# Diffing

Difftastic treats diff calculations as a route finding problem on a directed acyclic graph.

## Graph Representation

A vertex in the graph represents a position in two syntax trees.

The start vertex has both positions pointing to the first syntax node in both trees. The end vertex has both positions just after the last syntax node in both trees.

Consider comparing `A` with `X A`.

    START
    +---------------------+
    | Left: A  Right: X A |
    |       ^         ^   |
    +---------------------+

    END
    +---------------------+
    | Left: A  Right: X A |
    |        ^           ^|
    +---------------------+

From the start vertex, we have two options:

- we can mark the first syntax node on the left as novel, and advance to the next syntax node on the left (vertex 1 above), or
- we can mark the first syntax node on the right as novel, and advance to the next syntax node on the right (vertex 2 above).

                START
                +---------------------+
                | Left: A  Right: X A |
                |       ^         ^   |
                +---------------------+
                       /       \
         Novel atom L /         \ Novel atom R
    1                v       2   v
    +---------------------+  +---------------------+
    | Left: A  Right: X A |  | Left: A  Right: X A |
    |        ^        ^   |  |       ^           ^ |
    +---------------------+  +---------------------+

Choosing "novel atom R" to vertex 2 will turn out to be the best choice. From vertex 2, we can see three routes to the end vertex.

                2
                +---------------------+
                | Left: A  Right: X A |
                |       ^           ^ |
                +---------------------+
                       /    |   \
         Novel atom L /     |    \ Novel atom R
                     v      |     v
    +---------------------+ | +---------------------+
    | Left: A  Right: X A | | | Left: A  Right: X A |
    |        ^          ^ | | |       ^            ^|
    +---------------------+ | +---------------------+
      |                     |                    |
      | Novel atom R        | Nodes match        | Novel atom L
      |                     |                    |
      |         END         v                    |
      |         +---------------------+          |
      +-------->| Left: A  Right: X A |<---------+
                |        ^           ^|
                +---------------------+

## Comparing Routes

We assign a cost to each edge. Marking a syntax node as novel is worse than finding a matching syntax node, so the "novel atom" edge has a higher cost than the "syntax nodes match" edge.

The best route is the lowest cost route from the start vertex to the end vertex.

## Finding The Best Route

Difftastic uses Dijkstra's algorithm to find the best (i.e. lowest cost) route.

One big advantage of this algorithm is that we don't need to construct the graph in advance. Constructing the whole graph would require exponential memory relative to the number of syntax nodes. Instead, vertex neighbours are constructed as the graph is explored.

There are lots of resources explaining Dijkstra's algorithm online, but I particularly recommend the graph search section of Red Blob Games.