Counting possibilities in a char. combination using a consecutive repetition criterion - php

Question

In PHP, given
the final string length
the range of characters it can use
min consecutive repetition count possible
how can you calculate the number of matches that fits these criteria?To draw a better picture…
$range = array('a','b','c');
$length = 2; // looking for 2 digit results
$minRep = 2; // with >=2 consecutive characters
// aa,bb,cc = 3 possibilities
another one:
$range = array('a','b','c');
$length = 3; // looking for 3 digit results
$minRep = 2; // with >=2 consecutive characters
// aaa,aab,aac,baa,caa
// bbb,bba,bbc,abb,cbb
// ccc,cca,ccb,acc,bcc
// 5 + 5 + 5 = 15 possibilities
// note that combos like aa,bb,cc are not included
// because their length is smaller than $length
last one:
$range = array('a','b','c');
$length = 3; // looking for 3 digit results
$minRep = 3; // with >=3 consecutive characters
// aaa,bbb,ccc = 3 possibilities
So basically, in the 2nd example the 3rd criterion made it catch e.g. [aa]b in aab because a was repeating consecutively more than once, whereas [a]b[a] wouldn't be a match because those a's are separate.
Needless to say, none of the variables is static.

Answer 1

Got it. All credit to leonbloy #mathexchange.com.
/* The main function computes the number of words that do NOT contain
* a character repetition of length $minRep (or more). */
function countStrings($rangeLength, $length, $minRep, &$results = array())
{
if (!isset($results[$length]))
{
$b = 0;
if ($length < $minRep)
$b = pow($rangeLength, $length);
else
{
for ($i = 1; $i < $minRep; $i++)
$b += countStrings($rangeLength, $length - $i, $minRep, $results);
$b *= $rangeLength - 1;
}
$results[$length] = $b;
}
return $results[$length];
}
/* This one answers directly the question. */
function printNumStringsRep($rangeLength, $length, $minRep)
{
$n = (pow($rangeLength, $length)
- countStrings($rangeLength, $length, $minRep));
echo "Size of alphabet : $rangeLength<br/>"
. "Size of string : $length<br/>"
. "Minimal repetition : $minRep<br/>"
. "<strong>Number of words : $n</strong>";
}
/* Prints :
*
Size of alphabet : 3
Size of string : 3
Minimal repetition : 2
Number of words : 15
*
*/
printNumStringsRep(3, 3, 2);

Answer 2

I think it is best to handle this with math.
$range = array('a','b','c');
$length = 3; // looking for 3 digit results
$minRep = 2; // with >=2 consecutive characters
$rangeLength = count($range);
$count = (pow($rangeLength,$length-$minRep+1) * ($length-$minRep+1)) - ($rangeLength * ($length-$minRep)); // is the result
Now, $count is getting true result for three situation. But it may not be general formula and need to improve.
Try to explain it:
pow($rangeLength,$length-$minRep+1)
in this, we count repetitive characters like as one. For instance, in second example that you gave, we think in aab, aa is a one character. Because, two characters need to change together. We think now there is two character like xy. So there is same possibilities for both character a, b, and c namely 3 ($rangeLength) possible value for two characters($length-$minRep+1). So 3^2=9 is possible situations for second example.
We calculate 9 is for just xy not yx. For this, we multiply length of xy ($length-$minRep+1). And then we have 18.
It can be seemed that we calculated the result, but there is a repeat in our calculation. We didn't reckon with this situation: xy => aaa and yx => aaa. For this, we calculate and substract repeated results
- ($rangeLength * ($length-$minRep))
So after this, we get result.
As i said begining of the description, this formula may need to improve.

Answer 3

With Math, work becomes really complex. But, there is always a way, even not beautiful as much as Math. We can create all possible strings with php and control them with regexp like below:
$range = array('a','b','c');
$length = 3;
$minRep = 2;
$rangeLength = count($range);
$createdStrings = array();
$matchedStrings = array();
function calcIndex(){
global $range;
global $length;
global $rangeLength;
static $ret;
$addTrigger = false;
// initial values
if(is_null($ret)){
$ret = array_fill(0, $length, 0);
return $ret;
}
for($i=$length-1;$i>=0;$i--){
if($ret[$i] == ($rangeLength-1)) {
if($i==0) return false;
$ret[$i] = 0;
}
else {
$ret[$i]++;
break;
}
}
return $ret;
}
function createPattern()
{
global $minRep;
$patt = '/(.)\\1{'.($minRep-1).'}/';
return $patt;
}
$pattern = createPattern();
while(1)
{
$index = calcIndex();
if($index === false) break;
$string = '';
for($i=0;$i<$length;$i++)
{
$string .= $range[$index[$i]];
}
if(!in_array($string, $createdStrings)){
$createdStrings[] = $string;
if(preg_match($pattern, $string)){
$matchedStrings[] = $string;
}
}
}
echo count($createdStrings).' is created:';
var_dump($createdStrings);
echo count($matchedStrings).'strings is matched:';
var_dump($matchedStrings);