Dijkstra's algorithm. It picks the unvisited vertex with the lowest-distance, calculates the distance through it to each unvisited neighbor, and updates the neighbor's distance if smaller. Mark visited (set to red) when done with neighbors.
|Worst case performance|
|Graph and tree
Dijkstra's algorithm, conceived by computer scientist Edsger Dijkstra in 1956 and published in 1959, is an algorithm for finding the shortest paths between nodes in graph (which may represent, for example, road networks). The algorithm exists in many variants; Dijkstra's original variant found the shortest path between two nodes, but a more common variant fixes a single node as the "source" node and finds shortest paths from the source to all other nodes in the graph, producing a shortest path tree.
For a given source vertex (node) in the graph, the algorithm finds the path with lowest cost (i.e. the shortest path) between that vertex and every other vertex:196–206 (although Dijkstra originally only considered the shortest path between a given pair of nodes). It can also be used for finding costs of shortest paths from a single vertex to a single destination vertex by stopping the algorithm once the shortest path to the destination vertex has been determined. For example, if the vertices of the graph represent cities and edge path costs represent driving distances between pairs of cities connected by a direct road, Dijkstra's algorithm can be used to find the shortest route between one city and all other cities. As a result, the shortest path algorithm is widely used in network routing protocols, most notably IS-IS and OSPF (Open Shortest Path First). It is also employed as a subroutine in other algorithms such as Johnson's.
Dijkstra's original algorithm does not use a min-priority queue and runs in time (where is the number of vertices). The idea of this algorithm is also given in (Leyzorek et al. 1957). The implementation based on a min-priority queue implemented by a Fibonacci heap and running in (where is the number of edges) is due to (Fredman & Tarjan 1984). This is asymptotically the fastest known single-source shortest-path algorithm for arbitrary directed graphs with unbounded non-negative weights.
In some fields, in particular artificial intelligence, Dijkstra's algorithm or a variant of it is known as uniform-cost search and formulated as an instance of the more general idea of best-first search.
Dijkstra thought about shortest path problem when working at Mathematical Center in Amsterdam in 1956 as a program to demonstrate capabilities of the new computer called ARMAC. His objective was to choose both a problem as well as answer (that will be produced by computer) that non-computing people can understand. He designed the shortest path algorithm in about 20 minutes without aid of paper and pen and later implemented it for ARMAC for slightly simplified transportation map of 64 cities in Netherland (ARMAC was 6-bit computer and hence could hold 64 cities comfortably). A year later, he came across another problem from hardware engineers working on the institute's next computer: minimize the amount of wire needed to connect the pins on the back panel of the machine. As a solution, he re-discovered the algorithm known as Prim's minimal spanning tree algorithm in the same year that Prim re-discovered it. Dijkstra delayed publication of both algorithms until 1959.
Let the node at which we are starting be called the initial node. Let the distance of node Y be the distance from the initial node to Y. Dijkstra's algorithm will assign some initial distance values and will try to improve them step by step.
- Assign to every node a tentative distance value: set it to zero for our initial node and to infinity for all other nodes.
- Set the initial node as current. Mark all other nodes unvisited. Create a set of all the unvisited nodes called the unvisited set.
- For the current node, consider all of its unvisited neighbors and calculate their tentative distances. Compare the newly calculated tentative distance to the current assigned value and assign the smaller one. For example, if the current node A is marked with a distance of 6, and the edge connecting it with a neighbor B has length 2, then the distance to B (through A) will be 6 + 2 = 8. If B was previously marked with a distance greater than 8 then change it to 8. Otherwise, keep the current value.
- When we are done considering all of the neighbors of the current node, mark the current node as visited and remove it from the unvisited set. A visited node will never be checked again.
- If the destination node has been marked visited (when planning a route between two specific nodes) or if the smallest tentative distance among the nodes in the unvisited set is infinity (when planning a complete traversal; occurs when there is no connection between the initial node and remaining unvisited nodes), then stop. The algorithm has finished.
- Select the unvisited node that is marked with the smallest tentative distance, and set it as the new "current node" then go back to step 3.
- Note: For ease of understanding, this discussion uses the terms intersection, road and map — however, in formal notation these terms are vertex, edge and graph, respectively.
Suppose you would like to find the shortest path between two intersections on a city map, a starting point and a destination. The order is conceptually simple: to start, mark the distance to every intersection on the map with infinity. This is done not to imply there is an infinite distance, but to note that intersection has not yet been visited; some variants of this method simply leave the intersection unlabeled. Now, at each iteration, select a current intersection. For the first iteration, the current intersection will be the starting point and the distance to it (the intersection's label) will be zero. For subsequent iterations (after the first), the current intersection will be the closest unvisited intersection to the starting point—this will be easy to find.
From the current intersection, update the distance to every unvisited intersection that is directly connected to it. This is done by determining the sum of the distance between an unvisited intersection and the value of the current intersection, and relabeling the unvisited intersection with this value if it is less than its current value. In effect, the intersection is relabeled if the path to it through the current intersection is shorter than the previously known paths. To facilitate shortest path identification, in pencil, mark the road with an arrow pointing to the relabeled intersection if you label/relabel it, and erase all others pointing to it. After you have updated the distances to each neighboring intersection, mark the current intersection as visited and select the unvisited intersection with lowest distance (from the starting point) – or the lowest label—as the current intersection. Nodes marked as visited are labeled with the shortest path from the starting point to it and will not be revisited or returned to.
Continue this process of updating the neighboring intersections with the shortest distances, then marking the current intersection as visited and moving onto the closest unvisited intersection until you have marked the destination as visited. Once you have marked the destination as visited (as is the case with any visited intersection) you have determined the shortest path to it, from the starting point, and can trace your way back, following the arrows in reverse.
Of note is the fact that this algorithm makes no attempt to direct "exploration" towards the destination as one might expect. Rather, the sole consideration in determining the next "current" intersection is its distance from the starting point. This algorithm, therefore "expands outward" from the starting point, interactively considering every node that is closer in terms of shortest path distance until it reaches the destination. When understood in this way, it is clear how the algorithm necessarily finds the shortest path, however, it may also reveal one of the algorithm's weaknesses: its relative slowness in some topologies.
In the following algorithm, the code
u ← vertex in Q with min dist[u], searches for the vertex
u in the vertex set
Q that has the least
length(u, v) returns the length of the edge joining (i.e. the distance between) the two neighbor-nodes
v. The variable
alt on line 19 is the length of the path from the root node to the neighbor node
v if it were to go through
u. If this path is shorter than the current shortest path recorded for
v, that current path is replaced with this
alt path. The
previous array is populated with a pointer to the "next-hop" node on the source graph to get the shortest route to the source.
1 function Dijkstra(Graph, source): 2 3 dist[source] ← 0 // Distance from source to source 4 prev[source] ← undefined // Previous node in optimal path initialization 5 6 for each vertex v in Graph: // Initialization 7 if v ≠ source // Where v has not yet been removed from Q (unvisited nodes) 8 dist[v] ← infinity // Unknown distance function from source to v 9 prev[v] ← undefined // Previous node in optimal path from source 10 end if 11 add v to Q // All nodes initially in Q (unvisited nodes) 12 end for 13 14 while Q is not empty: 15 u ← vertex in Q with min dist[u] // Source node in first case 16 remove u from Q 17 18 for each neighbor v of u: // where v is still in Q. 19 alt ← dist[u] + length(u, v) 20 if alt < dist[v]: // A shorter path to v has been found 21 dist[v] ← alt 22 prev[v] ← u 23 end if 24 end for 25 end while 26 27 return dist, prev 28 29 end function
If we are only interested in a shortest path between vertices
target, we can terminate the search after line 15 if
u = target. Now we can read the shortest path from
target by reverse iteration:
1 S ← empty sequence 2 u ← target 3 while prev[u] is defined: // Construct the shortest path with a stack S 4 insert u at the beginning of S // Push the vertex onto the stack 5 u ← prev[u] // Traverse from target to source 6 end while
S is the list of vertices constituting one of the shortest paths from
target, or the empty sequence if no path exists.
A more general problem would be to find all the shortest paths between
target (there might be several different ones of the same length). Then instead of storing only a single node in each entry of
previous we would store all nodes satisfying the relaxation condition. For example, if both
source connect to
target and both of them lie on different shortest paths through
target (because the edge cost is the same in both cases), then we would add both
previous[target]. When the algorithm completes,
previous data structure will actually describe a graph that is a subset of the original graph with some edges removed. Its key property will be that if the algorithm was run with some starting node, then every path from that node to any other node in the new graph will be the shortest path between those nodes in the original graph, and all paths of that length from the original graph will be present in the new graph. Then to actually find all these shortest paths between two given nodes we would use a path finding algorithm on the new graph, such as depth-first search.
Using a priority queue
A min-priority queue is an abstract data structure that provides 3 basic operations :
extract_min(). As mentioned earlier, using such a data structure can lead to faster computing times than using a basic queue. Notably, Fibonacci heap (Fredman & Tarjan 1984) or Brodal queue offer optimal implementations for those 3 operations. As the algorithm is slightly different, we mention it here, in pseudo-code as well :
1 function Dijkstra(Graph, source): 2 dist[source] ← 0 // Initialization 3 for each vertex v in Graph: 4 if v ≠ source 5 dist[v] ← infinity // Unknown distance from source to v 6 prev[v] ← undefined // Predecessor of v 7 end if 8 Q.add_with_priority(v, dist[v]) 9 end for 10 11 while Q is not empty: // The main loop 12 u ← Q.extract_min() // Remove and return best vertex 13 for each neighbor v of u: 14 alt = dist[u] + length(u, v) 15 if alt < dist[v] 16 dist[v] ← alt 17 prev[v] ← u 18 Q.decrease_priority(v, alt) 19 end if 20 end for 21 end while 21 return prev
Instead of filling the priority queue with all nodes in the initialization phase, it is also possible to initialize it to contain only source; then, inside the
if alt < dist[v] block, the node must be inserted if not already in the queue (instead of performing a
It should be noted that other data structures can be used to achieve even faster computing times in practice.
Bounds of the running time of Dijkstra's algorithm on a graph with edges and vertices can be expressed as a function of the number of edges, denoted , and the number of vertices, denoted , using big-O notation. How tight a bound is possible depends on the way the vertex set is implemented. In the following, note that upper bounds can be simplified because for any graph, but that simplification disregards the fact that in some problems, other upper bounds on may hold.
For any implementation of the vertex set , the running time is in
where and are the complexities of the decrease-key and extract-minimum operations in , respectively. The simplest implementation of the Dijkstra's algorithm stores the vertex set as an ordinary linked list or array, and extract-minimum is simply a linear search through all vertices in . In this case, the running time is .
For sparse graphs, that is, graphs with far fewer than edges, Dijkstra's algorithm can be implemented more efficiently by storing the graph in the form of adjacency lists and using a self-balancing binary search tree, binary heap, pairing heap, or Fibonacci heap as a priority queue to implement extracting minimum efficiently. To perform decrease-key steps in a binary heap efficiently, it is necessary to use an auxiliary data structure that maps each vertex to its position in the heap, and to keep this structure up to date as the priority queue changes. With a self-balancing binary search tree or binary heap, the algorithm requires
time in the worst case; for connected graphs this time bound can be simplified to . The Fibonacci heap improves this to
When using binary heaps, the average case time complexity is lower than the worst-case: assuming edge costs are drawn independently from a common probability distribution, the expected number of decrease-key operations is bounded by , giving a total running time of:199–200
Practical optimizations and infinite graphs
In common presentations of Dijkstra's algorithm, initially all nodes are entered into the priority queue. This is, however, not necessary: the algorithm can start with a priority queue that contains only one item, and insert new items as they are discovered (instead of doing a decrease-key, check whether the key is in the queue; if it is, decrease its key, otherwise insert it).:198 This variant has the same worst-case bounds as the common variant, but maintains a smaller priority queue in practice, speeding up the queue operations.
Moreover, not inserting all nodes in a graph makes it possible to extend the algorithm to find the shortest path from a single source to the closest of a set of target nodes on infinite graphs or those too large to represent in memory. The resulting algorithm is called uniform-cost search (UCS) in the artificial intelligence literature and can be expressed in pseudocode as
procedure UniformCostSearch(Graph, start, goal) node ← start cost ← 0 frontier ← priority queue containing node only explored ← empty set do if frontier is empty return failure node := frontier.pop() if node is goal return solution explored.add(node) for each of node's neighbors n if n is not in explored if n is not in frontier frontier.add(n) else if n is in frontier with higher cost replace existing node with n
The complexity of this algorithm can be expressed in an alternative way for very large graphs: when C* is the length of the shortest path from the start node to any node satisfying the "goal" predicate, and each edge has cost at least ε, then the algorithm's worst-case time and space complexity are both in O(b⌈C* ⁄ ε⌉).
When arc weights are integers and bounded by a constant C, the usage of a special priority queue structure by Van Emde Boas etal.(1977) (Ahuja et al. 1990) brings the complexity to . Another interesting implementation based on a combination of a new radix heap and the well-known Fibonacci heap runs in time (Ahuja et al. 1990). Finally, the best algorithms in this special case are as follows. The algorithm given by (Thorup 2000) runs in time and the algorithm given by (Raman 1997) runs in time.
Also, for directed acyclic graphs, it is possible to find shortest paths from a given starting vertex in linear time, by processing the vertices in a topological order, and calculating the path length for each vertex to be the minimum length obtained via any of its incoming edges.
In the special case of integer weights and undirected graphs, the Dijkstra's algorithm can be completely countered with a linear complexity algorithm, given by (Thorup 1999).
Related problems and algorithms
The functionality of Dijkstra's original algorithm can be extended with a variety of modifications. For example, sometimes it is desirable to present solutions which are less than mathematically optimal. To obtain a ranked list of less-than-optimal solutions, the optimal solution is first calculated. A single edge appearing in the optimal solution is removed from the graph, and the optimum solution to this new graph is calculated. Each edge of the original solution is suppressed in turn and a new shortest-path calculated. The secondary solutions are then ranked and presented after the first optimal solution.
Unlike Dijkstra's algorithm, the Bellman–Ford algorithm can be used on graphs with negative edge weights, as long as the graph contains no negative cycle reachable from the source vertex s. The presence of such cycles means there is no shortest path, since the total weight becomes lower each time the cycle is traversed. It is possible to adapt Dijkstra's algorithm to handle negative weight edges by combining it with the Bellman-Ford algorithm (to remove negative edges and detect negative cycles), such an algorithm is called Johnson's algorithm.
The A* algorithm is a generalization of Dijkstra's algorithm that cuts down on the size of the subgraph that must be explored, if additional information is available that provides a lower bound on the "distance" to the target. This approach can be viewed from the perspective of linear programming: there is a natural linear program for computing shortest paths, and solutions to its dual linear program are feasible if and only if they form a consistent heuristic (speaking roughly, since the sign conventions differ from place to place in the literature). This feasible dual / consistent heuristic defines a non-negative reduced cost and A* is essentially running Dijkstra's algorithm with these reduced costs. If the dual satisfies the weaker condition of admissibility, then A* is instead more akin to the Bellman–Ford algorithm.
The process that underlies Dijkstra's algorithm is similar to the greedy process used in Prim's algorithm. Prim's purpose is to find a minimum spanning tree that connects all nodes in the graph; Dijkstra is concerned with only two nodes. Prim's does not evaluate the total weight of the path from the starting node, only the individual path.
Breadth-first search can be viewed as a special-case of Dijkstra's algorithm on unweighted graphs, where the priority queue degenerates into a FIFO queue.
Fast marching method can be viewed as a continuous version of Dijkstra's algorithm which computes the geodesic distance on a triangle mesh.
Dynamic programming perspective
From a dynamic programming point of view, Dijkstra's algorithm is a successive approximation scheme that solves the dynamic programming functional equation for the shortest path problem by the Reaching method.
In fact, Dijkstra's explanation of the logic behind the algorithm, namely
Problem 2. Find the path of minimum total length between two given nodes and .
We use the fact that, if is a node on the minimal path from to , knowledge of the latter implies the knowledge of the minimal path from to .
- A* search algorithm
- Bellman–Ford algorithm
- Euclidean shortest path
- Flood fill
- Floyd–Warshall algorithm
- Johnson's algorithm
- Longest path problem
- Dijkstra, Edsger; Thomas J. Misa, Editor (August 2010). "An Interview with Edsger W. Dijkstra". Communications of the ACM 53 (8): 41–47. doi:10.1145/1787234.1787249.
What is the shortest way to travel from Rotterdam to Groningen? It is the algorithm for the shortest path which I designed in about 20 minutes. One morning I was shopping with my young fiancée, and tired, we sat down on the café terrace to drink a cup of coffee and I was just thinking about whether I could do this, and I then designed the algorithm for the shortest path.
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