22 March 2015

Fortune’s Algorithm is a sweepline algorithm for computing the Voronoi Diagram of a set of points. I implemented an interactive Javascript version of this algorithm for the final project in my Theory of Algorithms (COS423) class at Princeton.

You can add points to the diagram in several different methods. You can also watch the execution of the algorithm interactively, controlling the sweepline position with the slider, or just animate the sweepline process.

Note that sometimes floating point error causes strange behavior with large numbers of points.

You can also click on the plane to add points manually.

Drag the slider to control the sweep line.

 Event queue: Beach line:

## Overview

This interactive javascript implementation of Fortune’s algorithm uses a left to right sweep line algorithm to generate a Voronoi Diagram using parabolic arcs.

It provides several methods for generating points, allowing user specified points entered textually as a list or visually on the plane, as well as randomly generating points in the unit square.

We provide interactive control of the sweep line allowing the user to see the parabolic arcs constituting the “beach line” as they sweep out the partial voronoi diagram; to the left of the beach line is the voronoi diagram so far. The sweep line can be animated to move from left to right, or can be dragged back and forth by the user using a slider underneath the plane. The user can manually set the bounds of the view plane using the controls above the plane.

More detailed information about the workings of the algorithm can be seen in the textboxes at the bottom right of the page, showing the priority queue of potential sweep line events and the current arcs

There are two cases in which this implementation will err: the obvious one is when the first two input points have the same x-coordinate (which results in no beach line intersection for the second one). The other case(s) are occasionally seen when dealing with large numbers of randomly generated points, where we see floating point error come into play when e.g. detecting arc events or when intersecting arcs. This may cause two points which should be coincident to be sensed as different points, or vice versa.

This project is implemented entirely in javascript. My implementation of Fortune’s algorithm relies on the data structures provided by Google’s Closure javascript library. I also use jQuery and jQueryUI libraries to provide the interaction widgets as well as helper functions for event handling.

## Implementation Notes

### Fortune’s Algorithm

• Data Structures: The two core data structures in Fortune's Algorithm are the Sweep Line Priority Queue and the Beach Line Tree
• The sweep line priority queue contains event objects. These objects contain an x-coordinate of the event, a type (either a SITE event or an ARC event), and a reference to a point or arc object depending on which type of event it is.
This is implemented as a Closure Priority Queue, which uses a standard heap implementation.
• The beach line tree maintains a sorted list of the arcs in the beach line. The tree key is actually a pseudo-index, since what we need for the beach line is essentially a random-access linked list of arcs. Thus, we take the first two elements to have indices e.g. -213, 213; then when we insert a new node between two existing nodes we simply take the index halfway in between the two existing indices (going to floating point indices if necessary), and new nodes at the "ends of the linked list" just get a new power of 2 as an index.
This allows easy retrieval of nodes by index, while still keeping the sorted order of nodes in a binary tree form so that our beach intersection is efficient
We could implement a comparator to directly index based on the arc, but this method turns out to be simpler.
This is implemented as a Closure AVL tree, which suffices as a Balanced Binary Search Tree.
One rather important and complex data structure stores the information about a parabolic arc. This data structure contains the following data elements:
• The coordinates of the point that is the focus of the parabola (i.e. the point that generated the arc)
• A pseudo-index into the beach line for this arc
• References to the next and previous arcs in the beach line to keep the linked-list linear time traversal
• An index into the array of diagram edges (see below). This index represents the edge being traced out by the upper breakpoint of the arc
• A static key used as a hash of this arc. This is simply implemented as a string containing the concatenation of the points that generated the previous arc, this arc, and next arc.
There are two supplementary data structures used for this algorithm:
• A hash table that maps a parabolic arc segment to the corresponding potential event in the Sweep Line priority queue. This is used to make deletion of invalid arc events efficient.
This is implemented as a Closure HashMap
• An array of edges in the final voronoi diagram. Array elements have two properties: one is a pair of edge endpoints and the other is the pair of points that the edge bisects. This is used to store the output of the algorithm (both partial and final)
• Geometry Module: A separate geometry module was necessary to abstract out the difficult math. This contained simple primitives such as line segment intersection and distance functions. There were two nontrivial functions in this module:
• Calculating the circumcircle of three points, in order to determine when an arc would disappear (resulting in an ARC event).
• Calculating the center of a circle tangent to a vertical line and intersecting two arbitray points. This was necessary in order to calculate the breakpoints between two adjacent arcs (since the center of this circle would be equidistant from the sweep line and the two points generating the parabolic arcs)
• Algorithm Implementation The main function in this implementation processes a single event from the Sweep Line priority queue. It examines events from the top of the queue, discarding previously invalidated ones. Once it finds a valid event, it proceeds as described in Barr et al. In processing SITE events (hitting a new point), there are two subtleties - searching the beach requires reaching into the "private" variables of the AVL tree for efficient binary search to determine where to insert the new arc for the new point, and keeping the sorted-linked-list structure while constructing the three new arcs requires some careful initialization. Processing ARC events was a little more tricky because the edges of the diagram needed to be tracked. This is achieved by keeping track of the edges being traced out by the beach line. A few edge cases needed to be taken into account when considering potential arc events. There are four edge cases for arc events: the most common is that an arc's generating point is in front of (i.e. has a greater x coordinate than) both of its neighbors, so that the arc never actually disappears and therefore does not generate an arc event. We also disregard events with three adjacent collinear sites, since the bisectors for these sites will never intersect. If the potential event would take place at a location that the sweep line has already passed, then it is certainly invalid, and finally if the circumcenter of three points does not actually lie at the intersection of the three parabolas (an odd geometric case that was only caught with visual debugging) then it is a false event as well.

### Graphical Output and Interaction

• Graphics primitives involved drawing circles, points, parabolic arcs, and lines onto a canvas. Drawing parabolic arcs was an interesting task since the HTML5 canvas only provides a quadratic bezier curve drawing function, and we used http://alecmce.com/as3/parabolas-and-quadratic-bezier-curves as a reference for this. Two edge cases for parabolic arcs were endpoints with the same y-coordinate as the focus, which required us to slightly perturb each endpoint for the quadratice bezier curve to render properly, and a degenerate arc, which is simply a line. The same object responsible for these primitives also maintains the list of points on the canvas as well as the location of the sweep line.
• Interaction code was very straightforward using jQuery and jQueryUI. The slider element and the overlay for entering a list of points were jQueryUI widgets
• Drawing the partial diagram on the left of the beach line was the hardest part of the visualization, since it was not described in the readings. The way in which we constructed the edges (one endpoint at a time) allows us to loop through the break points on the beach line and connect them with the appropriate edge endpoints as they trace out the edges; however, for an edge that is being traced out in both directions, we have to keep a reference between the arcs that are tracing out the edge so we can draw the partial edge. Any remaining edges already have both endpoints determined so those are easy to draw.

Note that sliding the slider leftwards usually results in completely recalculating the voronoi diagram up to that point, since there is no simple way to reverse the sweep line.