Compute dual graph from polygons

Question

Task

Compute the dual graph from a set of non-overlapping (but touching) polygons.

Example

Polygons A, B and C, their partially shared coordinates 1—22 (yellow) and the dual graph (blue).

Data

I have a set S of polygons. Every polygon P_i is represented as ordered list of coordinates. The edge a — b of a polygon P_i is denoted by P_i,(a,b)

Idea

The polygons represent the faces and hence the nodes of the dual graph. To identify adjacent faces of the polygon P_i just compare every edge of P_i with every edge of every other polygon P_j. If the edge is shared by the other polygon then P_i and P_j are adjacent.

This will create a significant amount of multiple edges which can be removed later.

Problem

The algorithm is not very efficient as it runs in O(E²) where E denotes the number of edges of the set S of polygons.

Improvement ideas

Create an edge-index in a first-step. This will reduce running-time to O(2×E) = O(E)

Remove every node with degree 2. (I think this won't affect the dual graph?)

Questions

Am I on the right track?
Are there any approved algorithms for this task?

I'd the purpose of the algorithm to find adjacent polygons (is *that* the required output)? — Amit, Oct 05 '16 at 20:27
Yes, the polygons don't overlap, they only touch. Hence, the graph is planar. — user2033412, Oct 05 '16 at 21:01

score 5 · Accepted Answer · answered Oct 05 '16 at 21:24

You can create a map from edge (as key) to polygon in O(n) time. Iterate all edges of all polygons (order is not important) and insert values into the map. When inserting a new value, if the key already exists, you've found an adjacent polygon - put this polygon pair in a set. When you're done, the set holds all adjacent polygon pairs.