I want to do the following, using as efficiently as possible.

I have a file with many lines (I'll call it 'big_file.txt').

Each line of 'big_file.txt' has the following format:

dir1/dir2/dir3/file_name.txt

Every line in 'big_file.txt' is unique. 'dir1' and 'dir2' are the same 
for every line. 'dir3' and 'file_name.txt' have many values, but some 
lines have duplicate values for 'dir3' and some lines have duplicate 
values for 'file_name.txt'. In other words, 'big_file.txt' is a list of 
paths describing a file systems with directories that hold multiple 
files, and some files exist in more than one directory. 'big_file.txt' 
is about 100MB.

I have another file with many lines (I'll call it 'smaller_file.txt'). 
'smaller_file.txt' has less lines than 'big_file.txt'.

Each line of 'smaller_file.txt' has the following format:

file_name.txt

Every line in 'smaller_file.txt' is unique. The 'file_name.txt' strings 
in 'smaller_file.txt' are a subset of the 'file_name.txt' strings in 
'big_file.txt'. 'smaller_file.txt' is about 20MB.

'smaller_file.txt' has been sorted. 'big_file.txt' has been sorted on 
the 'file_name.txt' substring (or field).

I want to go through 'big_file.txt' and 'smaller_file.txt' and write 
output to a third file, 'expanded_file.txt', such output being the full 
paths (taken from 'big_file.txt') to the files listed in 
'smaller_file.txt'.

I have written the following program, which works but is very slow:

#!/usr/local/bin/ruby

a=File.open("smaller_file.txt","r")
b=File.open("big_file.txt","r")
c=File.open("expanded_file.txt","a")

a.each { |file_name|

   b.rewind

   b.each {|path|

     c.puts(path) if path.include?("#{file_name}")
   }
}

a.close
b.close
c.close

I realize that, because the files are sorted, I should not need to 
rewind 'big_file.txt' to the beginning of the file for every iteration, 
but I don't know how to tell the program how far back to rewind once it 
gets to the end of 'big_file.txt' and is ready to iterate with another 
file_name.

Any thoughts, ideas, _code_, etc. would be welcome.

Thank you.