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Find file Copy path. Raw Blame History. You are free to use and extend these projects for educational purposes. To select an agent, use the '-p' option when running pacman.
Arguments can be passed to your agent using '-a'. Please only change the parts of the file you are asked to. Follow the project description for details.
Good luck and happy searching! WEST in state. WEST else : return Directions. STOP This portion is written for you, but will only work after you fill in parts of search. Here, we choose a path to the goal. In this phase, the agent should compute the path to the goal and store it in a local variable.
All of the work is done in this method! Return Directions. STOP if there is no further action to take. SearchProblem : """ A search problem defines the state space, start state, goal test, successor function and cost function. This search problem can be used to find paths to a particular point on the pacman board. The state space consists of x,y positions in a pacman game. Note: this search problem is fully specified; you should NOT change it.
As noted in search. EASTDirections. Time to write code! SearchProblem : """ This search problem finds paths through all four corners of a layout. If it doesn't, see if making the move would lead to a corner.
If so, give that move as a possible successor, and update the visited corners to reflect that the corner is visited if that move is mode. Otherwise, just update the position without changing the visited corners. If those actions include an illegal move, return This is implemented for you. This function should always return a number that is a lower bound on the shortest path from the state to a goal of the problem; i.
Use this as the heuristic because it is consistent will always choose the same corners for a given situation.To shift the balance between g n and h nyou can modify either one. The tradeoff between speed and accuracy can be exploited to make your game faster. Using a function that guarantees it never overestimates the cost means that it will sometimes underestimate the cost by quite a bit. Now it will search only twice as far along the flat terrain as along mountainous terrain. Either approach gives up ideal paths to get something quicker.
The choice between speed and accuracy does not have to be static. You can choose dynamically based on the CPU speed, the fraction of time going into pathfinding, the number of units on the map, the importance of the unit, the size of the group, the difficulty level, or any other factor. One way to make the tradeoff dynamic is to build a heuristic function that assumes the minimum cost to travel one grid space is 1 and then build a cost function that scales:.
If alpha is 0, then the modified cost function will always be 1. You can set alpha anywhere in between. You should also consider switching from the heuristic returning the absolute minimum cost to returning the expected minimum cost.
The choice between speed and accuracy does not have to be global. You can choose some things dynamically based on the importance of having accuracy in some region of the map. For example, it may be more important to choose a good path near the current location, on the assumption that we might end up recalculating the path or changing direction at some point, so why bother being accurate about the faraway part of the path?
To add two values, those two values need to be at the same scale. All nodes not on the right path will have a higher value of f than nodes that are on the right path. One way to construct an exact heuristic is to precompute the length of the shortest path between every pair of points.
This is not feasible for most game maps. However, there are ways to approximate this heuristic:. Then add in a heuristic h' that estimates the cost of going from any location to nearby waypoints.
The latter too can be precomputed if desired. The final heuristic will be:. In a special circumstance, you can make the heuristic exact without precomputing anything. If you have a map with no obstacles and no slow terrain, then the shortest path from the starting point to the goal should be a straight line. Multiply the distance in steps by the minimum cost for a step. The units meters, minutes, etc. The standard heuristic for a square grid is the Manhattan distance . Look at your cost function and find the minimum cost D for moving from one space to an adjacent space.
In the simple case, you can set D to be 1.This is one of the most well known difficult problems of time. A salesperson must visit n cities, passing through each city only once, beginning from one of the city that is considered as a base or starting city and returns to it.
The cost of the transportation among the cities is given. The problem is to find the order of minimum cost route that is, the order of visiting the cities in such a way that the cost is the minimum. Let's number the cities from 1 to n and city 1 be the start-city of the salesperson. Also let's assume that c ij is the visiting cost from any city i to any other city j.
Heuristic (computer science)
Here is the systematic way of solving this problem:. Clearly, this requires at least n -1! If we try to determine the solution of this problem systematically, we would ended up with n - 1! For example if there were 21 cities the steps required are n - 1! If each step required a 1msec we would need about centuries to calculate the minimum cost route. My life is too short for this crap. Clearly we cannot examine all possible solutions for minimum cost.
Whenever the salesman is in town i he chooses as his next city i. In case more than one cities give the minimum cost, the city with the smaller k will be chosen. This greedy algorithm selects the cheapest visit in every step and does not care whether this will lead to a wrong result or not.In the general case, finding the shortest path that visits every dot is NP-hard.
However, that does not mean it is hard in practice.
The reason is because there are fixed parameter tractable algorithms and the Pacman mazes provided fall under the case of graphs that are easy to solve. In particular, for any given branch-width, the shortest path can be found in time polynomial in the size of the graph but exponential in the branch width of the graph by a simple application of dynamic programming.
This does not violate the NP-hardness since arbitrary graphs can have a large branch width, but it means that the problem can be solved efficiently if you only care about graphs that have a low branch width. The Pacman mazes provided have poor connectivity, and hence a low branch width. For more details, see this post. I don't think the ones I mentioned will work if there're more than one fruits for pacman to find.
I need some good path-finding algorithms that PacMan can use to finish the maze with the least possible step-count. I've tried to search around for a guideline, but so far no luck. Note that real distances are not Manhattan distances, but real distances in maze - you can calculate that even precalculate if you want because you know the look of labyrinth you know all the walls, This information real distance between some two points in maze is static because walls don't change. If you are solving this as a part of Berkeley AI class project, for calculation of real distance between two points you could use function mazeDistance pos1, pos2, gameState which is already implemented and is using your implementation of bfs.
Also, this heuristic is admissible and consistentat least for their test cases. By the way, with this heuristic I managed to expand just nodes in trickySearch maze. I know this is old, but there are probably a lot of other people out there looking to solve this problem it's part of Berkeley's free AI class. You comment says you are looking for shortest path.
You can also use an edmissible heuristic, of fruits - but it will take a long time. If you are looking for optimal, well - it is hard. You can try all permutations of fruitsand check the total distance you need to travel. This solution is factorial in the number of fruitsand if it is greater then 20 - with naive bruteforce - it will take too long. You can somehow make it better by reducing the problem to TSPand use dynamic-programming solution, which is also exponential, or some heuristical solutions to TSP.
One can also improve the non-admissible heuristic solution to provide an any-time algorithm :. This guarantees that at some point, your heuristic function h will be admissible - and the solution found will be optimal. However, each iteration is expected to take much longer then the previous one [exponentially longer. If you are doing with AI course problem, the node expanded should be very low.
Another point to consider, is that you can even get a better heuristic if you consider the walls, this is much tougher problem so i didnt get into it much, but I noticed that if you bound pacman in a rectangle with the next optimal fruit, he will have to pay at least 2 more actions if theres some FULL vertical or horizontal wall line between them because he would have to exit the bounding rectangle and re-enter it again paying at least 2 actions while doing so for each such wall.
This can be further examined and you can also find more special features in this rectangle. PacMan: what kinds of heuristics are mainly used? BTW, to keep things simple, assuming that there're no ghosts around. Heuristic which worked for me if you know the look of labyrinth: Find real distance between two currently furthest fruits in labyrinth - let's call that x. Find real distance from current Pacman position to the closer of previous two fruits - let's call that y.The standard way to construct a heuristic function is to find a solution to a simpler problem, which is one with fewer constraints.
A problem with fewer constraints is often easier to solve and sometimes trivial to solve. An optimal solution to the simpler problem cannot have a higher cost than an optimal solution to the full problem because any solution to the full problem is a solution to the simpler problem.
In many spatial problems where the cost is distance and the solution is constrained to go via predefined arcs e. For many problems one can design a better heuristic function, as in the following examples. Consider the delivery robot of Example 3.
Suppose the cost function is the total distance traveled by the robot to deliver all the parcels. If the robot could carry multiple parcels, one possible heuristic function is the maximum of a and b :. This is not an overestimate because it is a solution to the simpler problem which is to ignore that it cannot travel though walls, and to ignore all but the most difficult parcel. Note that a maximum is appropriate here because the agent has to both deliver the parcels it is carrying and go to the parcels it is not carrying and deliver them to their destinations.
If the robot could only carry one parcel, one possible heuristic function is the sum of the distances that the parcels must be carried plus the distance to the closest parcel. Note that the reference to the closest parcel does not imply that the robot will deliver the closest parcel first, but is needed to guarantee that the heuristic is admissible. In route planning of Example 3.
A more sophisticated heuristic may take into account the different maximum speeds on highways and local roads. One admissible heuristic is the minimum of a and b :. The minimum is appropriate here because the agent can go via highways or local roads, whichever is quicker. In the above examples, determining the heuristic did not involve search. Once the problem is simplified, it could be solved using search, which should be simpler than the original problem.
Note the simpler search problem needs to be solved multiple times, even perhaps for all nodes. It is often useful to cache these results into a pattern database that maps the nodes of the simpler problem into the heuristic value. In the simpler problem, there are often fewer nodes, and so multiple original nodes are mapped into a single simpler node, so this may be feasible. The text is not for re-distribution, re-sale or use in derivative works. Poole and Alan K.
This book is published by Cambridge University Press. Artificial Intelligence 2E foundations of computational agents. Home Index Contents. Example 3. If the robot could carry multiple parcels, one possible heuristic function is the maximum of a and b : 1. One admissible heuristic is the minimum of a and b : 1.This is achieved by trading optimality, completeness, accuracyor precision for speed.
In a way, it can be considered a shortcut. A heuristic functionalso called simply a heuristicis a function that ranks alternatives in search algorithms at each branching step based on available information to decide which branch to follow. For example, it may approximate the exact solution.
The objective of a heuristic is to produce a solution in a reasonable time frame that is good enough for solving the problem at hand. This solution may not be the best of all the solutions to this problem, or it may simply approximate the exact solution.
But it is still valuable because finding it does not require a prohibitively long time. Heuristics may produce results by themselves, or they may be used in conjunction with optimization algorithms to improve their efficiency e. Results about NP-hardness in theoretical computer science make heuristics the only viable option for a variety of complex optimization problems that need to be routinely solved in real-world applications.
Heuristics underlie the whole field of Artificial Intelligence and the computer simulation of thinking, as they may be used in situations where there are no known algorithms.
The trade-off criteria for deciding whether to use a heuristic for solving a given problem include the following:. In some cases, it may be difficult to decide whether the solution found by the heuristic is good enough, because the theory underlying heuristics is not very elaborate. One way of achieving the computational performance gain expected of a heuristic consists of solving a simpler problem whose solution is also a solution to the initial problem. An example of approximation is described by Jon Bentley for solving the travelling salesman problem TSP :.
TSP is known to be NP-Hard so an optimal solution for even a moderate size problem is difficult to solve. Instead, the greedy algorithm can be used to give a good but not optimal solution it is an approximation to the optimal answer in a reasonably short amount of time. The greedy algorithm heuristic says to pick whatever is currently the best next step regardless of whether that prevents or even makes impossible good steps later.
It is a heuristic in that practice says it is a good enough solution, theory says there are better solutions and even can tell how much better in some cases. Another example of heuristic making an algorithm faster occurs in certain search problems.
Initially, the heuristic tries every possibility at each step, like the full-space search algorithm.
But it can stop the search at any time if the current possibility is already worse than the best solution already found. In such search problems, a heuristic can be used to try good choices first so that bad paths can be eliminated early see alpha-beta pruning.
Simon discuss the heuristic search hypothesis: a physical symbol system will repeatedly generate and modify known symbol structures until the created structure matches the solution structure. Each following step depends upon the step before it, thus the heuristic search learns what avenues to pursue and which ones to disregard by measuring how close the current step is to the solution.
Therefore, some possibilities will never be generated as they are measured to be less likely to complete the solution. A heuristic method can accomplish its task by using search trees. However, instead of generating all possible solution branches, a heuristic selects branches more likely to produce outcomes than other branches. It is selective at each decision point, picking branches that are more likely to produce solutions. Antivirus software often uses heuristic rules for detecting viruses and other forms of malware.
The dark mode beta is finally here. Change your preferences any time. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. I don't think the ones I mentioned will work if there're more than one fruits for pacman to find. I need some good path-finding algorithms that PacMan can use to finish the maze with the least possible step-count.
I've tried to search around for a guideline, but so far no luck. You comment says you are looking for shortest path. You can also use an edmissible heuristic, of fruits - but it will take a long time.
An Effective Heuristic Algorithm for the Traveling-Salesman Problem
If you are looking for optimal, well - it is hard. You can try all permutations of fruitsand check the total distance you need to travel. This solution is factorial in the number of fruitsand if it is greater then 20 - with naive bruteforce - it will take too long.
You can somehow make it better by reducing the problem to TSPand use dynamic-programming solution, which is also exponential, or some heuristical solutions to TSP.The Travelling Salesman (3 of 3: Ant Colonisation Heuristic)
One can also improve the non-admissible heuristic solution to provide an any-time algorithm :. This guarantees that at some point, your heuristic function h will be admissible - and the solution found will be optimal. However, each iteration is expected to take much longer then the previous one [exponentially longer. Note that real distances are not Manhattan distances, but real distances in maze - you can calculate that even precalculate if you want because you know the look of labyrinth you know all the walls, This information real distance between some two points in maze is static because walls don't change.
If you are solving this as a part of Berkeley AI class project, for calculation of real distance between two points you could use function mazeDistance pos1, pos2, gameState which is already implemented and is using your implementation of bfs.
Also, this heuristic is admissible and consistentat least for their test cases. By the way, with this heuristic I managed to expand just nodes in trickySearch maze. I found the closest approx food using manhattan distances but for my heuristic I used the actual distance from my position to the closest food.
To this I added 1 for all those food points that dont share row or column with my position or closest food point. Because the food points that do share row or col with my position or closest food position would be eaten while going from my position to the closest food and Ive already counted the cost of this in the actual distance I mentioned in the second line. This was admissible and consistent. I know this is old, but there are probably a lot of other people out there looking to solve this problem it's part of Berkeley's free AI class.
If you are doing with AI course problem, the node expanded should be very low. In the general case, finding the shortest path that visits every dot is NP-hard. However, that does not mean it is hard in practice.