# Diff, part 7: String ’em all together? (1 of 2)

Continuing the Diff for the Complete Goober series (after a brief hiatus on account of Deleted Sleep Time):

The core problem of diff is the building of the longest common subsequence, which we accomplish by finding all the matches in the two lists, then assembling match strings that match our rule (which filters out the match strings that are not common subsequences).

Does it have to be that complicated? Let’s just try it. Say we have these two lists:

A is (1 2 3 4)
B is (1 3 4)
Matches: [0,0], [2,1], and [3,2].

Do we really need to go through all the rigamarole? You can build a match string out of all three matches in their original order, and it’ll be a common subsequence — and at length 3, it’ll be the longest common subsequence. So our LCS has to be [0,0] [2,1] [3,2].

Well, that was easy. Grab all the matches, put them together, and there’s the LCS. It can’t be that easy, can it? Well… no, actually, it can’t:

A is (1 2 3)
B is (1 3 2)
Matches: [0,0], [1,2], and [2,1].

If we try stringing these all together, we get [0,0] [1,2] [2,1], which isn’t a valid common subsequence. In this case, the real LCS is going to be length 2: either [0,0] [1,2] or [0,0] [2,1]; take your pick.

Can we adapt our simple “string ’em all together” strategy to work with these inputs? Well, we could start with [0,0], and keep adding matches on, as long as the result is still a valid CS; and just throw away any matches that we can’t append to our longest-running-common-subsequence. Let’s try that.

• First match: [0,0]. This gives us a length-1 match string that is a valid CS. (Actually, any length-1 match string will always be a valid CS.)
• Next match: [1,2]. Add that to our best CS so far. Result: [0,0] [1,2]. This is a valid CS; keep it.
• Next match: [2,1]. Add that to our best CS so far. Result: [0,0] [1,2] [2,1]. This is not a valid CS; throw it out. Our best CS so far is still [0,0] [1,2].

That looks promising, doesn’t it? Let’s try it with a slightly different input:

A is (1 2 3 4)
B is (1 3 2 4)
Matches: [0,0], [1,2], [2,1], and [3,3].

The lists are the same as above, but with one more match. So we do the same thing as above, but with one more step at the end:

• Next match: [3,3]. Add that to our best CS so far. Result: [0,0] [1,2] [3,3]. This is a valid CS; keep it.

Again, we get the right LCS. Looking good! One more example:

A is (1 2 3 4 5)
B is (5 2 3 4 1)
Matches: [0,4], [1,1], [2,2], [3,3], and [4,0].

Steps:

• First match: [0,4]. This gives us a length-1 match string that is a valid CS.
• Next match: [1,1]. Add that to our best CS so far. Result: [0,4] [1,1]. This is not a valid CS; throw it out. Our best CS so far is still [0,4].
• Next match: [2,2]. Add that to our best CS so far. Result: [0,4] [2,2]. This is not a valid CS; throw it out. Our best CS so far is still [0,4].
• Next match: [3,3]. Add that to our best CS so far. Result: [0,4] [3,3]. This is not a valid CS; throw it out. Our best CS so far is still [0,4].
• Next match: [4,0]. Add that to our best CS so far. Result: [0,4] [4,0]. This is not a valid CS; throw it out. Our best CS so far is still [0,4].

So our modified “string ’em all together” algorithm gives us an LCS of [0,4]. But wait — that’s not right. There are three matches right in a row, in the middle of the inputs. The LCS should be [1,1] [2,2] [3,3]. What does that mean?

Well, it means that our algorithm isn’t right yet. Next time, I’ll add a refinement that would make it work for all these inputs.

(Notice, though, that I said these inputs. I’ve still got several more articles to go.)