If I'm right in thinking that your are only using a hand-coded bubblesort so that you can sort two parallel arrays together, then you are right in thinking that there is a better way. Actually several.

1. One way would be to avoid using parallel arrays.

   Instead of:

                   push @sort_array, $trial_1;
                 push @pointer_array, $key_r;
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   You build an Array of Arrays:

                  push @sort_array, [ $trial_1, $key_r ];
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   You can then sort both datasets using the (much faster) built-in sort routine like so:

              my @sorted = sort{ $a->[ 0 ] <=> $b->[ 0 ] } @sort_array;
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   And get your $closest and $pointer values like this:

   my( $closest, $pointer ) = @{ $sorted[ 0 ] };
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2. The second way is to sort the indices by the values in the first array. These will then be applicable to the second array also. See the code below:
   <Reveal this spoiler or all in this thread>

But you don't need to sort at all!

As you are only after the key associated with the closest pairing, you don't need to build any arrays, nor sort. You can just keep track of the lowest value (and associated key) as you traverse the datasets.

Also, as you are comparing all the keys from both datasets with each other, there is no need to sort the keys either. The following should run substantially more quickly:

#!/usr/bin/perl -slw
use strict;
use integer;
use vars qw( %tf1 %tf2 );

## Establish an array of all chromosome files.
my @files = ("chr1", "chr2", "chr3", "chr4", "chr5", "chr6", "chr7", "
+chr8", "chr9",
             "chr10", "chr11", "chr12", "chr13", "chr14", "chr15", "ch
+r16", "chr17",
             "chr18", "chr19", "chr20", "chr21", "chr22", "chrX", "chr
+Y");

## Loop through each file, calculate the overlaps, and output to file.
foreach (@files) {
    print STDOUT scalar localtime;
    open TF1, "<", "TF1.$_.gff" or die "Cannot open the TF1 hit file!"
+;  ## tf1 hit file
    open TF2, "<", "TF2.$_.gff" or die "Cannot open the TF2 hit file!"
+; ## tf2 hit file
    open OUT, ">", "both.$_.gff" or die "Cannot open the output file!"
+;    ## output file
    
    while (<TF1>) {
        chomp;
        my ($location, undef, $type, $start, $end, $score, $strand, un
+def) = split "\t"; ## Parse
        my ($chr, $location_numbers) = split "_", $location;          
+                          ## Get the chromosome
        my ($target_start, $target_end) = split "-", $location_numbers
+; ## Get the start site for the 10kb sequence "blasted" against
        my $new_start = $target_start + $start;  ## Use this for true 
+genomic location
        my $new_end = $target_start + $end;      ## Use thie for true 
+genomic location
        
        ## Build a hash and put the data inside;
        $tf1{ $new_start }[0] = $chr;
        $tf1{ $new_start }[1] = $type; ## Transcription factor
        $tf1{ $new_start }[2] = $new_end;
        $tf1{ $new_start }[3] = $score; ## Alignment score from PWM bl
+ast
        $tf1{ $new_start }[4] = $strand; ## Strand where the match is 
+found
    }
    
    while (<TF2>) {
        chomp;
        my ($location, undef, $type, $start, $end, $score, $strand, un
+def) = split "\t"; ## Parse
        my ($chr, $location_numbers) = split "_", $location;          
+                          ## Get the chromosome
        my ($target_start, $target_end) = split "-", $location_numbers
+; ## Get the start site for the 10kb sequence "blasted" against
        my $new_start = $target_start + $start;  ## Use this for true 
+genomic location
        my $new_end = $target_start + $end;      ## Use thie for true 
+genomic location
        
        ## Build a hash and put the data inside;
        $tf2{ $new_start }[0] = $chr;
        $tf2{ $new_start }[1] = $type; ## Transcription factor
        $tf2{ $new_start }[2] = $new_end;
        $tf2{ $new_start }[3] = $score; ## Alignment score from PWM bl
+ast
        $tf2{ $new_start }[4] = $strand; ## Strand where the match is 
+found
    }
    
    ## Close the unused files
    close TF1;
    close TF2;
    
    foreach my $key_c ( keys %tf2 ) {
        my( $best, $best_key ) = ( 1e99, '' ); 
        my $start_c = $key_c;
        my $end_c = $tf2{ $key_c }[2];
        
        foreach my $key_r ( keys %tf1) {
            my $start_r = $key_r;
            my $end_r = $tf1{ $key_r }[2];
            my $trial_1 = $start_r - $end_c;
            my $trial_2 = $start_c - $end_r;

            if ($trial_1 >= 0) {
                if( $trial_1 < $best ) {
                    $best = $trial_1;
                    $best_key = $key_r;
                }
            }
            elsif ($trial_2 >=0) {
                if( $trial_2 < $best ) {
                    $best = $trial_2;
                    $best_key = $key_r;
                }
            }
        }

        if( $best < 50 ) {
            print OUT join "\t", 
                $tf2{ $key_c }[0], $tf2{ $key_c }[1], 'tf2', $start_c,
+ $end_c,
                $tf2{ $key_c }[3], $tf2{ $key_c }[4], 'tf1', $best_key
+,
                $tf1{ $best_key}[2], $tf1{ $best_key}[3], $tf1{ $best_
+key}[4];
        }
    }

    ## Close the output file
    close OUT;
    print STDOUT scalar localtime;

} ## cycle through again til done with all files
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