Rush Hour - An A* Implementation

Team

• Fabian Pirklbauer
• Sascha Zarhuber

Abstract

As an exercise for the course Artificial Intelligence at the University of Applied Sciences Upper Austria Campus Hagenberg we implemented the A* algorithm to solve the rush hour puzzle.

To reduce the number of nodes visited as well as the branching factor of the search, heuristics are used to make an educated guess on whether it is a good idea to visit a given node or not.

The Game

Rush Hour is a sliding puzzle game. The red car has to reach the exit, while every car on the board may be moved forward or backward as far as it can in each turn.

> Image source: http://www.cs.princeton.edu/courses/archive/fall04/cos402/assignments/rushhour/

The A* Algorithm

The A* algorithm uses an open and closed list to ensure, that nodes are not visited twice. Also the open list is sorted, that nodes with a lower heuristic value are visited first. In addition, when adding a node to the open list with the same depth, only the one with the better heuristic value will be kept.

It is worth to mention, that A* doesn't simply use the value returned by the heuristic, but also adds the previous path costs (in this case the depth) to it. The value that is actually used to sort the open list is node_depth + node_heuristic_value.

Take a look at our implementation in AStar/AStar.java

Heuristics

Heuristics should make a good guess on how many moves would be required when visiting a given node. It is important to never overestimate the moves required, as the heuristic would not be admissible any more. As a result the solution found by A* may not be optimal.

Every move required counts a value of one.

Zero Heuristic

The zero heuristic returns the same value for every node, which doesn't improve the algorithm at all.

Blocking Heuristic

The blocking heuristic counts the number of cars, blocking the red car (referred to as "goal car") that has to reach the exit. This value reflects the minimum required moves for a given node's state.

For getting the desired value, we go through each car on the board, ignore all cars that are behind the goal car and those that are aligned in the same way. If there were an equally aligned car blocking the goal car, the puzzle would not be solvable.

Every remaining car intersecting with the goal car counts one. This value plus one for the goal car (to move the goal car to the exit, one move is required) is the result for a state.

The implementation can be found in Heuristics/BlockingHeuristic.java.

Advanced Heuristic

Idea

The blocking heuristic already improves the algorithm by simply counting cars blocking the goal car. It doesn't take cars into account, that are blocking the blocking cars, or cars that are blocking the blocking blocking cars, or... Well, let's stop here.

That's exactly what our heuristic does. It tries to count every blocking car recursively.

Challenges

The idea did sound simple at a first glance. It soon turned out, that there were several situations that simply cannot occur or are not relevant for cars blocking the goal car:

• How much space would a car need to move, to clear the way for another car?
• How much space is actually available?
• How many cars are blocking (there could me more than one, in two directions)?
• Cars cannot be considered twice.
• Cars could be aligned in the same way. If there are multiple cars aligned in the same way, the space required to move, has to be passed on recursively and reduced by the space available between equally aligned cars.
• There are walls.

Solution

Code is worth a thousand words...
But please note, that the following code snippets are not actual code, it's simplified dummy code.

Entry point of the heuristic:

getValue(State state) {
  return 0 if state.isGoal()

  visited.clear()
  visited.add(goalCar)

  value = 1;

  for (getInitialBlockingCars() as car) {
    value += getBlockingValue(car, getRequiredSpace(goalCar, car))
  }

  return value;
}

Recursive identification of blocking cars:

getBlockingValue(car, requiredSpace) {
  visited.add(car)

  value = 1

  for (allCars as next) {
    continue if next == car
    continue if visited.contains(next)
    continue unless car.intersectsWith(next)

    forwardCosts = 0, backwardCosts = 0

    if (!canMoveForward(car, next, requiredSpace.forward) {
      forwardCosts = getBlockingValue(car, getRequiredSpace(car, next))
    } else if (isWallBlockingAhead(car, requiredSpace.forward)) {
      forwardCosts = INFINITY
    }

    if (!canMoveBackward(car, next, requiredSpace.backward) {
      backwardCosts = getBlockingValue(car, getRequiredSpace(car, next))
    } else if (isWallBlockingBehind(car, requiredSpace.backward)) {
      backwardCosts = INFINITY
    }

    value += min(forwardCosts, backwardCosts);
  }

  return value;
}

You can see the full implementation in Heuristics/AdvancedHeuristic.java.

Drawbacks

• The time required to calculate the heuristic value for a state is higher than that of the the blocking heuristic.

• Cars never be counted twice. If a car was already considered as blocking, it must not count as blocking again, as the heuristic may overestimate in that case. If the car was already counted, it is considered as "moved", so it may, or may not be blocking another car.

Results

We could accomplish to implement an admissible and optimal heuristic, that visits fewer nodes than the blocking heuristic in most cases, but never visits more nodes.

The results can also be found in a separate file RESULTS.txt as plain text.

	Zero			Blocking			Advanced
name	nodes	dpth	br.fac	nodes	dpth	br.fac	nodes	dpth	br.fac
Jam-1	11587	8	3.066	8678	8	2.950	4026	8	2.661
Jam-2	24178	8	3.380	6201	8	2.820	4634	8	2.712
Jam-3	7814	14	1.789	5007	14	1.728	3931	14	1.695
Jam-4	3491	9	2.326	1303	9	2.061	378	9	1.762
Jam-5	24040	9	2.928	8353	9	2.583	1999	9	2.173
Jam-6	16046	9	2.792	7339	9	2.543	3993	9	2.364
Jam-7	56222	13	2.215	24529	13	2.068	24529	13	2.068
Jam-8	6470	12	1.957	5864	12	1.940	3588	12	1.854
Jam-9	5913	12	1.941	3704	12	1.860	3704	12	1.860
Jam-10	16197	17	1.677	12787	17	1.651	11865	17	1.643
Jam-11	6848	25	1.349	5918	25	1.340	4847	25	1.328
Jam-12	11686	17	1.642	7247	17	1.591	5325	17	1.560
Jam-13	72952	16	1.923	33949	16	1.827	23274	16	1.781
Jam-14	116381	17	1.901	44014	17	1.787	44014	17	1.787
Jam-15	3197	23	1.338	3171	23	1.337	3148	23	1.337
Jam-16	23072	21	1.534	17662	21	1.513	16119	21	1.506
Jam-17	19580	24	1.436	18287	24	1.432	16920	24	1.427
Jam-18	13881	25	1.392	12024	25	1.383	7246	25	1.352
Jam-19	3585	22	1.366	3533	22	1.365	3533	22	1.365
Jam-20	13881	10	2.464	4925	10	2.203	3635	10	2.131
Jam-21	1690	21	1.334	1590	21	1.329	1266	21	1.313
Jam-22	32371	26	1.423	22704	26	1.402	14757	26	1.376
Jam-23	19901	29	1.342	11978	29	1.316	10794	29	1.311
Jam-24	46313	25	1.468	43306	25	1.464	43306	25	1.464
Jam-25	82852	27	1.457	64644	27	1.443	60849	27	1.439
Jam-26	40855	28	1.397	34202	28	1.387	33998	28	1.387
Jam-27	21342	28	1.362	17927	28	1.352	16426	28	1.348
Jam-28	15591	30	1.316	11162	30	1.299	8022	30	1.283
Jam-29	38355	31	1.345	38111	31	1.345	38082	31	1.345
Jam-30	8610	32	1.264	7930	32	1.260	7316	32	1.257
Jam-31	32852	37	1.270	31236	37	1.268	29594	37	1.266
Jam-32	3292	37	1.184	2853	37	1.178	2788	37	1.177
Jam-33	37590	40	1.250	24507	40	1.235	19364	40	1.227
Jam-34	37544	43	1.229	36076	43	1.227	34943	43	1.226
Jam-35	34045	43	1.225	33326	43	1.225	32983	43	1.224
Jam-36	22760	44	1.207	18627	44	1.201	16928	44	1.198
Jam-37	15270	47	1.179	15231	47	1.179	13888	47	1.177
Jam-38	28560	48	1.192	24130	48	1.187	23497	48	1.187
Jam-39	24877	50	1.179	24361	50	1.179	24277	50	1.178
Jam-40	24467	51	1.174	22288	51	1.172	18337	51	1.167