Fastest way to get the set of convex polygons formed by Voronoi line segments

Question

I used Fortune's Algorithm to find the Voronoi diagram of a set of points. What I get back is a list of line segments, but I need to know which segments form closed polygons, and put them together in an object hashed by the original point they surround.

What might be the fastest way to find these?? Should I save some crucial information from from the algorithm? If so what?

Here is my implementation of fortune's algorithm in Java ported from a C++ implementation here

class Voronoi {

// The set of points that control the centers of the cells
private LinkedList<Point> pts;
// A list of line segments that defines where the cells are divided
private LinkedList<Edge> output;
// The sites that have not yet been processed, in acending order of X coordinate
private PriorityQueue sites;
// Possible upcoming cirlce events in acending order of X coordinate
private PriorityQueue events;
// The root of the binary search tree of the parabolic wave front
private Arc root;

void runFortune(LinkedList pts) {

    sites.clear();
    events.clear();
    output.clear();
    root = null;

    Point p;
    ListIterator i = pts.listIterator(0);
    while (i.hasNext()) {
        sites.offer(i.next());
    }

    // Process the queues; select the top element with smaller x coordinate.
    while (sites.size() > 0) {
        if ((events.size() > 0) && ((((CircleEvent) events.peek()).xpos) <= (((Point) sites.peek()).x))) {
            processCircleEvent((CircleEvent) events.poll());
        } else {
            //process a site event by adding a curve to the parabolic front
            frontInsert((Point) sites.poll());
        }
    }

    // After all points are processed, do the remaining circle events.
    while (events.size() > 0) {
        processCircleEvent((CircleEvent) events.poll());
    }

    // Clean up dangling edges.
    finishEdges();

}

private void processCircleEvent(CircleEvent event) {
    if (event.valid) {
        //start a new edge
        Edge edgy = new Edge(event.p);

        // Remove the associated arc from the front.
        Arc parc = event.a;
        if (parc.prev != null) {
            parc.prev.next = parc.next;
            parc.prev.edge1 = edgy;
        }
        if (parc.next != null) {
            parc.next.prev = parc.prev;
            parc.next.edge0 = edgy;
        }

        // Finish the edges before and after this arc.
        if (parc.edge0 != null) {
            parc.edge0.finish(event.p);
        }
        if (parc.edge1 != null) {
            parc.edge1.finish(event.p);
        }

        // Recheck circle events on either side of p:
        if (parc.prev != null) {
            checkCircleEvent(parc.prev, event.xpos);
        }
        if (parc.next != null) {
            checkCircleEvent(parc.next, event.xpos);
        }

    }
}

void frontInsert(Point focus) {
    if (root == null) {
        root = new Arc(focus);
        return;
    }

    Arc parc = root;
    while (parc != null) {
        CircleResultPack rez = intersect(focus, parc);
        if (rez.valid) {
            // New parabola intersects parc.  If necessary, duplicate parc.

            if (parc.next != null) {
                CircleResultPack rezz = intersect(focus, parc.next);
                if (!rezz.valid){
                    Arc bla = new Arc(parc.focus);
                    bla.prev = parc;
                    bla.next = parc.next;
                    parc.next.prev = bla;
                    parc.next = bla;
                }
            } else {
                parc.next = new Arc(parc.focus);
                parc.next.prev = parc;
            }
            parc.next.edge1 = parc.edge1;

            // Add new arc between parc and parc.next.
            Arc bla = new Arc(focus);
            bla.prev = parc;
            bla.next = parc.next;
            parc.next.prev = bla;
            parc.next = bla;

            parc = parc.next; // Now parc points to the new arc.

            // Add new half-edges connected to parc's endpoints.
            parc.edge0 = new Edge(rez.center);
            parc.prev.edge1 = parc.edge0;
            parc.edge1 = new Edge(rez.center);
            parc.next.edge0 = parc.edge1;

            // Check for new circle events around the new arc:
            checkCircleEvent(parc, focus.x);
            checkCircleEvent(parc.prev, focus.x);
            checkCircleEvent(parc.next, focus.x);

            return;
        }

        //proceed to next arc
        parc = parc.next;
    }

    // Special case: If p never intersects an arc, append it to the list.
    parc = root;
    while (parc.next != null) {
        parc = parc.next; // Find the last node.
    }
    parc.next = new Arc(focus);
    parc.next.prev = parc;
    Point start = new Point(0, (parc.next.focus.y + parc.focus.y) / 2);
    parc.next.edge0 = new Edge(start);
    parc.edge1 = parc.next.edge0;

}

void checkCircleEvent(Arc parc, double xpos) {
    // Invalidate any old event.
    if ((parc.event != null) && (parc.event.xpos != xpos)) {
        parc.event.valid = false;
    }
    parc.event = null;

    if ((parc.prev == null) || (parc.next == null)) {
        return;
    }

    CircleResultPack result = circle(parc.prev.focus, parc.focus, parc.next.focus);
    if (result.valid && result.rightmostX > xpos) {
        // Create new event.
        parc.event = new CircleEvent(result.rightmostX, result.center, parc);
        events.offer(parc.event);
    }

}

// Find the rightmost point on the circle through a,b,c.
CircleResultPack circle(Point a, Point b, Point c) {
    CircleResultPack result = new CircleResultPack();

    // Check that bc is a "right turn" from ab.
    if ((b.x - a.x) * (c.y - a.y) - (c.x - a.x) * (b.y - a.y) > 0) {
        result.valid = false;
        return result;
    }

    // Algorithm from O'Rourke 2ed p. 189.
    double A = b.x - a.x;
    double B = b.y - a.y;
    double C = c.x - a.x;
    double D = c.y - a.y;
    double E = A * (a.x + b.x) + B * (a.y + b.y);
    double F = C * (a.x + c.x) + D * (a.y + c.y);
    double G = 2 * (A * (c.y - b.y) - B * (c.x - b.x));

    if (G == 0) { // Points are co-linear.
        result.valid = false;
        return result;
    }

    // centerpoint of the circle.
    Point o = new Point((D * E - B * F) / G, (A * F - C * E) / G);
    result.center = o;

    // o.x plus radius equals max x coordinate.
    result.rightmostX = o.x + Math.sqrt(Math.pow(a.x - o.x, 2.0) + Math.pow(a.y - o.y, 2.0));

    result.valid = true;
    return result;
}

// Will a new parabola at point p intersect with arc i?
CircleResultPack intersect(Point p, Arc i) {
    CircleResultPack res = new CircleResultPack();
    res.valid = false;
    if (i.focus.x == p.x) {
        return res;
    }

    double a = 0.0;
    double b = 0.0;
    if (i.prev != null) // Get the intersection of i->prev, i.
    {
        a = intersection(i.prev.focus, i.focus, p.x).y;
    }
    if (i.next != null) // Get the intersection of i->next, i.
    {
        b = intersection(i.focus, i.next.focus, p.x).y;
    }

    if ((i.prev == null || a <= p.y) && (i.next == null || p.y <= b)) {
        res.center = new Point(0, p.y);

        // Plug it back into the parabola equation to get the x coordinate
        res.center.x = (i.focus.x * i.focus.x + (i.focus.y - res.center.y) * (i.focus.y - res.center.y) - p.x * p.x) / (2 * i.focus.x - 2 * p.x);

        res.valid = true;
        return res;
    }
    return res;
}

// Where do two parabolas intersect?
Point intersection(Point p0, Point p1, double l) {
    Point res = new Point(0, 0);
    Point p = p0;

    if (p0.x == p1.x) {
        res.y = (p0.y + p1.y) / 2;
    } else if (p1.x == l) {
        res.y = p1.y;
    } else if (p0.x == l) {
        res.y = p0.y;
        p = p1;
    } else {
        // Use the quadratic formula.
        double z0 = 2 * (p0.x - l);
        double z1 = 2 * (p1.x - l);

        double a = 1 / z0 - 1 / z1;
        double b = -2 * (p0.y / z0 - p1.y / z1);
        double c = (p0.y * p0.y + p0.x * p0.x - l * l) / z0 - (p1.y * p1.y + p1.x * p1.x - l * l) / z1;

        res.y = (-b - Math.sqrt((b * b - 4 * a * c))) / (2 * a);
    }
    // Plug back into one of the parabola equations.
    res.x = (p.x * p.x + (p.y - res.y) * (p.y - res.y) - l * l) / (2 * p.x - 2 * l);
    return res;
}

void finishEdges() {
    // Advance the sweep line so no parabolas can cross the bounding box.
    double l = gfx.width * 2 + gfx.height;

    // Extend each remaining segment to the new parabola intersections.
    Arc i = root;
    while (i != null) {
        if (i.edge1 != null) {
            i.edge1.finish(intersection(i.focus, i.next.focus, l * 2));
        }
        i = i.next;
    }
}

class Point implements Comparable<Point> {

    public double x, y;
    //public Point goal;

    public Point(double X, double Y) {
        x = X;
        y = Y;
    }

    public int compareTo(Point foo) {
        return ((Double) this.x).compareTo((Double) foo.x);
    }
}

class CircleEvent implements Comparable<CircleEvent> {

    public double xpos;
    public Point p;
    public Arc a;
    public boolean valid;

    public CircleEvent(double X, Point P, Arc A) {
        xpos = X;
        a = A;
        p = P;
        valid = true;
    }

    public int compareTo(CircleEvent foo) {
        return ((Double) this.xpos).compareTo((Double) foo.xpos);
    }
}

class Edge {

    public Point start, end;
    public boolean done;

    public Edge(Point p) {
        start = p;
        end = new Point(0, 0);
        done = false;
        output.add(this);
    }

    public void finish(Point p) {
        if (done) {
            return;
        }
        end = p;
        done = true;
    }
}

class Arc {
    //parabolic arc is the set of points eqadistant from a focus point and the beach line

    public Point focus;
    //these object exsit in a linked list
    public Arc next, prev;
    //
    public CircleEvent event;
    //
    public Edge edge0, edge1;

    public Arc(Point p) {
        focus = p;
        next = null;
        prev = null;
        event = null;
        edge0 = null;
        edge1 = null;
    }
}

class CircleResultPack {
    // stupid Java doesnt let me return multiple variables without doing this

    public boolean valid;
    public Point center;
    public double rightmostX;
}
}

(I know it wont compile, the data structures need to be initialized, and its missing imports)

What I want is this:

LinkedList<Poly> polys;
//contains all polygons created by Voronoi edges

class Poly {
    //defines a single polygon
    public Point locus;
    public LinkedList<Points> verts;
}

The most immediate brute force way I can think of to do this is to create an undirected graph of the points in the diagram (the endpoints of the edges), with a single entry for each point, and a single connection for each edge between a point (no duplicates) then go find all the loops in this graph, then for each set of loops that share 3 or more points, throw away everything but the shortest loop. However this would be way too slow.

Answer 1

The Voronoi diagram's dual is the Delaunay triangulation. That means each vertex on the Voroni diagram is connected to three edges - meaning each vertex belongs to three regions.

My algorithm to use this would be:

for each vertex in Voronoi Diagram
    for each segment next to this point
       "walk around the perimeter" (just keep going counter-clockwise)
                     until you get back to the starting vertex

That should be O(N) as there are only 3 segments for each vertex. You also have to do some bookkeeping to make sure you don't do the same region twice (a simple way is to just keep a bool for each outgoing edge, and as you walk, mark it off), and keeping in mind the point at infinity, but the idea should be enough.

Answer 2

Nathan - did you find an efficient way to associate your points with the Voronoi polygons that surround them? I already have a Delauney triangulation of my points - like you, I'm now considering how best to get from that to a structure that has (point, polygon).