Given a sequence of positive integers, find the longest monotone subsequence. That is, moving through the sequence from beginning to end, select from the sequence, while maintaining the original order, a sequence that is either strictly ascending or strictly descending.
For example, if the given sequence is 4 7 3 8 12 21 9 0 2 5 6 then 4 7 8 12 21 is the longest ascending subsequence.
I used a modified version of my answer for the ascending sequence problem to solve this.
PHP:
<? // generate a string of random numbers $max = 1; for ($n = 0; $n < mt_rand(3, 5); $n++) { // add a random number $sequence[] = mt_rand(1, 10); } // show the original sequence echo 'Random sequence of numbers: ' . implode($sequence, ' '); // 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 sequences $cur_run_asc = array(); $cur_run_desc = array(); // loop through each digit in the binary sequence for ($b = 0; $b < strlen($binary); $b++) { // check on if ($binary[$b]) { // if the number is higher than the previous number checked // also true for the first number if (is_array($cur_run_asc) && $sequence[$b] > $cur_run_asc[count($cur_run_asc)-1]) // add this to the current sequence $cur_run_asc[] = $sequence[$b]; else // if lower, no need to check any more $cur_run_asc = NULL; // if the number is lower than the previous number checked // also true for the first number if (is_array($cur_run_desc) && ($sequence[$b] < $cur_run_desc[count($cur_run_desc)-1] || !count($cur_run_desc))) // add this to the current sequence $cur_run_desc[] = $sequence[$b]; else // if higher, no need to check any more $cur_run_desc = NULL; } } // if the longest sequence generated from this binary number is more than the previous maximum // - ascending if (count($cur_run_asc) > count($max_run)) // make this the current leader $max_run = $cur_run_asc; // - descending if (count($cur_run_desc) > count($max_run)) // make this the current leader $max_run = $cur_run_desc; } // show the longest sequence that can be made echo '<br/>Longest ascending or descending sequence: ' . implode($max_run, ' ') . ' (' . count($max_run) . ')';?>Which produces:
Random sequence of numbers: 10 2 1
Longest ascending or descending sequence: 10 2 1 (3)