Caveats:

I am not certain of your file formats. Here are my assumptions:

1. For FILE A, it is unclear if you have multiple ID strings on one line, or one per line over multiple lines. I'll make this solution work with either.
2. For FILE B, are your lines "key <whitespace> letters" or "key <newline> letters <newline>", ie key and values on separate lines?

Solution:

Perl hashes are very robust and often a great solution for simple to medium complexity problems. For this solution I'll read all the entries from the first file, parse out the IDs, and insert each ID into a hash. Then we will parse each entry in fileB and check if that ID is in the hash we built while walking fileA. In the case of a match, we print the ID and LETTERS joined with a < tab %gt; character.

With this solution, we look at each line of fileA and fileB exactly once, and we use a hash lookup on IDs which is fast. This reduces our complexity from O(n^2)+ from the previous solution to something closer to O(n log n), possibly close to O(n) if we're lucky with our ID hashing.

The Code

#!/usr/bin/perl

use warnings; use strict;

# open filea and parse all id strings. 
# Add id strings as keys to %wanted array.
my %wanted;
{
    open my $file, '<', "filea" 
      || die "failed to open filea : $!";
    while( <$file>)
    {
        chomp;
        @ids = split( /\s+/, $_);
        $wanted{ $_ }++ for @ids;
    }
    close $file;
}

#read fileb, parse lines of the form "id <whitespace> letters" 
#and print lines that match the id strings from filea.
{
    open my $file, '<', 'fileb'
      || die "failed to open fileb : $!";
    while (<$file>)
    {
        chomp;
        my ($id, $letters) = split( /\s+/, $_);
        print "$id\t$letters\n" if $wanted{$id};
    }
}

#OR

#read fileb, parse lines of the form "id <newline> letters" 
#and print lines that match the id strings from filea.
{
    open my $file, '<', 'fileb'
      || die "failed to open fileb : $!";
    while (<$file>)
    {
        my $id = $_;
        my $letters = <$file>;
        chomp($id);
        chomp($letters);
        
        print "$id\t$letters\n" if $wanted{$id};
    }
}
__END__
FileA:
1DWK 2RFK 
4ERH

FileB:
1DWK HRSDKKDAHJKLSDLDLLJDGHDFJJE
4ERH DFSKFHADFSBVHFWIHFWJBFS
2RFK DADUHRQWERKBNJAIJDLAJDKAKDNAKDJKSADJKAHDJASHRWEUB

FileB (alternate):
1DWK
HRSDKKDAHJKLSDLDLLJDGHDFJJE
4ERH
DFSKFHADFSBVHFWIHFWJBFS
2RFK
DADUHRQWERKBNJAIJDLAJDKAKDNAKDJKSADJKAHDJASHRWEUB
[download]

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