Robson » Little Red Book » Ascending subsequence

Chapter 7 - Question 7

Given a sequence of positive integers, find the longest ascending subsequence. That is, moving from the beginning to end, select from the sequence, while maintaining order, a sequence that is strictly ascending.

For example, if the sequence is 4 7 3 8 21 21 9 0 2 5 6 then 4 7 8 12 21 is the longest subsequence.

Solution 1

<?    // generate a string of random numbers    $max = 1;    for ($n = 0; $n < mt_rand(3, 5); $n++)    {        // add a random number        $sequence[] = mt_rand(1, 10);    }    // loop through all binary numbers,    // until the length of the binary sequence is longer than the original sequence    for ($a = 1; $a < pow(2, count($sequence)); $a++)    {        // make a complete sequence, with leading zeros        // 001, 010, 011, etc        $binary = str_repeat('0', count($sequence)-strlen(decbin($a))) . decbin($a);        // reset sequence        $cur_run = array();        // loop through each digit in the binary sequence        for ($b = 0; $b < strlen($binary); $b++)        {            // check on            if ($binary[$b])                // if the number is higher than the previous number checked                // also true for the first number                if ($sequence[$b] > $cur_run[count($cur_run)-1])                    // add this to the current sequence                    $cur_run[] = $sequence[$b];                else                    // if lower, no need to check any more                    break;        }        // if the longest sequence generated from this binary number is more than the previous maximum        if (count($cur_run) > count($max_run))            // make this the current leader            $max_run = $cur_run;    }    // show the original sequence    echo 'Random sequence of numbers: ' . implode($sequence, ' ') . '<br/>';    // show the longest sequence that can be made    echo 'Longest ascending sequence: ' . implode($max_run, ' ') . ' (' . count($max_run) . ')';   ?>

Which produces:

Random sequence of numbers: 7 3 6 6 6
Longest ascending sequence: 3 6 (2)

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