How can I find the best fit subsequences of a large string?

Question

Say I have one large string and an array of substrings that when joined equal the large string (with small differences).

For example (note the subtle differences between the strings):

large_str = "hello, this is a long string, that may be made up of multiple
 substrings that approximately match the original string"

sub_strs = ["hello, ths is a lng strin", ", that ay be mad up of multiple",
 "subsrings tat aproimately ", "match the orginal strng"]

How can I best align the strings to produce a new set of sub strings from the original large_str? For example:

["hello, this is a long string", ", that may be made up of multiple",
 "substrings that approximately ", "match the original string"]

Additional Info

The use case for this is to find the page breaks of the original text from the existing page breaks of text extracted from a PDF document. Text extracted from the PDF is OCR'd and has small errors compared to the original text, but the original text does not have page breaks. The goal is to accurately page break the original text avoiding the OCR errors of the PDF text.

Answer 1

1. Concatenate the substrings
2. Align the concatenation with the original string
3. Keep track of which positions in the original string are aligned with the boundaries between the substrings
4. Split the original string on the positions aligned with those boundaries

---

An implementation using Python's difflib:

from difflib import SequenceMatcher
from itertools import accumulate

large_str = "hello, this is a long string, that may be made up of multiple substrings that approximately match the original string"

sub_strs = [
  "hello, ths is a lng strin",
  ", that ay be mad up of multiple",
  "subsrings tat aproimately ",
  "match the orginal strng"]

sub_str_boundaries = list(accumulate(len(s) for s in sub_strs))

sequence_matcher = SequenceMatcher(None, large_str, ''.join(sub_strs), autojunk = False)

match_index = 0
matches = [''] * len(sub_strs)

for tag, i1, i2, j1, j2 in sequence_matcher.get_opcodes():
  if tag == 'delete' or tag == 'insert' or tag == 'replace':
    matches[match_index] += large_str[i1:i2]
    while j1 < j2:
      submatch_len = min(sub_str_boundaries[match_index], j2) - j1
      while submatch_len == 0:
        match_index += 1
        submatch_len = min(sub_str_boundaries[match_index], j2) - j1
      j1 += submatch_len
  else:
    while j1 < j2:
      submatch_len = min(sub_str_boundaries[match_index], j2) - j1
      while submatch_len == 0:
        match_index += 1
        submatch_len = min(sub_str_boundaries[match_index], j2) - j1
      matches[match_index] += large_str[i1:i1+submatch_len]
      j1 += submatch_len
      i1 += submatch_len

print(matches)

Output:

['hello, this is a long string', 
 ', that may be made up of multiple ', 
 'substrings that approximately ', 
 'match the original string']

Answer 2

(The additional info makes a lot of the following unnecessary. It was written for a situation where the substrings provided might be any permutation of the order in which they occur in the main string)

There will be a dynamic programming solution for a problem very close to this. In the dynamic programming algorithm that gives you edit distance, the state of the dynamic program is (a, b) where a is the offset into the first string and b is the offset into the second string. For each pair (a, b) you work out the smallest possible edit distance that matches the first a characters of the first string with the first b characters of the second string, working out (a, b) from (a-1, b-1), (a-1, b), and (a, b-1).

You can now write a similar algorithm with state (a, n, m, b) where a is the total number of characters consumed by substrings so far, n is the index of the current substring, m is the position within the current substring, and b is the number of characters matched in the second string. This solves the problem of matching b against a string composed by pasting together any number of copies of any of the available substrings.

This is a different problem, because if you are trying to reconstitute a long string from fragments, you might get a solution that uses the same fragment more than once, but if you are doing this you might hope that the answer is obvious enough that the collection of substrings it produces happens to be a permutation of the collection given to it.

Because the edit distance returned by this method will always be at least as good as the best edit distance when you force a permutation, you could also use this to compute a lower bound on the best possible edit distance for a permutation, and run a branch and bound algorithm to find the best permutation.

Answer 3

You are trying to solve sequence alignment problem. In your case it is a "local" sequence alignment. It can be solved with Smith-Waterman approach. One possible implementation is here. If you run it, you receive:

large_str = "hello, this is a long string, that may be made up of multiple substrings that approximately match the original string"
sub_strs = ["hello, ths is a lng sin", ", that ay be md up of mulple", "susrings tat aproimately ", "manbvch the orhjgnal strng"]

for sbs in sub_strs:
    water(large_str, sbs)


 >>>

Identity = 85.185 percent
Score = 210
hello, this is a long strin
hello, th s is a l ng s  in
hello, th-s is a l-ng s--in

Identity = 84.848 percent
Score = 255
, that may be made up of multiple
, that  ay be m d  up of mul  ple
, that -ay be m-d- up of mul--ple

Identity = 83.333 percent
Score = 225
substrings that approximately 
su s rings t at a pro imately 
su-s-rings t-at a-pro-imately 

Identity = 75.000 percent
Score = 175
ma--tch the or-iginal string
ma   ch the or  g nal str ng
manbvch the orhjg-nal str-ng

The middle line shows matching characters. If you need the positions, look for max_i value to get ending position in original string. The starting position will be the value of i at the end of water() function.