A* search algorithm
From Wikipedia, the free encyclopedia
Categories: Graph algorithms | Routing algorithms | Search algorithms | Combinatorial optimization | Game artificial intelligence
In computer science, A* (pronounced "A star") is a best-first, graph search algorithm that finds the least-cost path from a given initial node to one goal node (out of one or more possible goals). It uses a distance-plus-cost heuristic function (usually denoted Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) ) to determine the order in which the search visits nodes in the tree. The distance-plus-cost heuristic is a sum of two functions: the path-cost function (usually denoted Failed to parse (Missing texvc executable; please see math/README to configure.): g(x) , which may or may not be a heuristic) and an admissible "heuristic estimate" of the distance to the goal (usually denoted Failed to parse (Missing texvc executable; please see math/README to configure.): h(x) ). The path-cost function Failed to parse (Missing texvc executable; please see math/README to configure.): g(x) is the cost from the starting node to the current node. Since the Failed to parse (Missing texvc executable; please see math/README to configure.): h(x) part of the Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) function must be an admissible heuristic, it must underestimate the distance to the goal. Thus for an application like routing, Failed to parse (Missing texvc executable; please see math/README to configure.): h(x) might represent the straight-line distance to the goal, since that is physically the smallest possible distance between any two points (or nodes for that matter). The algorithm was first described in 1968 by Peter Hart, Nils Nilsson, and Bertram Raphael. In their paper, it was called algorithm A. Since using this algorithm yields optimal behavior for a given heuristic, it has been called A*. This algorithm has been generalized into a bidirectional heuristic search algorithm; see bidirectional search.
Algorithm descriptionImage:A+ Pathfinding Algorithm.png
A single-step simulation in a Visual Basic GUI (link see below)
A* incrementally searches all routes leading from the starting point until it finds the shortest path to a goal. Like all informed search algorithms, it searches first the routes that appear to be most likely to lead towards the goal. What sets A* apart from a greedy best-first search is that it also takes the distance already traveled into account (the Failed to parse (Missing texvc executable; please see math/README to configure.): g(x) part of the heuristic is the cost from the start, and not simply the local cost from the previously expanded node). Starting with a given node, the algorithm expands the node with the lowest Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) value—the node that has the highest cost-per-benefit. A* maintains a set of partial solutions—unexpanded leaf nodes of expanded nodes—stored in a priority queue. The priority assigned to a path Failed to parse (Missing texvc executable; please see math/README to configure.): x is determined by the function Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = g(x) + h(x) . The function continues until a goal has a lower Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) value than any node in the queue (or until the tree is fully traversed). Multiple goals may be passed over if there is a path that may lead to a lower-cost goal. The lower Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) , the higher the priority (so a min-heap could be used to implement the queue). <source lang="matlab">
function A*(start,goal)
var closed := the empty set
var q := make_queue(path(start))
while q is not empty
var p := remove_first(q)
var x := the last node of p
if x in closed
continue
if x = goal
return p
add x to closed
foreach y in successors(x)
enqueue(q, p, y)
return failure
</source> The closed set can be omitted (yielding a tree search algorithm) if either a solution is guaranteed to exist, or if the PropertiesLike breadth-first search, A* is complete in the sense that it will always find a solution if there is one. If the heuristic function Failed to parse (Missing texvc executable; please see math/README to configure.): h is admissible, meaning that it never overestimates the actual minimal cost of reaching the goal, then A* is itself admissible (or optimal) if we do not use a closed set. If a closed set is used, then Failed to parse (Missing texvc executable; please see math/README to configure.): h must also be monotonic (or consistent) for A* to be optimal. Admissible means that the heuristic function never overestimates the cost of getting from a node to its neighbor, while being monotonic means that if there is a connection from node A to node C, and a connection from A to B to C, the estimated cost from A to C will always be less than or equal to the estimated cost from A to B + the estimated cost from B to C. (Monotonicity is known as the triangle inequality, for one side of a triangle cannot be longer than the sum of the other two.) Formally, for all paths Failed to parse (Missing texvc executable; please see math/README to configure.): x,y where Failed to parse (Missing texvc executable; please see math/README to configure.): y is a successor of Failed to parse (Missing texvc executable; please see math/README to configure.): x
exactly predicts the cost of the optimal path. While optimal in arbitrary graphs, it is not guaranteed to perform better than simpler search algorithms that are more informed about the problem domain. For example, in a maze-like environment, the only way to reach the goal might be to first travel one way (away from the goal) and eventually double back. In this case trying nodes closer to your destination first may cost you more time. Special casesGenerally speaking, depth-first search and breadth-first search are two special cases of A* algorithm. Dijkstra's algorithm, as another example of a best-first search algorithm, is the special case of A* where Failed to parse (Missing texvc executable; please see math/README to configure.): h(x) = 0 for all Failed to parse (Missing texvc executable; please see math/README to configure.): x . For depth-first search, we may consider that there is a global counter C initialized with a very big value. Every time we process a node we assign C to all of its newly discovered neighbors. After each single assignment, we decrease the counter C by one. Thus the earlier a node is discovered, the higher its h(x) value. Why A* is admissible and computationally optimalA* is both admissible and considers fewer nodes than any other admissible search algorithm with the same heuristic, because A* works from an “optimistic” estimate of the cost of a path through every node that it considers — optimistic in that the true cost of a path through that node to the goal will be at least as great as the estimate. But, critically, as far as A* “knows”, that optimistic estimate might be achievable. When A* terminates its search, it has, by definition, found a path whose actual cost is lower than the estimated cost of any path through any open node. But since those estimates are optimistic, A* can safely ignore those nodes. In other words, A* will never overlook the possibility of a lower-cost path and so is admissible. Suppose now that some other search algorithm A terminates its search with a path whose actual cost is not less than the estimated cost of a path through some open node. Algorithm A cannot rule out the possibility, based on the heuristic information it has, that a path through that node might have a lower cost. So while A might consider fewer nodes than A*, it cannot be admissible. Accordingly, A* considers the fewest nodes of any admissible search algorithm that uses a no more accurate heuristic estimate. ComplexityThe time complexity of A* depends on the heuristic. In the worst case, the number of nodes expanded is exponential in the length of the solution (the shortest path), but it is polynomial when the heuristic function h meets the following condition:
is the optimal heuristic, i.e. the exact cost to get from Failed to parse (Missing texvc executable; please see math/README to configure.): x to the goal. In other words, the error of h should not grow faster than the logarithm of the “perfect heuristic” Failed to parse (Missing texvc executable; please see math/README to configure.): h^* that returns the true distance from x to the goal (Russell and Norvig 2003, p. 101). More problematic than its time complexity is A*’s memory usage. In the worst case, it must also remember an exponential number of nodes. Several variants of A* have been developed to cope with this, including iterative deepening A* (IDA*), memory-bounded A* (MA*) and simplified memory bounded A* (SMA*) and recursive best-first search (RBFS). References
External links
es:Algoritmo de búsqueda A* fr:Algorithme A* ko:A* 알고리즘 it:A* nl:A*-algoritme ja:A* pl:Algorytm A* pt:Algoritmo A* fi:A*-algoritmi vi:Giải thuật tìm kiếm A* |
