Algorithm for determining largest covered area

Question

I'm looking for an algorithm which I'm sure must have been studied, but I'm not familiar enough with graph theory to even know the right terms to search for.

In the abstract, I'm looking for an algorithm to determine the set of routes between reachable vertices [x1, x2, xn] and a certain starting vertex, when each edge has a weight and each route can only have a given maximum total weight of x.

In more practical terms, I have road network and for each road segment a length and maximum travel speed. I need to determine the area that can be reached within a certain time span from any starting point on the network. If I can find the furthest away points that are reachable within that time, then I will use a convex hull algorithm to determine the area (this approximates enough for my use case).

So my question, how do I find those end points? My first intuition was to use Dijkstra's algorithm and stop once I've 'consumed' a certain 'budget' of time, subtracting from that budget on each road segment; but I get stuck when the algorithm should backtrack but has used its budget. Is there a known name for this problem?

Answer 1

If I understood the problem correctly, your initial guess is right. Dijkstra's algorithm, or any other algorithm finding a shortest path from a vertex to all other vertices (like A*) will fit.

In the simplest case you can construct the graph, where weight of edges stands for minimum time required to pass this segment of road. If you have its length and maximum allowed speed, I assume you know it. Run the algorithm from the starting point, pick those vertices with the shortest path less than x. As simple as that.

If you want to optimize things, note that during the work of Dijkstra's algorithm, currently known shortest paths to the vertices are increasing monotonically with each iteration. Which is kind of expected when you deal with graphs with non-negative weights. Now, on each step you are picking an unused vertex with minimum current shortest path. If this path is greater than x, you may stop. There is no chance that you have any vertices with shortest path less than x from now on.

If you need to exactly determine points between vertices, that a vehicle can reach in a given time, it is just a small extension to the above algorithm. As a next step, consider all (u, v) edges, where u can be reached in time x, while v cannot. I.e. if we define shortest path to vertex w as t(w), we have t(u) <= x and t(v) > x. Now use some basic math to interpolate point between u and v with the coefficient (x - t(u)) / (t(v) - t(u)).