MST from an array of 2D triangles, but with a little twist

Question

Here's an illustration of the steps taken thus far:

1. Pseudo-random rectangle generation
2. "Central node" insertion, rect separation, and final node selections
3. Delaunay triangulation (shown with previously selected nodes)
4. Rendering of triangle edges

At this point (Step 5), I would like to use this data to form a Minimum Spanning Tree , but there's a slight catch...

Somewhere in the graph (likely near the center, but not always) will be a node that requires between 3-5 connections to it from other unique nodes. This complicates things, since every other node should only contain a single connection, and the data structures being used make it difficult to determine "what's connected to what" in a solid, traversable format.

So, given an array of triangles in the above format, and a random vertex to use as the "root node", how could I properly traverse the network to create an MST where there are at least 3 connections to our "central node", but no more than 5 connections to it? Is this possible?

Answer 1

Since it's rare to have a vertex in a Delaunay triangulation have much more than 6 edges, you can use brute force: there are only 20+15+6 ways to select 3, 4, or 5 edges out of 6 (respectively), so try all of them. For each of the 41 (up to 336 for degree 9) small trees (the root and a few edges) thus created, run either Kruskal's algorithm or Prim's algorithm starting with that tree already "found" to be part of the MST. (Ignore the root's other edges so as not to increase its degree further.) Then just pick the best one (including the cost of the seed tree).

As for the general neighbor information problem, it seems you just need to build a standard graph representation first. For example, you can make an adjacency list by scanning over all your Edge objects and storing each endpoint in a list associated with the other (in a map<Vector2<T>,vector<Vector2<T>>> or an equivalent based on whatever identifiers for your vertices).

Answer 2

I've taken a workaround approach...

After step 3 of my algorithm, I simply remove all edges which connect to the "central node", keeping track of which edges form the "ring" (aka "edge loop") around it, and run the MST on all remaining edges.

For the MST, I went with the boost graph library. That made it easy to loop through the triangles I had, adding each of its three edges to an adjacency_list . Then a simple call to whichever boost-provided MST algorithm took care of the rest.

Finally, I readd the edges that were previously taken out. The shortest path is whatever it was in the previous step, plus the length of whichever readded edge that connects to another edge on the "ring" is shortest.

I can then add (or remove) an arbitrary number of previous edges to ensure there are between 3 to 5 edges connecting from the edge loop to the "central node".

Doing things in this order also allows me to know as soon as step 3 if we'll even have a valid number of edges, so we don't waste cycles running a MST.