Run-length encoding is a data compression algorithm that helps us encode large runs of repeating items by only sending one item from the run and a counter showing how many times this item is repeated. Unfortunately this technique is useless when trying to compress natural language texts, because they don’t have long runs of repeating elements. In the other hand RLE is useful when it comes to image compression, because images happen to have long runs pixels with identical color.

As you can see on the following picture we can compress consecutive pixels by only replacing each run with one pixel from it and a counter showing how many items it contains.

In this case we can save only counters for pixels that are repeated more than once. Such the input stream “aaaabbaba” will be compressed as “[4]a[2]baba”.

Actually there are several ways run-length encoding can be used for image compression. A possible way of compressing a picture can be either row by row or column by column, as it is shown on the picture below.

The problem in practice is that sometimes compressing row by row may be effective, while in other cases the same approach is very ineffective. This is illustrated by the image below.

Obviously run-length encoding is a very good approach when compressing images, however when we talk about big images with millions of pixels it’s somehow natural to come with some lossy compression.

Overview

Lossy RLE is a very suitable algorithm when it comes to images, because in most of the cases large images do appear to have big spaces of identical pixel colors, i.e. when the half of the picture is the blue sky. By using lossy compression we can skip very short runs.

First we’ve to say how long will be the shortest run that we will keep in the compression. For instance if 3 is the shortest run, then runs of 2 consecutive elements will be skipped.

Of course if we set the shortest run to be only one element long, this will make our compression completely lossless, which isn’t very effective. However when we talk about millions of pixels even runs of three or more elements are very short, so it’s up to the developer to decide how long will be the shortest run.

Some Examples

Let’s first define the shortest run that we will keep untouched to be at least three element long.

The above image is compressed more effectively than the lossless pixel row from the previous picture.

The thing is how to merge short runs. For instance the following three runs have to be blended into one color run.

We can choose the middle color (option #1) or not, but this will always depend on the picture and it will be effective in some cases and ineffective in other.

Implementation

Implementing run-length encoding is easy in general. Here’s a simple PHP code that shows a lossy RLE.

/**
 * Compresses an input list of objects by losing some data using
 * run-length encoding
 * 
 * @param mixed $objectList
 * @param int $minLength
 */
function lossyRLE($objectList, $minLength)
{
	$len 		= is_string($objectList) 
			? strlen($objectList)		// string as an input stream
			: count($objectList);		// array as an input stream
	$j 		= 1;
	$compressed = array();				// compressed output
	
	for ($i = 0; $i < $len; $i++) {
		if (isset($objectList[$i+1]) && $objectList[$i] == $objectList[$i+1]) {
			$j++;
		} else {
			$l = count($compressed);
			// This is where RLE is converted to a lossy algorithm!
			// In case the run is shorter than a predefined length the
			// algorithm will skip these elements and will stretch the last
			// saved run.
			// NOTE: this logic can be changed in order to take other
			// decisions depending on the goals.
			if ($j < $minLength && $j < $l) {
				$compressed[$l-1]['count'] += $j;
			} else {
				$compressed[] = array('count' => $j, $objectList[$i]);
			}
			$j = 1;
		}
	}
	
	return $compressed;
}

$input = 'aaaabbaabbbbba';

// aaaaaaaabbbbbb
lossyRLE($input, 3);

The code above can be modified in order to work with more complex data. Let’s say we have a “pixel” abstraction as on the example above.

/**
 * Pixel abstraction
 */
class Pixel
{
	private $_color = null;
	
	public function __construct($color = '')
	{
		$this->_color = $color;
	}
	
	public function getColor() 
	{
		return $this->_color;
	}
}

/**
 * Inits the pixels array
 * 
 * @param array $pixels
 */
function init(array &$pixels = array())
{
	$colors = array('red', 'green', 'blue');
	
	for ($i = 0; $i < 100; $i++) {
		$pixels[] = new Pixel($colors[mt_rand(0, 2)]);
	}
}

/**
 * Compresses an input list of objects by losing some data using
 * run-length encoding
 * 
 * @param mixed $objectList
 * @param int $minLength
 */
function lossyRLE($objectList, $minLength)
{
	$len 		= is_string($objectList) 
			? strlen($objectList)		// string as an input stream
			: count($objectList);		// array as an input stream
	$j 		= 1;
	$compressed = array();				// compressed output
	
	for ($i = 0; $i < $len; $i++) {
		if (isset($objectList[$i+1]) && $objectList[$i] == $objectList[$i+1]) {
			$j++;
		} else {
			$l = count($compressed);
			// This is where RLE is converted to a lossy algorithm!
			// In case the run is shorter than a predefined length the
			// algorithm will skip these elements and will stretch the last
			// saved run.
			// NOTE: this logic can be changed in order to take other
			// decisions depending on the goals.
			if ($j < $minLength && $l > $j) {
				$compressed[$l-1]['count'] += $j;
			} else {
				$compressed[] = array('count' => $j, $objectList[$i]);
			}
			$j = 1;
		}
	}
	
	return $compressed;
}

$pixels = array();

// initializes the pixels array
init($pixels);

$compressed = lossyRLE($pixels, 3);

print_r($compressed);

Complexity

In general lossless RLE compelxity is linear – O(n) where n is the number of items from the input stream. Even with the small modification above the complexity remains linear. However we can modify the compression in a slightly different manner (in order to get the middle value from consecutive short runs). This will somehow affect the complexity of the algorithm, of course.

Application

Run-length encoding isn’t a very effective option when compressing texts, but for images where long runs of the identical pixels happen to occur it is quite useful.

Nevertheless RLE is easy to convert into a lossy algorithm, that makes it very suitable for image compression.

Relative encoding is another data compression algorithm. While run-length encoding, bitmap encoding and diagram and pattern substitution were trying to reduce repeating data, with relative encoding the goal is a bit different. Indeed run-length encoding was searching for long runs of repeating elements, while pattern substitution and bitmap encoding were trying to “map” where the repetitions happen to occur.

The only problem with these algorithms is that not always the input stream of data is constructed out of repeating elements. It is clear that if the input stream contains many repeating elements there must be some way of reducing them. However that doesn’t mean that we cannot compress data if there are no repetitions. It all depends on the data. Let’s say we have the following stream to compress.

1, 2, 3, 4, 5, 6, 7

We can hardly imagine how this stream of data can be compressed. The same problem may occur when trying to compress the alphabet. Indeed the alphabet letters the very base of the words so it is the minimal part for word construction and it’s hard to compress them.

Fortunately this isn’t true always. An algorithm that tryies to deal with non repeating data is relative encoding. Let’s see the following input stream – years from a given decade (the 90’s).

1991,1991,1999,1998,1991,1993,1992,1992

Here we have 39 characters and we can reduce them. A natural approach is to remove the leading “19” as we humans often do.

91,91,99,98,91,93,92,92

Now we have a shorter string, but we can go even further with keeping only the first year. All other years will as relative to this year.

91,0,8,7,0,2,1,1

Now the volume of transferred data is reduced a lot (from 39 to 16 – more than 50%). However there are some questions we need to answer first, because the stream wont be always formatted in such pretty way. How about the next character stream?

91,94,95,95,98,100,101,102,105,110

We see that the value 100 is somehow in the middle of the interval and it is handy to use it as a base value for the relative encoding. Thus the stream above will become:

-9,-6,-5,-5,-2,100,1,2,5,10

The problem is that we can’t decide which value will be the base value so easily. What if the data was dispersed in a different way.

96,97,98,99,100,101,102,103,999,1000,1001,1002

Now the value of “100” isn’t useful, because compressing the stream will get something like this:

-4,-3,-2,-1,100,1,2,3,899,900,901,902

To group the relative values around “some” base values will be far more handy.

(-4,-3,-2,-1,100,1,2,3)(-1,1000,1,2)

However to decide which value will be the base value isn’t that easy. Also the encoding format is not so trivial. In the other hand this type of encoding can be useful in som specific cases as we can see bellow.

Implementation

The implementation of this algorithm depends on the specific task and the format of the data stream. Assuming that we’ve to transfer the stream of years in JSON from a web server to a browser, here’s a short PHP snippet.

// JSON: [1991,1991,1999,1998,1999,1998,1995,1997,1994,1993]
$years = array(1991,1991,1999,1998,1999,1998,1995,1997,1994,1993);

function relative_encoding($input)
{
	$output = array();
	$inputLength = count($input);
	
	$base = $input[0];
	
	$output[] = $base;
	
	for ($i = 1; $i < $inputLength; $i++) {
		$output[] = $input[$i] - $base;
	}
	
	return $output;
}

// JSON: [1991,0,8,7,8,7,4,6,3,2]
echo json_encode(relative_encoding($years));

Application

This algorithm may be very useful in many cases, but here’s one of them. There are plenty of map applications around the web. Some products as Google Maps, Yahoo! Maps, Bing Maps are quite famous, while there are very useful open source projects as OpenStreetMap. The web sites using these apps are thousands.

A typical use case is to transfer lots of Geo coordinates from web server to a browser using JSON. Indeed any GEO point on Earth is relative to the point (0,0), which is located near the west coast of Africa, however on large zoom levels, when there are tons of markers we can transfer the information with relative encoding.

For instance the following diagram shows San Francisco with some markers on it. Their coordinates are be relative to the point (0,0) on Earth.

Far more useful may be to encode those markers, relative to the center of the city, thus we can save some space.

However this type of compression can be tricky, for example when dragging the map and updating the marker array. In the other hand we must group markers if we have to load more than one city. That’s why we must be careful when implementing it. But in the other hand it can be very useful - for instance on initial load of the map we can reduce data and speed up the load time.

The thing is that with relative encoding we can save only changes to base value (data) - something like version control systems and thus reducing data transfer and load. Here's a graphical example. In the first case on the diagram bellow we can see that each item is stored on its own. It doesn't depend on the adjacent items and it can be completely independent of them.

However we can keep full info only for the first item and any other item will be relative to it, like on the diagram bellow.

In my previous post we saw how to compress data consisting of very long runs of repeating elements. This type of compression is known as “run-length encoding” and can be very handy when transferring data with no loss. The problem is that the data must follow a specific format. Thus the string “aaaaaaaabbbbbbbb” can be compressed as “a8b8”. Now a string with length 16 can be compressed as a string with length 4, which is 25% of its initial length without loosing any information. There will be a problem in case the characters (elements) were dispersed in a different way. What would happen if the characters are the same, but they don’t form long runs? What if the string was “abababababababab”? The same length, the same characters, but we cannot use run-length encoding! Indeed using this algorithm we’ll get at best the same string.

In this case, however, we can see another fact. The string consists of too many repeating elements, although not arranged one after another. We can compress this string with a bitmap. This means that we can save the positions of the occurrences of a given element with a sequence of bits, which can be easily converted into a decimal value. In the example above the string “abababababababab” can be compressed as “1010101010101010”, which is 43690 in decimals, and even better AAAA in hexadecimal. Thus the long string can be compressed. When decompressing (decoding) the message we can convert again from decimal/hexadecimal into binary and match the occurrences of the characters. Well, the example above is too simple, but let’s say only one of the characters is repeating and the rest of the string consists of different characters like this: “abacadaeafagahai”. Then we can use bitmap only for the character “a” – “1010101010101010” and compress it as “AAAA bcdefghi”. As you can see all the example strings are exactly 16 characters and that is a limitation. To use bitmaps with variable length of the data is a bit tricky and it is not always easy (if possible) to decompress it.

$data = array(
	0 	=> 1991,
	1 	=> 1992,
	2 	=> 1993,
	3 	=> 1994,
	4 	=> 1991,
	5 	=> 1992,
	6 	=> 1993,
	7 	=> 1992,
	8 	=> 1991,
	9 	=> 1991,
	10 	=> 1991,
	11 	=> 1992,
	12 	=> 1992,
	13 	=> 1991,
	14 	=> 1991,
	15 	=> 1992,	
	...
);

The JSON will encoded message will be the following (a simple but yet very large javascript array).

[1991,1992,1993,1994,1991,1992,1993,1992,1991,1991,1991,1992,1992,1991,1991,1992, ...]

However if we use bitmap compression we’ll get a “shorter” array.

$data = array(
	0 => array(1991, '1000100011100110'),
	1 => array(1992, '0100010100011001'),
	2 => array(1993, '0010001000000000'),
	3 => array(1994, '0001000000000000'),
);

Now the JSON is:

[[1991,"1000100011100110"],[1992,"0100010100011001"],[1993,"0010001000000000"],[1994,"0001000000000000"]]

It is obvious that the compression ratio is getting better and better as the uncompressed data grows. In fact, most of us know bitmap compression from images, because this algorithm is largely used for image compression. We can imagine how successful it can be when compressing black and white images (as black and white can be represented as 0 and 1s). Actually it is used for more than two colors (256 for instance) and again the level of compression is very high.

Implementation

The following implementation on PHP aims only to illustrate the bitmap compressing algorithm. As we know this algorithm can be applicable for any kind of data structures.

// too many repeating "a" characters
$msg = 'aazahalavaatalawacamaahakafaaaqaaaiauaacaaxaauaxaaaaaapaayatagaaoafaawayazavaaaazaaabararaaaaakakaaqaarazacajaazavanazaaaeanaaoajauaaaaaxalaraaapabataaavaaab';

function bitmap($message) 
{
	$i = 0;
	$bits = $rest = '';
	
	while ($v = $message[$i]) {
		if ($v == 'a') {
			$bits .= '1';
		} else {
			$bits .= '0';
			$rest .= $v;
		}
		$i++;
	}
	
	return number_format(bindec($bits), 0, '.', '') . $rest;;
}

echo bitmap($msg);

// uncompressed: 
acaaaaadaaaabalaaeaaaaganaaxakaavawamaasavajawaaaayaauaaadalanagaeaeamaarafalaazaaaiasaanaahaaazaraxaalaahaaawaaajasamahaajaakarapanaakaoakaanawalaacamauaamaal
// compressed:
152299251941730035874325065523548237677352452096zhlvtlwcmhkfqiucxuxpytgofwyzvzbrrkkqrzcjzvnzenojuxlrpbtvb

Application

This algorithm is very useful when there is an element in our data that repeats very often, so you need to investigate the nature of the data you want to compress. Actually because of this fact this algorithm is used for image compression as PNG8 or GIF.

No matter how fast today’s computers and networks are, the users will constantly need faster and faster services. To reduce the volume of the transferred data we usually use some sort of compression. That is why this computer sciences area will be always interesting to research and develop.

There are many data compression algorithms, some of them lossless, others lossy, but their main goal aways will be to spare storage space and traffic. These algorithms are very useful when talking about data transfer between two distant places. Perhaps the best example is the transfer between a web server and a browser.

In the last few years a lot of research has been done on compressing files, executed on the client side. Such files are javascript, css, htmls and images. In fact servers and clients already have some techniques to compress data, like using GZIP for instance, that can dramatically decrease the transfer. In the other hand there are lots of tools and tricks in order to decrease the size of the data.

Actually when a file is executed by the client’s virtual machine, it doesn’t matter how “beautifully” it is formatted from a programmer’s point of view. Thus the spaces, tabs and the new lines don’t bring any significant information for the environment. That is why such compressing tools like YUI Compressor, Google Closure Compiler, etc. remove those symbols. Well, they can achieve even more in order to improve the compression rate. In this post I won’t cover this, but this shows how important data compression algorithms are.

It would be great if we could just compress data with some tool. Unfortunately this is not the case and usually the compression rate depends on the data itself. It is obvious that the choice of data compression algorithm depends mainly on the data and first of all we must explore the data.

Here I’ll cover one very simple lossless data compression algorithm called “run-length encoding” that can be very useful in some cases.

Overview

This algorithm consists of replacing large sequences of repeating data with only one item of this data followed by a counter showing how many times this item is repeated. To become clearer let’s see a string example.

aaaaaaaaaabbbaxxxxyyyzyx

This string’s length is 24 and as we can see there are lots of repetitions. Using the run-length algorithm, we replace any run with shorter string followed by a counter.

a10b3a1x4y3z1y1x1

The length of this string is 17, which is approximately 70% of the initial length. Obviously this is not the optimal way to compress the given string. For instance we don’t need to use the digit “1” when the character is repeated only once. In some cases this approach can increase the length of the initial string which is exactly the opposite of what we need. In this case we’ll get the string bellow.

a10b3ax4y3zyx

Now the length of the resulting string is 13, which is 54% of the initial length! A variation of the example above is not to keep a counter of the repetitions of the character, but their position instead. Thus the initial string will be compressed as follows.

a0b10a13x14y18z21y22x23

Which of these two approaches you’ll use depends on the goal. In the second case we can achieve a good optimization of binary search.

It is clear that this algorithm is not only applicable on strings. We can achieve very good results on arrays. A typical example is the transfer of JSON from a server to a client. Then if there are large sequences of repeating data we can achieve great results.

Implementation

The implementation bellow is assuming that we’re compressing a string and it’s written on PHP. However the nature of this algorithm doesn’t restrict us to use only strings. As I said before with slight modifications we can use it with other data structures. It is important only to understand that the run-length algorithm is very useful on large sequences of repeating elements, no matter characters or array items.

$message = 'aaaaaaaaaabbbaxxxxyyyzyx';

function run_length_encode($msg)
{
	$i = $j = 0;
	$prev = '';
	$output = '';
	
	while ($msg[$i]) {
		if ($msg[$i] != $prev) {
			
			if ($i) 
				$output .= $j;
				
			$output .= $msg[$i];
				
			$prev = $msg[$i];
			
			$j = 0;
		}
		$j++;
		$i++;
	}
	
	$output .= $j;
	
	return $output;
}

// a10b3a1x4y3z1y1x1
echo run_length_encode($message);

And slightly optimized.

$message = 'aaaaaaaaaabbbaxxxxyyyzyx';

function run_length_encode($msg)
{
	$i = $j = 0;
	$prev = '';
	$output = '';
	
	while ($msg[$i]) {
		if ($msg[$i] != $prev) {
			
			if ($i && $j > 1) 
				$output .= $j;
				
			$output .= $msg[$i];
				
			$prev = $msg[$i];
			
			$j = 0;
		}
		$j++;
		$i++;
	}
	
	if ($j > 1)
		$output .= $j;
	
	return $output;
}

// a10b3ax4y3zyx
echo run_length_encode($message);

Finally a small change – now we store the position of the character.

$message = 'aaaaaaaaaabbbaxxxxyyyzyx';

function run_length_encode($msg)
{
	$i = 0;
	$prev = '';
	$output = '';
	
	while ($msg[$i]) {
		if ($msg[$i] != $prev) {
				
			$output .= $msg[$i] . $i;
				
			$prev = $msg[$i];
			
		}

		$i++;
	}
	
	return $output;
}

// a0b10a13x14y18z21y22x23
echo run_length_encode($message);

Complexity and Data Compression

We’re used to talk about complexity of an algorithm measuring time and we usually try to find the fastest implementation, like in search algorithms. Here it is not so important to compress data quickly, but to compress as much as possible so the output is as small as possible without lossing data. A great feature of run-length encoding is that this algorithm is easy to implement.

Application

We can use run-length encoding in many cases. It is commonly used to compress images and is very successful when we deal only with black and white images. Here I’ll cover another use case that I only mentioned above. Let’s say we have to transfer a very large array of data to our AJAX-powered application using JSON. Let’s say also that the data are some years, for instance the years of the premiere of a movie. There are lots of movies with a premiere in the same year, thus although the data is sorted, we actually can’t have any benefit. More important is that we have large sequences of data. Here we can use run-length encoding.

$data = array(
	0 	=> 1991,
	1 	=> 1991,
	...
	2223 	=> 1991,
	2224 	=> 1992,
	...
	19298 	=> 1995,
	19299 	=> 1996,
	...
);

As you can see to transfer the whole array can be a nightmare, especially on slow networks. It is better to compress it (i.e. with PHP’s json_encode).

// {"0":1991,"1":1991, ..., "2223":1991,"2224":1992, ..., "19298":1995,"19299":1996, ...}
echo json_encode($data);

After running run-length encoding we can receive something like the following array (note that these are only sample data and it’s up to you to decide which is the best format to store data).

$data = array(
	0 => array(1991, 2224),
	1 => array(1992, 3948),
	2 => array(1995, 2398),
	3 => array(1996, 3489),
);

And the JSON output.

// [[1991,2224],[1992,3948],[1995,2398],[1996,3489]]
echo json_encode($data);

Note that if the data is sorted we can achieve great success compressing it!!! This approach can be used for images, graphics or map coordinates.

This is only one example of how data compression can be useful in our daily work. Although the communication between the server and the client can be optimized and compressed, we can improve it. In other words we’re not always sure that the opposite side supports compression.

Well, it’s true that the client has to decompress the data, which can also be slow. Now in the first case we have only the time to transfer, as on the diagram bellow.

In the second case, we should sum the time for compression, transfer and decompression.

All this is important, but in general data compression can be handy in many cases in our daily work.