Maze-solving algorithm: Difference between revisions

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{{Short description|Automated method for solving mazes}}

[[File:Cyclope robot.jpg|thumb|right|Robot in a wooden maze]]

There are a number of differentA '''maze-solving algorithmsalgorithm''', that is, an [[algorithm|automated methodsmethod]] for the solving ofa [[maze]]s. The random mouse, wall follower, Pledge, and Trémaux's [[algorithm]]s are designed to be used inside the maze by a traveler with no prior knowledge of the maze, whereas the [[Cul-de-sac|dead-end]] filling and [[shortest path algorithm]]s are designed to be used by a person or computer program that can see the whole maze at once.

Mazes containing no loops are known as "simply connected", or "perfect" mazes, and are equivalent to a [[Tree (graph theory)|''tree'']] in graph theory. Thus many mazeMaze-solving algorithms are closely related to [[graph theory]]. Intuitively, if one pulled and stretched out the paths in the maze in the proper way, the result could be made to resemble a tree.<ref>{{youtubeyouTube|k1tSK5V1pds|Maze to Tree}}</ref>

== Random mouse algorithm ==

This is a trivialsimple method that can be implemented by a very unintelligent [[robot]] or perhaps a mouse., Itbecause isit simplydoes tonot require any memory. The robot proceedproceeds following the current passage until a junction is reached, and then to makemakes a random decision about the next direction to follow. Although such a method would always [[Las Vegas algorithm|eventually find the right solution]], thisthe algorithm can be extremelyvery slow.<ref name="Aleliunas1979">{{cite conference | last1=Aleliunas | first1=Romas | last2=Karp | first2=Richard M | last3=Lipton | first3=Richard J | last4=Lovász | first4=László | last5=Rackoff | first5=Charles | title=Random walks, universal traversal sequences, and the complexity of maze problems | book-title=20th Annual Symposium on Foundations of Computer Science (sfcs 1979) | pages=218–223 | year=1979 | publisher=IEEE Computer Society }}</ref>

== Hand On Wall followerRule ==

[[File:maze01-02.pngsvg|left|framethumb|164px|Traversal using ''right-hand rule'' ([[Media:Maze01-02.svg|animation]])]]

TheOne best-knowneffective rule for traversing mazes is the ''wallHand followerOn Wall Rule'', also known as either the ''left-hand rule'' or the ''right-hand rule''. If the maze is [[Simply connected space|''simply connected'']], that is, all its walls are connected together or to the maze's outer boundary, then by keeping one hand in contact with one wall of the maze the solver is guaranteed not to get lost and will reach a different exit if there is one; otherwise, the algorithm will return to the entrance having traversed every corridor next to that connected section of walls at least once. The algorithm is a depth-first in-order [[tree traversal]].

Another perspective into why wall following works is topological. If the walls are connected, then they may be deformed into a loop or circle.<ref>{{youtubeyouTube|IIBwiGrUgzc|Maze Transformed}}</ref> Then wall following reduces to walking around a circle from start to finish. To further this idea, notice that by grouping together connected components of the maze walls, the boundaries between these are precisely the solutions, even if there is more than one solution (see figures on the right).

If the maze is not simply-connected (i.e. if the start or endpoints are in the center of the structure surrounded by passage loops, or the pathways cross over and under each other and such parts of the solution path are surrounded by passage loops), this method will not necessarily reach the goal.

Another concern is that care should be taken to begin wall-following at the entrance to the maze. If the maze is not simply-connected and one begins wall-following at an arbitrary point inside the maze, one could find themselves trapped along a separate wall that loops around on itself and containing no entrances or exits. Should it be the case that wall-following begins late, attempt to mark the position in which wall-following began. Because wall-following will always lead you back to where you started, if you come across your starting point a second time, you can conclude the maze is not simply-connected, and you should switch to an alternative wall not yet followed. See the ''Pledge Algorithm'', below, for an alternative methodology.

Wall-following can be done in 3D or higher-dimensional mazes if its higher-dimensional passages can be projected onto the 2D plane in a deterministic manner. For example, if in a 3D maze "up" passages can be assumed to lead Northwest, and "down" passages can be assumed to lead southeast, then standard wall following rules can apply. However, unlike in 2D, this requires that the current orientation is known, to determine which direction is the first on the left or right.

A simulation of this algorithm working can be found [https://scratch.mit.edu/projects/1049044916/ here].

== Pledge algorithm ==

[[File:Pledge Algorithm.png|left|thumb| Left: Left-turn solver trapped <br /> Right: Pledge algorithm solution]]

Disjoint{{huh?|date=March 2017}}(where walls are not connected to the outer boundary/boundary is not closed) mazes can be solved with the wall follower method, so long as the entrance and exit to the maze are on the outer walls of the maze. If however, the solver starts inside the maze, it might be on a section disjoint from the exit, and wall followers will continually go around their ring. The Pledge algorithm (named after JohnJon Pledge of [[Exeter]]) can solve this problem.<ref>{{citation|title=Turtle Geometry: the computer as a medium for exploring mathematics|last1=Abelson|last2=diSessa|year=1980|publisher=MIT Press |isbn=9780262510370|url=https://books.google.com/books?id=3geYp44hJVcC}}</ref><ref>Seymour Papert, [https://web.archive.org/web/20170706130427/ftp://publications.ai.mit.edu/ai-publications/pdf/AIM-298.pdf "Uses of Technology to Enhance Education"], ''MIT Artificial Intelligence Memo No. 298'', June 1973</ref>

The Pledge algorithm, designed to circumvent obstacles, requires an arbitrarily chosen direction to go toward, which will be preferential. When an obstacle is met, one hand (say the right hand) is kept along the obstacle while the angles turned are counted (clockwise turn is positive, counter-clockwise turn is negative). When the solver is facing the original preferential direction again, and the angular sum of the turns made is 0, the solver leaves the obstacle and continues moving in its original direction.

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== Trémaux's algorithm ==

[[File:Tremaux Maze Solving Algorithm.gif|thumb| Trémaux's algorithm. The large green dot shows the current position, the small blue dots show single marks on pathsentrances, and the red crosses show double marks. Once the exit is found, the route is traced through the singly-marked pathsentrances. <br/><br/> Note that two marks are placed simultaneously each time the green dot arrives at a junction. This is a quirk of the illustration; each mark should in actuality be placed whenever the green dot passes through the ___location of the mark.]]

Trémaux's algorithm, invented by [[Charles Pierre Trémaux]],<ref>Public conference, December 2, 2010 – by professor Jean Pelletier-Thibert in Academie de Macon (Burgundy – France) – (Abstract published in the Annals academic, March 2011 – {{ISSN|0980-6032}}) <br/>Charles Tremaux (° 1859 – † 1882) Ecole Polytechnique of Paris (X:1876), French engineer of the telegraph</ref> is an efficient method to find the way out of a maze that requires drawing lines on the floor to mark a path, and is guaranteed to work for all mazes that have well-defined passages,<ref name="Récréations Mathématiques">Édouard Lucas: ''Récréations Mathématiques'' Volume I, 1882.</ref> but it is not guaranteed to find the shortest route.

An entrance of a passage is either unvisited, marked once or marked twice. Note that marking an entrance is not the same as marking a junction or a passage, because a junction may have multiple entrances, and a passage has an entrance at both ends. Dead ends can be thought of as junctions with one entrance (imagine there being a room at each dead end).

A path from a junction is either unvisited, marked once or marked twice. The algorithm works according to the following rules:

* Mark each path once, when you follow it. The marks need to be visible at both ends of the path. Therefore, if they are being made as physical marks, rather than stored as part of a computer algorithm, the same mark should be made at both ends of the path.

* Never enter a path which has two marks on it.

* If you arrive at a junction that has no marks (except possibly for the one on the path by which you entered), choose an arbitrary unmarked path, follow it, and mark it.

* Otherwise:

** If the path you came in on has only one mark, turn around and return along that path, marking it again. In particular this case should occur whenever you reach a dead end.

** If not, choose arbitrarily one of the remaining paths with the fewest marks (zero if possible, else one), follow that path, and mark it.

A path from a junction is either unvisited, marked once or marked twice. The algorithm works according to the following rules:

The "turn around and return" rule effectively transforms any maze with loops into a simply connected one; whenever we find a path that would close a loop, we treat it as a dead end and return. Without this rule, it is possible to cut off one's access to still-unexplored parts of a maze if, instead of turning back, we arbitrarily follow another path.

* Whenever you pass through an entrance of a passage, whether it is to enter or exit a junction, leave a mark at the entrance as you pass.

* When you are at a junction, use the first applicable rule below to pick an entrance to exit through:

*# If only the entrance you just came from is marked, pick an arbitrary unmarked entrance, if any. This rule also applies if you're just starting in the middle of the maze and there are no marked entrances at all.

*# If all entrances are marked, go back through the entrance you just came from, unless it is marked twice. This rule will apply whenever you reach a dead end.

*# Pick any entrance with the fewest marks (zero if possible, else one).

The "turn around and return" rule effectively transforms any maze with loops into a simply connected one; whenever we find a path that would close a loop, we treat it as a dead end and return. Without this rule, it is possible to cut off one's access to still-unexplored parts of a maze if, instead of turning back, we arbitrarily followpick another pathentrance.

When you finally reach the solution, paths marked exactly once will indicate a way back to the start. If there is no exit, this method will take you back to the start where all paths are marked twice.

In this case each path is walked down exactly twice, once in each direction. The resulting [[Glossary of graph theory#Walks|walk]] is called a bidirectional double-tracing.<ref name="Eulerian Graphs and related Topics">H. Fleischner: ''Eulerian Graphs and related Topics.'' In: ''Annals of Discrete Mathematics'' No. 50 Part 1 Volume 2, 1991, page X20.</ref>

When you finally reach the solution, pathsentrances marked exactly once will indicate a way back to the start. If there is no exit, this method will take you back to the start where all pathsentrances are marked twice.

In this case each pathpassage is walked down exactly twice, once in each direction. The resulting [[Glossary of graph theory#Walks|walk]] is called a bidirectional double-tracing.<ref name="Eulerian Graphs and related Topics">H. Fleischner: ''Eulerian Graphs and related Topics.'' In: ''Annals of Discrete Mathematics'' No. 50 Part 1 Volume 2, 1991, page X20.</ref>

Essentially, this algorithm, which was discovered in the 19th century, has been used about a hundred years later as [[depth-first search]].<ref>{{citation|title=Graph Algorithms|first=Shimon|last=Even|authorlink=Shimon Even|edition=2nd|publisher=Cambridge University Press|year=2011|isbn=978-0-521-73653-4|pages=46–48|url=https://books.google.com/books?id=m3QTSMYm5rkC&pg=PA46}}.</ref><ref>{{citation|title=Algorithms in C++: Graph Algorithms|first=Robert|last=Sedgewick|edition=3rd|publisher=Pearson Education|year=2002|isbn=978-0-201-36118-6}}.</ref>

== Dead-end filling ==

{{External media

Dead-end filling is an algorithm for solving mazes that fills all dead ends, leaving only the correct ways unfilled. It can be used for solving mazes on paper or with a computer program, but it is not useful to a person inside an unknown maze since this method looks at the entire maze at once. The method is to 1) find all of the dead-ends in the maze, and then 2) "fill in" the path from each dead-end until the first junction is met. Note that some passages won't become parts of dead end passages until other dead ends are removed first. A video of dead-end filling in action can be seen here: [https://www.youtube.com/watch?v=yqZDYcpCGAI][https://www.youtube.com/watch?v=FkueaIT6RSU].

|video1=[https://www.youtube.com/watch?v=yqZDYcpCGAI Maze Strategy: Dead End Filling]

|video2=[https://www.youtube.com/watch?v=FkueaIT6RSU Maze Solving algorithm]

}}

Dead-end filling is an algorithm for solving mazes that fills all dead ends, leaving only the correct ways unfilled. It can be used for solving mazes on paper or with a computer program, but it is not useful to a person inside an unknown maze since this method looks at the entire maze at once. The method is to 1) find all of the dead-ends in the maze, and then 2) "fill in" the path from each dead-end until the first junction is met. Note that some passages won't become parts of dead end passages until other dead ends are removed first. A video of dead-end filling in action can be seen here: [https://www.youtube.com/watch?v=yqZDYcpCGAI][https://www.youtube.com/watch?v=FkueaIT6RSU].

# find all of the dead-ends in the maze, and then

# "fill in" the path from each dead-end until the first junction is met.

Note that some passages won't become parts of dead end passages until other dead ends are removed first. A video of dead-end filling in action can be seen to the right.

Dead-end filling cannot accidentally "cut off" the start from the finish since each step of the process preserves the topology of the maze. Furthermore, the process won't stop "too soon" since the end result cannot contain any dead-ends. Thus if dead-end filling is done on a perfect maze (maze with no loops), then only the solution will remain. If it is done on a partially braid maze (maze with some loops), then every possible solution will remain but nothing more. [http://www.astrolog.org/labyrnth/algrithm.htm]

== Recursive algorithm ==

If given an omniscient view of the maze, a simple recursive algorithm can tell one how to get to the end. The algorithm will be given a starting X and Y value. If the X and Y values are not on a wall, the method will call itself with all adjacent X and Y values, making sure that it did not already use those X and Y values before. If the X and Y values are those of the end ___location, it will save all the previous instances of the method as the correct path.

This is in effect a depth-first search expressed in term of grid points. The omniscient view prevents entering loops by memoizationmemorization. Here is a sample code in [[Java (programming language)|Java]]:

<syntaxhighlight lang="java">

boolean[][] maze = new boolean[width][height]; // The maze

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public void solveMaze() {

maze = generateMaze(); // Create Maze (false = path, true = wall)

// Below assignment to false is redundant as Java assigns array elements to false by default, but it is included because other languages may not behave the same.

for (int row = 0; row < maze.length; row++)

// Sets boolean Arrays to default values

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== Maze-routing algorithm ==

The maze-routing algorithm <ref>{{cite journalbook|last1=Fattah|first1=Mohammad|last2=et|first2=al.|title=Proceedings of the 9th International Symposium on Networks-on-Chip |chapter=A Low-Overhead, Fully-Distributed, Guaranteed-Delivery Routing Algorithm for Faulty Network-on-Chips |journalseries=NOCSNocs '15 Proceedings of the 9th International Symposium on Networks-on-Chip|date=2015-09-28|pages=1–8|doi=10.1145/2786572.2786591|isbn=9781450333962|s2cid=17741498|chapter-url=http://dl.acm.org/citation.cfm?id=2786591}}</ref> is a low overhead method to find the way between any two locations of the maze. The algorithm is initially proposed for [[Chipchip multiprocessor|chip multiprocessors]]s (CMPs) ___domain and guarantees to work for any grid-based maze. In addition to finding paths between two ___locationlocations of the grid (maze), the algorithm can detect when there is no path between the source and destination. Also, the algorithm is to be used by an inside traveler with no prior knowledge of the maze with fixed memory complexity regardless of the maze size; requiring 4 variables in total for finding the path and detecting the unreachable locations. Nevertheless, the algorithm is not to find the shortest path.

Maze-routing algorithm uses the notion of [[Manhattan distance]] (MD) and relies on the property of grids that the MD increments/decrements ''exactly'' by 1 when moving from one ___location to any 4 neighboring locations. Here is the pseudocode without the capability to detect unreachable locations.

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== Shortest path algorithm ==

{{Further|Pathfinding#Algorithms}}

[[File:MAZE 40x20 DFS no deadends.png|thumb|A maze with many solutions and no dead-ends, where it may be useful to find the shortest path]]

When a maze has multiple solutions, the solver may want to find the shortest path from start to finish. There are several algorithms to find shortest paths, most of them coming from [[graph theory]]. One such algorithm finds the shortest path by implementing a [[breadth-first search]], while another, the [[A* algorithm]], uses a [[heuristic]] technique. The breadth-first search algorithm uses a [[queue (data structure)|queue]] to visit cells in increasing distance order from the start until the finish is reached. Each visited cell needs to keep track of its distance from the start or which adjacent cell nearer to the start caused it to be added to the queue. When the finish ___location is found, follow the path of cells backwards to the start, which is the shortest path. The breadth-first search in its simplest form has its limitations, like finding the shortest path in weighted graphs.

== Multi-agent maze-solving ==

Collective exploration refers to the exploration of an unknown environment by multiple mobile agents that move at the same speed. This model was introduced

to study the [[Parallel computing|paralellizability]] of maze-solving,<ref name="Fraigniaud2006">{{cite journal | last1=Fraigniaud | first1=Pierre | last2=Gasieniec | first2=Leszek | last3=Kowalski | first3=Dariusz R | last4=Pelc | first4=Andrzej | title=Collective tree exploration | journal=Networks| volume=48 | issue=3 | pages=166–177 | year=2006 | publisher=Wiley Online Library | doi=10.1002/net.20127 }}</ref> especially in the case of [[Tree (graph theory)|trees]]. The study depends on the model of communication between the agents. In the centralized communication model, the agents are allowed to communicate at all times with one another. In the [[Distributed computing|distributed]] communication model, the agents can communicate only by reading and writing on the walls of the maze. For trees with <math>n</math> nodes and depth <math>D</math>, with <math>k</math> robots, the current-best algorithm is in <math>O\left(\frac{n}{k} + kD\right)</math> in the centralized communication model and in <math>O\left(\frac{n}{\log k} + D\right)</math> in the distributed communication model.<ref name="Fraigniaud2006"/>

==See also==

* [[MazesMaze]]

* [[Maze generation algorithm]]