A* search algorithm
In computer science, A* is a computer algorithm, used in pathfinding and graph traversal, the process of finding a path between multiple points, called "nodes". It enjoys widespread use due to its accuracy. However, in practical travel-routing systems, it is outperformed by algorithms which can pre-process the graph to attain better performance, although other work has found A* to be superior to other approaches. Peter Hart, Nils Nilsson and Bertram Raphael of Stanford Research Institute first published the algorithm in 1968, it can be seen as an extension of Edsger Dijkstra's 1959 algorithm. A* achieves better performance by using heuristics to guide its search. A* was created as part of the Shakey project, which had the aim of building a mobile robot that could plan its own actions. Nils Nilsson proposed using the Graph Traverser algorithm for Shakey's path planning. Graph Traverser is guided by a heuristic function h, the estimated distance from node n to the goal node, it ignores g, the distance from the start node to n.
Bertram Raphael suggested using the sum, g + h. Peter Hart invented the concepts we now call admissibility and consistency of heuristic functions. A* was designed for finding least-cost paths when the cost of a path is the sum of its edge costs, but it has been shown that A* can be used to find optimal paths for any problem satisfying the conditions of a cost algebra; the original 1968 A* paper contained a theorem that no A*-like algorithm could expand fewer nodes than A* if the heuristic function is consistent and A*’s tie-breaking rule is suitably chosen. A correction was published a few years showing that consistency was not required, just admissibility. However, what was shown was that no algorithm can be more efficient than A* with respect to the set of expanded nodes; as shown in Dechter and Pearl’s definitive study of A*'s optimality, it is possible for a different correct algorithm to outperform A* on some graphs while losing to A* on other graphs, or to expand a set of nodes that neither contains the set of nodes expanded by A*, nor is contained in it.
A* is an informed search algorithm, or a best-first search, meaning that it is formulated in terms of weighted graphs: starting from a specific starting node of a graph, it aims to find a path to the given goal node having the smallest cost. It does this by maintaining a tree of paths originating at the start node and extending those paths one edge at a time until its termination criterion is satisfied. At each iteration of its main loop, A * needs to determine, it does so based on the cost of the path and an estimate of the cost required to extend the path all the way to the goal. A* selects the path that minimizes f = g + h where n is the next node on the path, g is the cost of the path from the start node to n, h is a heuristic function that estimates the cost of the cheapest path from n to the goal. A* terminates when the path it chooses to extend is a path from start to goal or if there are no paths eligible to be extended; the heuristic function is problem-specific. If the heuristic function is admissible, meaning that it never overestimates the actual cost to get to the goal, A* is guaranteed to return a least-cost path from start to goal.
Typical implementations of A* use a priority queue to perform the repeated selection of minimum cost nodes to expand. This priority queue is known as fringe. At each step of the algorithm, the node with the lowest f value is removed from the queue, the f and g values of its neighbors are updated accordingly, these neighbors are added to the queue; the algorithm continues. The f value of the goal is the cost of the shortest path, since h at the goal is zero in an admissible heuristic; the algorithm described so far gives us only the length of the shortest path. To find the actual sequence of steps, the algorithm can be revised so that each node on the path keeps track of its predecessor. After this algorithm is run, the ending node will point to its predecessor, so on, until some node's predecessor is the start node; as an example, when searching for the shortest route on a map, h might represent the straight-line distance to the goal, since, physically the smallest possible distance between any two points.
If the heuristic h satisfies the additional condition h ≤ d + h for every edge of the graph h is called monotone, or consistent. In such a case, A* can be implemented more efficiently—roughly speaking, no node needs to be processed more than once —and A* is equivalent to running Dijkstra's algorithm with the reduced cost d' = d + h − h; the following pseudocode describes the algorithm: Remark: the above pseudocode assumes that the heuristic function is monotonic, a frequent case in many practical problems, such as the Shortest Distance Path in road networks. However, if
Peter Norvig is an American computer scientist. He is used to be its director of search quality. Norvig received a Bachelor of Science in applied mathematics from Brown University and a Ph. D. in computer science from the University of California, Berkeley. Norvig is an AAAI Fellow and councilor of the Association for the Advancement of Artificial Intelligence and co-author, with Stuart Russell, of Artificial Intelligence: A Modern Approach, now the leading college text in the field, he was head of the Computational Sciences Division at NASA Ames Research Center, where he oversaw a staff of 200 scientists performing NASA's research and development in autonomy and robotics, automated software engineering and data analysis, collaborative systems research, simulation-based decision-making. Before that he was chief scientist at Junglee, where he helped develop one of the first Internet comparison shopping services. Norvig has served an assistant professor at the University of Southern California and a research faculty member at Berkeley.
He has over fifty publications in various areas of computer science, concentrating on artificial intelligence, natural language processing, information retrieval and software engineering, including the books Artificial Intelligence: A Modern Approach, Paradigms of AI Programming: Case Studies in Common Lisp, Verbmobil: A Translation System for Face-to-Face Dialog, Intelligent Help Systems for UNIX. Norvig is one of the creators of JScheme. In 2006 he was inducted as a fellow of the Association for Computing Machinery. Norvig is listed under "Academic Advisors" for the Singularity University. In 2011, Norvig worked with Sebastian Thrun to develop a popular online course in Artificial Intelligence that had more than 160,000 students enrolled, he teaches an online course via the Udacity platform. He believes. In 2001, Norvig published a short article titled Teach Yourself Programming in Ten Years, arguing against the fashionable introductory programming textbooks that purported to teach programming in days or weeks.
The article was shared and discussed, has attracted contributed translations to over 20 languages. Norvig is known for his Gettysburg Powerpoint Presentation, a satire about bad presentation practices using Abraham Lincoln's famous Gettysburg Address; the Prospects for AI, featuring Neil Jacobstein, Patrick Lincoln, Peter Norvig, Bruno Olshausen An experiment by Norvig on Scientific opinion on climate change Teach Yourself Programming in Ten Years
Donald Michie was a British researcher in artificial intelligence. During World War II, Michie worked for the Government Code and Cypher School at Bletchley Park, contributing to the effort to solve "Tunny," a German teleprinter cipher. Michie was born in Burma, he won a scholarship to study classics at Balliol College, Oxford. In Spring 1943, looking for some way to contribute to the war effort, Michie instead attempted to enroll on a Japanese language course in Bedford for intelligence officers. On arrival, it transpired that he had been misinformed, instead he trained in cryptography, displaying a natural aptitude for the subject. Six weeks he was recruited to Bletchley Park and was assigned to the "Testery," a section which tackled a German teleprinter cipher. During his time at Bletchley Park he worked with Max Newman and Jack Good. Between 1945 and 1952 he studied at Balliol College, Oxford. In 1960, he developed the Machine Educable Noughts And Crosses Engine, one of the first programs capable of learning to play a perfect game of Tic-Tac-Toe.
Since computers were not available at this time, Michie implemented his program with about 304 matchboxes, each representing a unique board state. Each matchbox was filled with coloured beads, each representing a different move in that board state; the quantity of a colour indicated the "certainty" that playing the corresponding move would lead to a win. The program was trained by playing hundreds of games and updating the quantities of beads in each matchbox depending on the outcome of each game. Michie was director of the University of Edinburgh's Department of Machine Intelligence and Perception from its establishment in 1965; the machine intelligence unit predated the university's computer science unit. He remained at Edinburgh until 1985. Active in the research community into his eighties, he devoted the last decade of his life to the UK charity The Human Computer Learning Foundation, worked with Stephen Muggleton, Claude Sammut, Richard Wheeler, others on natural language systems and theories of intelligence.
In 2007 he was completing a series of scientific articles on the Sophie Natural Language System and a book manuscript entitled "Jehovah's Creatures". Michie invented the memoisation technique, he was founder and Treasurer of the Human-Computer Learning Foundation, a charity registered in the UK. He was awarded numerous fellowships and honours during his career including: Fellow of the Royal Society of Edinburgh Honorary Fellow of the American National Academy of Sciences Honorary Fellow of the Slovenian Academy of Sciences. Fellow of the British Computer Society Michie was married three times, the second to biologist Anne McLaren from 1952 to 1959, he had four children, one by his first wife, three by Prof. McLaren, including economist Jonathan Michie and health psychologist Susan Michie. Michie and McLaren remained friends after their divorce, became close again after the death of his third wife. On 7 July 2007 Michie and McLaren were killed in a car crash, while travelling from Cambridge to London
In computer science, graph traversal refers to the process of visiting each vertex in a graph. Such traversals are classified by the order. Tree traversal is a special case of graph traversal. Unlike tree traversal, graph traversal may require that some vertices be visited more than once, since it is not known before transitioning to a vertex that it has been explored; as graphs become more dense, this redundancy becomes more prevalent, causing computation time to increase. Thus, it is necessary to remember which vertices have been explored by the algorithm, so that vertices are revisited as infrequently as possible; this may be accomplished by associating each vertex of the graph with a "color" or "visitation" state during the traversal, checked and updated as the algorithm visits each vertex. If the vertex has been visited, it is ignored and the path is pursued no further. Several special cases of graphs imply the visitation of other vertices in their structure, thus do not require that visitation be explicitly recorded during the traversal.
An important example of this is a tree: during a traversal it may be assumed that all "ancestor" vertices of the current vertex have been visited. Both the depth-first and breadth-first graph searches are adaptations of tree-based algorithms, distinguished by the lack of a structurally determined "root" vertex and the addition of a data structure to record the traversal's visitation state. Note. — If each vertex in a graph is to be traversed by a tree-based algorithm the algorithm must be called at least once for each connected component of the graph. This is accomplished by iterating through all the vertices of the graph, performing the algorithm on each vertex, still unvisited when examined. A depth-first search is an algorithm for traversing a finite graph. DFS visits the child vertices before visiting the sibling vertices. A stack is used when implementing the algorithm; the algorithm begins with a chosen "root" vertex. The algorithm backtracks along visited vertices, until it finds a vertex connected to yet more uncharted territory.
It will proceed down the new path as it had before, backtracking as it encounters dead-ends, ending only when the algorithm has backtracked past the original "root" vertex from the first step. DFS is the basis for many graph-related algorithms, including topological sorts and planarity testing. Input: A graph G and a vertex v of G. Output: A labeling of the edges in the connected component of v as discovery edges and back edges.1 procedure DFS: 2 label v as explored 3 for all edges e in G.incidentEdges do 4 if edge e is unexplored 5 w ← G.adjacentVertex 6 if vertex w is unexplored 7 label e as a discovered edge 8 recursively call DFS 9 else 10 label e as a back edge A breadth-first search is another technique for traversing a finite graph. BFS visits the sibling vertices before visiting the child vertices, a queue is used in the search process; this algorithm is used to find the shortest path from one vertex to another. Input: A graph G and a vertex v of G. Output: The closest vertex to v satisfying some conditions, or null if no such vertex exists.1 procedure BFS: 2 create a queue Q 3 enqueue v onto Q 4 mark v 5 while Q is not empty do 6 w ← Q.dequeue 7 if w is what we are looking for 8 return w 9 for all edges e in G.adjacentEdges do 12 x ← G.adjacentVertex 13 if x is not marked 14 mark x 15 enqueue x onto Q 16 return null Breadth-first search can be used to solve many problems in graph theory, for example: finding all vertices within one connected component.
The problem of graph exploration can be seen as a variant of graph traversal. It is an online problem, meaning that the information about the graph is only revealed during the runtime of the algorithm. A common model is as follows: given a connected graph G = with non-negative edge weights; the algorithm starts at some vertex, knows all incident outgoing edges and the vertices at the end of these edges—but not more. When a new vertex is visited again all incident outgoing edges and the vertices at the end are known; the goal is to visit all n vertices and return to the starting vertex, but the sum of the weights of the tour should be as small as possible. The problem can be understood as a specific version of the travelli
In numerical analysis, hill climbing is a mathematical optimization technique which belongs to the family of local search. It is an iterative algorithm that starts with an arbitrary solution to a problem attempts to find a better solution by making an incremental change to the solution. If the change produces a better solution, another incremental change is made to the new solution, so on until no further improvements can be found. For example, hill climbing can be applied to the travelling salesman problem, it is easy to find an initial solution that visits all the cities but will be poor compared to the optimal solution. The algorithm starts with such a solution and makes small improvements to it, such as switching the order in which two cities are visited. A much shorter route is to be obtained. Hill climbing finds optimal solutions for convex problems – for other problems it will find only local optima, which are not the best possible solution out of all possible solutions. Examples of algorithms that solve convex problems by hill-climbing include the simplex algorithm for linear programming and binary search.
To attempt to avoid getting stuck in local optima, one could use restarts, or more complex schemes based on iterations, or on memory, or on memory-less stochastic modifications. The relative simplicity of the algorithm makes it a popular first choice amongst optimizing algorithms, it is used in artificial intelligence, for reaching a goal state from a starting node. Different choices for next nodes and starting nodes are used in related algorithms. Although more advanced algorithms such as simulated annealing or tabu search may give better results, in some situations hill climbing works just as well. Hill climbing can produce a better result than other algorithms when the amount of time available to perform a search is limited, such as with real-time systems, so long as a small number of increments converges on a good solution. At the other extreme, bubble sort can be viewed as a hill climbing algorithm, yet this approach is far from efficient for modest N, as the number of exchanges required grows quadratically.
Hill climbing is an anytime algorithm: it can return a valid solution if it's interrupted at any time before it ends. Hill climbing attempts to maximize a target function f, where x is a vector of continuous and/or discrete values. At each iteration, hill climbing will adjust a single element in x and determine whether the change improves the value of f. With hill climbing, any change that improves f is accepted, the process continues until no change can be found to improve the value of f. X is said to be "locally optimal". In discrete vector spaces, each possible value for x may be visualized as a vertex in a graph. Hill climbing will follow the graph from vertex to vertex, always locally increasing the value of f, until a local maximum x m is reached. In simple hill climbing, the first closer node is chosen, whereas in steepest ascent hill climbing all successors are compared and the closest to the solution is chosen. Both forms fail if there is no closer node, which may happen if there are local maxima in the search space which are not solutions.
Steepest ascent hill climbing is similar to best-first search, which tries all possible extensions of the current path instead of only one. Stochastic hill climbing does not examine all neighbors before deciding. Rather, it selects a neighbor at random, decides whether to move to that neighbor or to examine another. Coordinate descent does a line search along one coordinate direction at the current point in each iteration; some versions of coordinate descent randomly pick a different coordinate direction each iteration. Random-restart hill climbing is a meta-algorithm built on top of the hill climbing algorithm, it is known as Shotgun hill climbing. It iteratively does hill-climbing, each time with a random initial condition x 0; the best x m is kept: if a new run of hill climbing produces a better x m than the stored state, it replaces the stored state. Random-restart hill climbing is a effective algorithm in many cases, it turns out that it is better to spend CPU time exploring the space, than optimizing from an initial condition.
Hill climbing will not find the global maximum, but may instead converge on a local max
The 15-puzzle is a sliding puzzle that consists of a frame of numbered square tiles in random order with one tile missing. The puzzle exists in other sizes the smaller 8-puzzle. If the size is 3×3 tiles, the puzzle is called the 8-puzzle or 9-puzzle, if 4×4 tiles, the puzzle is called the 15-puzzle or 16-puzzle named for the number of tiles and the number of spaces; the object of the puzzle is to place the tiles in order by making sliding moves that use the empty space. The n-puzzle is a classical problem for modelling algorithms involving heuristics. Used heuristics for this problem include counting the number of misplaced tiles and finding the sum of the taxicab distances between each block and its position in the goal configuration. Note that both are admissible, i.e. they never overestimate the number of moves left, which ensures optimality for certain search algorithms such as A*. Johnson & Story used a parity argument to show that half of the starting positions for the n-puzzle are impossible to resolve, no matter how many moves are made.
This is done by considering a function of the tile configuration, invariant under any valid move, using this to partition the space of all possible labeled states into two equivalence classes of reachable and unreachable states. The invariant is the parity of the permutation of all 16 squares plus the parity of the taxicab distance of the empty square from the lower right corner; this is an invariant because each move changes both the parity of the permutation and the parity of the taxicab distance. In particular, if the empty square is in the lower right corner the puzzle is solvable if and only if the permutation of the remaining pieces is even. Johnson & Story showed that the converse holds on boards of size m×n provided m and n are both at least 2: all permutations are solvable; this is straightforward but a little messy to prove by induction on m and n starting with m=n=2. Archer gave another proof, based on defining equivalence classes via a hamiltonian path. Wilson studied the analogue of the 15 puzzle on arbitrary finite connected and non-separable graphs.
He showed that, except for polygons, one exceptional graph on 7 vertices, it is possible to obtain all permutations unless the graph is bipartite, in which case the permutations can be obtained. The exceptional graph is a regular hexagon with one diagonal and a vertex at the center added. For larger versions of the n-puzzle, finding a solution is easy, but the problem of finding the shortest solution is NP-hard, it is NP-hard to approximate the fewest slides within an additive constant, but there is a polynomial-time constant-factor approximation. For the 15-puzzle, lengths of optimal solutions range from 0 to 80 single-tile moves or 43 multi-tile moves; the multi-tile metric counts subsequent moves of the empty tile in the same direction as one. The number of possible positions of the 24-puzzle is 25!/2 ≈ 7.76×1024, too many to calculate God's number. In 2011, lower bounds of 152 single-tile moves or 41 multi-tile moves had been established, as well as upper bounds of 208 single-tile moves or 109 multi-tile moves.
In 2016, the upper bound was improved to 205 single-tile moves. The transformations of the fifteen puzzle form a groupoid. In an alternate view of the problem, we can consider the invariant to be the sum of the parity of the number of inversions in the current order of the 15 numbered pieces and the parity of the difference in the row number of the empty square from the row number of the last row; this is an invariant because each column move, when we move a piece within the same column, changes both the parity of the number of inversions and the parity of the row distance from the last row. Now if we look at the solved state of the puzzle, this sum is even. Hence it is easy to prove by induction that any state of the puzzle for which the above sum is odd cannot be solvable. In particular, if the empty square is in the lower right corner the puzzle is solvable if and only if the number of inversions of the numbered pieces is even; the puzzle was "invented" by Noyes Palmer Chapman, a postmaster in Canastota, New York, said to have shown friends, as early as 1874, a precursor puzzle consisting of 16 numbered blocks that were to be put together in rows of four, each summing to 34.
Copies of the improved Fifteen Puzzle made their way to Syracuse, New York, by way of Noyes' son and from there, via sundry connections, to Watch Hill, Rhode Island, to Hartford, where students in the American School for the Deaf started manufacturing the puzzle and, by December 1879, selling them both locally and in Boston, Massachusetts. Shown one of these, Matthias Rice, who ran a fancy woodworking business in Boston, started manufacturing the puzzle sometime in December 1879 and convinced a "Yankee Notions" fancy goods dealer to sell them under the name of "Gem Puzzle". In late January 1880, Dr. Charles Pevey, a dentist in Worcester, garnered so