Algorithm to find two points furthest away from each other

Question

Im looking for an algorithm to be used in a racing game Im making. The map/level/track is randomly generated so I need to find two locations, start and goal, that makes use of the most of the map.

• The algorithm is to work inside a two dimensional space
• From each point, one can only traverse to the next point in four directions; up, down, left, right
• Points can only be either blocked or nonblocked, only nonblocked points can be traversed

Regarding the calculation of distance, it should not be the "bird path" for a lack of a better word. The path between A and B should be longer if there is a wall (or other blocking area) between them.

Im unsure on where to start, comments are very welcome and proposed solutions are preferred in pseudo code.

Edit: Right. After looking through gs's code I gave it another shot. Instead of python, I this time wrote it in C++. But still, even after reading up on Dijkstras algorithm, the floodfill and Hosam Alys solution, I fail to spot any crucial difference. My code still works, but not as fast as you seem to be getting yours to run. Full source is on pastie. The only interesting lines (I guess) is the Dijkstra variant itself on lines 78-118.

But speed is not the main issue here. I would really appreciate the help if someone would be kind enough to point out the differences in the algorithms.

• In Hosam Alys algorithm, is the only difference that he scans from the borders instead of every node?
• In Dijkstras you keep track and overwrite the distance walked, but not in floodfill, but thats about it?

Answer 1

Your description sounds to me like a maze routing problem. Check out the Lee Algorithm. Books about place-and-route problems in VLSI design may help you - Sherwani's "Algorithms for VLSI Physical Design Automation" is good, and you may find VLSI Physical Design Automation by Sait and Youssef useful (and cheaper in its Google version...)

Answer 2

Assuming the map is rectangular, you can loop over all border points, and start a flood fill to find the most distant point from the starting point:

bestSolution = { start: (0,0), end: (0,0), distance: 0 };
for each point p on the border
    flood-fill all points in the map to find the most distant point
    if newDistance > bestSolution.distance
        bestSolution = { p, distantP, newDistance }
    end if
end loop

I guess this would be in O(n^2). If I am not mistaken, it's (L+W) * 2 * (L*W) * 4, where L is the length and W is the width of the map, (L+W) * 2 represents the number of border points over the perimeter, (L*W) is the number of points, and 4 is the assumption that flood-fill would access a point a maximum of 4 times (from all directions). Since n is equivalent to the number of points, this is equivalent to (L + W) * 8 * n, which should be better than O(n²). (If the map is square, the order would be O(16n^1.5).)

Update: as per the comments, since the map is more of a maze (than one with simple obstacles as I was thinking initially), you could make the same logic above, but checking all points in the map (as opposed to points on the border only). This should be in order of O(4n²), which is still better than both F-W and Dijkstra's.

Note: Flood filling is more suitable for this problem, since all vertices are directly connected through only 4 borders. A breadth first traversal of the map can yield results relatively quickly (in just O(n)). I am assuming that each point may be checked in the flood fill from each of its 4 neighbors, thus the coefficient in the formulas above.

Update 2: I am thankful for all the positive feedback I have received regarding this algorithm. Special thanks to @gs for his review.

P.S. Any comments or corrections are welcome.

Answer 3

If your objects (points) do not move frequently you can perform such a calculation in a much shorter than O(n^3) time.

All you need is to break the space into large grids and pre-calculate the inter-grid distance. Then selecting point pairs that occupy most distant grids is a matter of simple table lookup. In the average case you will need to pair-wise check only a small set of objects.

This solution works if the distance metrics are continuous. Thus if, for example there are many barriers in the map (as in mazes), this method might fail.

Answer 4

Finished a python mockup of the dijkstra solution to the problem. Code got a bit long so I posted it somewhere else: http://refactormycode.com/codes/717-dijkstra-to-find-two-points-furthest-away-from-each-other

In the size I set, it takes about 1.5 seconds to run the algorithm for one node. Running it for every node takes a few minutes.

Dont seem to work though, it always displays the topleft and bottomright corner as the longest path; 58 tiles. Which of course is true, when you dont have obstacles. But even adding a couple of randomly placed ones, the program still finds that one the longest. Maybe its still true, hard to test without more advanced shapes.

But maybe it can at least show my ambition.

Answer 5

Follow up to the question about Floyd-Warshall or the simple algorithm of Hosam Aly:

I created a test program which can use both methods. Those are the files:

• maze creator
• find longest distance

In all test cases Floyd-Warshall was by a great magnitude slower, probably this is because of the very limited amount of edges that help this algorithm to achieve this.

These were the times, each time the field was quadruplet and 3 out of 10 fields were an obstacle.

Size         Hosam Aly      Floyd-Warshall
(10x10)      0m0.002s       0m0.007s     
(20x20)      0m0.009s       0m0.307s
(40x40)      0m0.166s       0m22.052s
(80x80)      0m2.753s       -
(160x160)    0m48.028s      -

The time of Hosam Aly seems to be quadratic, therefore I'd recommend using that algorithm. Also the memory consumption by Floyd-Warshall is n², clearly more than needed. If you have any idea why Floyd-Warshall is so slow, please leave a comment or edit this post.

PS: I haven't written C or C++ in a long time, I hope I haven't made too many mistakes.

Answer 6

I deleted my original post recommending the Floyd-Warshall algorithm. :(

gs did a realistic benchmark and guess what, F-W is substantially slower than Hosam Aly's "flood fill" algorithm for typical map sizes! So even though F-W is a cool algorithm and much faster than Dijkstra's for dense graphs, I can't recommend it anymore for the OP's problem, which involves very sparse graphs (each vertex has only 4 edges).

For the record:

• An efficient implementation of Dijkstra's algorithm takes O(Elog V) time for a graph with E edges and V vertices.
• Hosam Aly's "flood fill" is a breadth first search, which is O(V). This can be thought of as a special case of Dijkstra's algorithm in which no vertex can have its distance estimate revised.
• The Floyd-Warshall algorithm takes O(V^3) time, is very easy to code, and is still the fastest for dense graphs (those graphs where vertices are typically connected to many other vertices). But it's not the right choice for the OP's task, which involves very sparse graphs.

Answer 7

It sounds like what you want is the end points separated by the graph diameter. A fairly good and easy to compute approximation is to pick a random point, find the farthest point from that, and then find the farthest point from there. These last two points should be close to maximally separated.

For a rectangular maze, this means that two flood fills should get you a pretty good pair of starting and ending points.

Answer 8

Raimund Seidel gives a simple method using matrix multiplication to compute the all-pairs distance matrix on an unweighted, undirected graph (which is exactly what you want) in the first section of his paper On the All-Pairs-Shortest-Path Problem in Unweighted Undirected Graphs [pdf].

The input is the adjacency matrix and the output is the all-pairs shortest-path distance matrix. The run-time is O(M(n)*log(n)) for n points where M(n) is the run-time of your matrix multiplication algorithm.

The paper also gives the method for computing the actual paths (in the same run-time) if you need this too.

Seidel's algorithm is cool because the run-time is independent of the number of edges, but we actually don't care here because our graph is sparse. However, this may still be a good choice (despite the slightly-worse-than n^2 run-time) if you want the all pairs distance matrix, and this might also be easier to implement and debug than floodfill on a maze.

Here is the pseudocode:

Let A be the nxn (0-1) adjacency matrix of an unweighted, undirected graph, G

All-Pairs-Distances(A)
    Z = A * A
    Let B be the nxn matrix s.t. b_ij = 1 iff i != j and (a_ij = 1 or z_ij > 0)
    if b_ij = 1 for all i != j return 2B - A //base case
    T = All-Pairs-Distances(B)
    X = T * A
    Let D be the nxn matrix s.t. d_ij = 2t_ij if x_ij >= t_ij * degree(j), otherwise d_ij = 2t_ij - 1
    return D

To get the pair of points with the greatest distance we just return argmax_ij(d_ij)

Answer 9

Ok, "Hosam's algorithm" is a breadth first search with a preselection on the nodes. Dijkstra's algorithm should NOT be applied here, because your edges don't have weights.

The difference is crucial, because if the weights of the edges vary, you need to keep a lot of options (alternate routes) open and check them with every step. This makes the algorithm more complex. With the breadth first search, you simply explore all edges once in a way that garantuees that you find the shortest path to each node. i.e. by exploring the edges in the order you find them.

So basically the difference is Dijkstra's has to 'backtrack' and look at edges it has explored before to make sure it is following the shortest route, while the breadth first search always knows it is following the shortest route.

Also, in a maze the points on the outer border are not guaranteed to be part of the longest route. For instance, if you have a maze in the shape of a giant spiral, but with the outer end going back to the middle, you could have two points one at the heart of the spiral and the other in the end of the spiral, both in the middle!

So, a good way to do this is to use a breadth first search from every point, but remove the starting point after a search (you already know all the routes to and from it). Complexity of breadth first is O(n), where n = |V|+|E|. We do this once for every node in V, so it becomes O(n^2).