$ perl  predecessor.pl
Data structure:
$VAR1 = {
          '123' => [
                     '234',
                     '345',
                     '456',
                     '567'
                   ],
          '131' => [
                     '131'
                   ],
          '234' => [],
          '213' => [
                     '324',
                     '435'
                   ],
          '345' => [],
          '324' => [],
          '372' => [
                     '372'
                   ],
          '117' => [
                     '228',
                     '339'
                   ]
        };


Result:

123 => 123
234 => 123
345 => 123
456 => 123
567 => 123
131 => 131
213 => 213
324 => 213
435 => 213
372 => 372
117 => 117
228 => 117
339 => 117