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use warnings;
use strict;
...
sub RegexSort {
    return (length ((sort {length $b <=> length $a} $str =~ m[((.)\2+)
+]g)[0]), -1);
}

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Index: Run from 701117 for 30
Linear: Run from 701117 for 30
...
Linear    1.05/s        --      -77%      -82%
RegexSort 4.56/s      334%        --      -21%
Index     5.77/s      450%       27%        --