The underlying problem is path search. A 2D canvas can be treated as a Graph, where each pixel is represented by a node (lets say we represent a pixel (x,y) with node [x,y]). Obviously, pixels that collide with chart components or other flowlines will have no corresponding nodes. Since we are interested only in orthogonal paths, we create edges from each node to its four orthogonal neighbours (ie. [x,y-1], [x-1,y], [x,y+1], [x+1,y]). The problem of finding orthogonal flow line from point (x,y) to (x',y') can now be treated as the problem of finding a path from node [x,y] to [x',y'], which can be easily solved by Graph Search Algorithms.
We would also like the flowlines to look aesthetically pleasing. Although its slightly difficult to define, mathematically, what could be aesthetically pleasing, i reason that the following two parameters can create good flowlines.
- Minimize Path Length : Obviously, shorter lines connecting chart components are easier to read.
- Minimize Bends : Connecting lines which have too many bends are hard to follow. The problem with orthogonal paths is that the shortest distance between any two points is given by manhattan distance. This means that there could be several paths with shortest length. In the absence of obstacles, any of the shortest path can have atmost about min(abs(x'-x), abs(y'-y)) bends (blind guess formula no proofs :P), therefore minimizing bends is important.
For a given path, if we define path cost function g(n) as the sum of distance to node n from starting point along the given path and the number of bends along the path from starting point to n, then we would like our algorithm the choose the path with least path cost function for the end point. We use the dijkstra's algorithm with path cost = g(n) = distance + #of bends to do this.
The problem with this approach is that, standard Dijkstra algorithm works by extending optimal partial paths. In simple graph search, when there are several optimal partial paths of the same cost, the choice between one of these for extension doesnt affect the optimality of the final path. But however, in our case, this is not true (again, due to the Taxicab Geometry). Consider the following:
A----(B) | | | | | | D-----C | | | (E)
We are trying to connect B to E. At point D, there are two optimal partial paths BAD and BCD of the same cost (cost=length+bends=2+1). If the algorithm chooses path BCD for extension the final path BCDE(cost=length+bends=3+2) is suboptimal since there is path BADE(cost=length+bends=3+1) with fewer bends. Thus we need to teach our algorithm to make the choice in such a way that the extended path is also optimal.
In Dijkstra, we extend a path to a node only if the optimal partial path (if one has aleady been computed) at that point, from the start, has greater cost than the new path that would be generated after extension. If we introduce an estimate of the minimum number of bends required to reach the target also in the above condition ie. we extend only if the sum of cost of optimal partial path and estimate of min bends to target is greater than that for the new path that would be generated, then we can avoid the previously mentioned problem. Thus, in the previous example, if path BAD has already been computed and BC is being considered, then we do not extend BC to BCD since atleast 1 more bend is required to reach E along BCD wheras BAD requires no more bends to reach E. Conversely, if BCD has already been computed and BA is being considered, then we detach D from BCD and extend BA to BAD since it has lower bends estimate.
The minimum bends-to-target estimate can be easily computed by checking the direction of the partial path at the end point and the relative position of the target eg: if target is along same direction and in front of the end point then min-bends-to-target is 0, there are three other such cases.
Making it Faster
Dijkstra's algorithm is sort of like hitting the pinata, it'd go about searching all over the canvas to find the end point. If we have some more information about how warm or cold each point is, with respect to the target, we can avoid searching in directions that we know to be less likely to lead us to the endpoint. We do this by implementing an A* Search, which is a generalization of Dijkstra. We use a heuristic function h(n) = manhattan-distance(n,target)+min-bends(n,target), so that the search would try find paths within or close to a rectangle with the start and end points at opposite corners and are likely to have fewer bends.
This is a hacky little demo with animation. Hit the play button to run the animation.
Red dots are nodes which have been considered and Yellow dots are the nodes in the queue.
n.heuristic = 0; line in dijkstra function to switch to simple dijkstra without
A very interesting optimization to this is to reduce the search to the orthogonal visibility Graph (a graph containing points of interest, such as vertices, with arcs drawn between nodes that can see each other). This and a more formal approach to this problem is detailed in this paper: Orthogonal Connector Routing. The Authors of the paper have a C++ implementation of their technique in libavoid libaray.
I dot not have proofs for the correctness, optimality etc. for anything, though it seems to work in the simple prototype. Any comments are welcome.