Shortest path given correct order of colours?

Question

I have a directed graph $G=(V,E)$ where each vertex is a 4-D coordinate $v: (x, y, z, c)$ representing spatial coordinates $x, y, z \in \mathbb{R}$ and the non-physical parameter colour $c \in (c_{1}, c_{2},... c_{n})$. The edge weights $\omega_{v_{a} \to v_{b}}: f(x, y, z)$ are purely a function of three of the four dimensions, as this is a physical problem.

Trouble is, I wish to find the shortest path given a specific sequence of colors (eg. $c_{2}, c_{5}, c_{1}, c_{1}, c_{3}, c_{4}$). It's difficult to incorporate this into the weight function as it is non-physical. Are there well-established path traversal algorithms that find the shortest path through coloured vertices, given the starting vertex, ending vertex, and the sequence it needs to pass through? I wonder if there is already a family of algorithms that describe this problem?

I've considered something like a Hidden Markov Model that abstracts vertices of colours into hidden states, but I am not sure how to prevent one state from being visited again (whereas in a directed acyclic graph I can prevent revisiting the same vertex).

Answer 1

Use a product construction, to construct a new graph whose vertices are given $(x,y,z,k)$, where $k$ counts the index into the sequence of colors (i.e., we are currently at $k$th vertex in that sequence). Then, find the shortest path in this new graph using any standard shortest-path algorithm. If all weights are non-negative, you can use Dijkstra's algorithm on this graph; otherwise you can use Bellman-Ford on this graph.

See How hard is finding the shortest path in a graph matching a given regular language?. Your problem is a special case of the problem solved there. (Possibly also useful: Product construction for given two finite automata .)

Specifically, here is how the product construction works in your setting. Let $(c_1,c_2,\dots,c_q)$ be the desired sequence of colors. Then, for each $k=1,2,\dots,q-1$, you'll have an edge $(x,y,z,k) \to (x,y,z,k+1)$ in the new graph if there is an edge $(x,y,z,c_k) \to (x,y,z,c_{k+1})$ in the original graph; the weight of the edge is copied over from the corresponding edge in the original graph. Now, find the shortest path from $(x_0,y_0,z_0,1)$ to $(x_1,y_1,z_1,q)$ in the new graph, where $(x_0,y_0,z_0,c_1)$ is the starting vertex in the original graph and $(x_1,y_1,z_1,c_q)$ is the ending vertex in the original graph.