Ricochet Robots Mitch Powell Daniel Tilgner

Abstract Ricochet robots is a board game created in Germany in A player is given 30 seconds to find a best solution to the presented board. Our goal is to develop an algorithm that finds a minimal solution as quickly as possible.

So what exactly is ricochet robots? A board game in which the object is to move a specific robot to an end location in as few steps as possible using the following restrictions: ▫A robot can only move in the four cardinal directions, not diagonally. ▫When moved, the robot will continue moving until it reaches a wall or another robot at which time it will stop. This constitutes one step.

The Problem Given an directed graph G=(V, E) let V be the set of all vertices contained in the graph, each representing a square on the board and E be the set of all edges connecting V where each vertex w contained in V can be defined as w = (Occ., Targ.) where Occ. is a temporal variable describing the occupation of the square and can be either a 1 describing an occupied square or a 0 describing an unoccupied square and Targ. can be either a 0 representing that the square is not the target of the objective robot or a 1 representing that the square is the target of the objective robot. Given a target robot on a square v in V such that v = (1,0) and a target square t in V such that t = (0,1) let a function P(i,j) describes a number of edge traversals on a path between two squares i and j in V. Find a P’(v, t) such that P’(v, t) ≤ P (v, w) for all P. A stipulation on the problem is that there may exist more than one occupied square w in V such that w = (1,0). The robots occupying each square w where w ≠ v may traverse their own paths under the condition that the traversal of such a robot from its original square w to another square t means that w = (0,0) and t = (1,0). This change makes t unreachable.

Wait what.

Informally, please. For a given board configuration, find the path for the specified robot to reach the target space in as few steps as possible. As each robot can be moved at any time, available paths are updated after each movement. The movement of any robot counts as one step.

Contributions Michael Fogleman ▫Wrote a solver using iterative deepening. ▫We used this solver as a benchmark for checking solutions. Randy Coulman ▫Using pre-computing to create estimates for an A* algorithm.

Statement and implementation Breadth-First Search ▫Generate a state tree where any state of the board was given by moving a robot in any available direction. ▫Breadth-first traversal of state tree until a solution is found. ▫As soon as a solution is found it is guaranteed to be a minimal path (at which point the algorithm stops).

BFS cont. Optimization: ▫No-memoization ▫Visited states with linear search ▫Visited states with binary search ▫Visited states with hash set Board states were condensed to 8-byte strings where non-object robots were indistinct from one another.

Branch and Bound Generating a state tree of all possible candidate solutions. Initial “best” path found by attempting to move objective robot towards the goal without moving the other robots. While the step count for each branch is less than the current best solution, the branch is traversed depth- first. If the goal is reached, that new path becomes the best solution and the next branch is explored until all branches have been checked. Guaranteed to find the shortest path by exhaustion.

A* and Iterative Deepening Many alternative solutions exist. Michael Fogleman used iterative deepening to create a fast solver while Randy Coulman used an A* algorithm. Iterative deepening is a depth-first search in which the depth is increased with each iteration. It is essentially equivalent to a breadth-first search, but on each iteration it instead visits the nodes in the same order as a depth first search. A* traverses a graph and builds a tree of partial paths as it does so. The leaves are stored in a priority queue where the cost is the distance traveled plus an estimate of the distance to the goal.

Evidence Maximum number of steps through all trials: 16 (i.e. we encountered no board configuration that took more than 16 steps to solve) Average number of steps through all trials: 6.3 BFS with Hash table average time to solve: 829 milliseconds BFS with Binary search average time to solve: 2160 milliseconds

Holes Lack of evidence that prioritization of the objective bot results in a minimal number of visited states. Not distinguishing between objective robots has the same issue. While no unsolvable boards have been found, this doesn’t preclude the possibility that some could exist. There could be methods/tools for increasing the speed of the process.

Interpretation Breadth-first search with a hash algorithm is the best solution we found for average case or longer solutions. As the number of visited states gets higher, it gets more and more important to efficiently search the list of visited states. Without dynamic programming, the BFS becomes significantly less useful. So far, no solution longer than 7 steps has been able to be returned without dynamic programming.

Interpretation cont. Branch and bound runs very quickly if lower bounded solution is found very early into the search. Otherwise it is forced to run an exhaustive search with a large upper bound, increasing time needed dramatically. Could be useful in application purposes to test a solution by giving it a small bound.

Conclusions Breadth-first search seems to be the best solution to the problem. It guarantees a best solution and for boards where the best solution is small, it is incredibly fast. Other algorithms which rely on estimates and predictions such as A* or branch and bound depend heavily on initial conditions.

Future Work Generating pre-computed boards Maintain a list of known board setups with large step solutions Other algorithms to try (A*, B*, alpha-beta pruning, etc.) Buying a physical copy and clowning my friends

Questions What algorithm design technique made breadth- first search viable? What is the worst-case starting branching factor? What algorithm would you use to input and test your own solution?

Answers What algorithm design technique made breadth- first search viable? ▫Dynamic Programming What is the worst-case starting branching factor? ▫16 (4 robots with 4 directions each) What algorithm would you use to input and test your own solution? ▫Branch and Bound