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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 a graph search algorithm that solves the single-source shortest path problem for a graph with non-negative edge path costs, producing a shortest path tree. This algorithm is often used in routing and as a subroutine in other graph algorithms.
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. 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 first is widely used in network routing protocols, most notably IS-IS and OSPF (Open Shortest Path First).
Dijkstra's original algorithm does not use a min-priority queue and runs in (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.
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.
- Mark all nodes unvisited. Set the initial node as current. Create a set of the unvisited nodes called the unvisited set consisting of all the nodes.
- For the current node, consider all of its unvisited neighbors and calculate their tentative distances. 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.
- 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, formally 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 smallest dist, searches for the vertex
u in the vertex set
Q that has the least
dist[u] value. That vertex is removed from the set
Q and returned to the user.
dist_between(u, v) calculates the length between the two neighbor-nodes
v. The variable
alt on lines 17 & 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 for each vertex v in Graph: // Initializations 3 dist[v] := infinity; // Mark distances from source to v as not yet computed 4 visited[v] := false; // Mark all nodes as unvisited 5 previous[v] := undefined; // Previous node in optimal path from source 6 end for 7 8 dist[source] := 0; // Distance from source to itself is zero 9 insert source into Q; // Start off with the source node 10 11 while Q is not empty: // The main loop 12 u := vertex in Q with smallest distance in dist and has not been visited; // Source node in first case 13 remove u from Q; 14 visited[u] := true // mark this node as visited 15 16 for each neighbor v of u: 17 alt := dist[u] + dist_between(u, v); // accumulate shortest dist from source 18 if alt < dist[v]: 19 dist[v] := alt; // keep the shortest dist from src to v 20 previous[v] := u; 21 if !visited[v]: 22 insert v into Q; // Add unvisited v into the Q to be processed 23 end if 24 end if 25 end for 26 end while 27 return dist; 28 endfunction
If we are only interested in a shortest path between vertices
target, we can terminate the search at line 12 if
u = target. Now we can read the shortest path from
target by reverse iteration:
1 S := empty sequence 2 u := target 3 while previous[u] is defined: // Construct the shortest path with a stack S 4 insert u at the beginning of S // Push the vertex into the stack 5 u := previous[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 fastest 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 // Initializations 3 for each vertex v in Graph: 4 if v ≠ source 5 dist[v] := infinity // Unknown distance from source to v 6 previous[v] := undefined // Predecessor of v 7 end if 8 PQ.add_with_priority(v,dist[v]) 9 end for 10 11 12 while PQ is not empty: // The main loop 13 u := PQ.extract_min() // Remove and return best vertex 14 for each neighbor v of u: // where v has not yet been removed from Q. 15 alt = dist[u] + length(u, v) 16 if alt < dist[v] // Relax the edge (u,v) 17 dist[v] := alt 18 previous[v] := u 19 PQ.decrease_priority(v,alt) 20 end if 21 end for 22 end while 23 return previous
It should be noted that other data structures can be used to achieve even faster computing times in practice.
An upper bound of the running time of Dijkstra's algorithm on a graph with edges and vertices can be expressed as a function of and using big-O notation.
For any implementation of vertex set the running time is in , where and are times needed to perform decrease key and extract minimum operations in set , respectively.
The simplest implementation of the Dijkstra's algorithm stores vertices of set in an ordinary linked list or array, and extract minimum from 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. With a self-balancing binary search tree or binary heap, the algorithm requires time (which is dominated by , assuming the graph is connected). To avoid look-up in decrease-key step on a vanilla binary heap, it is necessary to maintain a supplementary index mapping each vertex to the heap's index (and keep it up to date as priority queue changes), making it take only time instead. The Fibonacci heap improves this to .
Note that 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.
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.
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 .
- Dijkstra, Edsger; Thomas J. Misa, Editor (2010-08). "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."
- Dijkstra 1959
- Chen, M.; Chowdhury, R. A.; Ramachandran, V.; Roche, D. L.; Tong, L. (2007). Priority Queues and Dijkstra’s Algorithm — UTCS Technical Report TR-07-54 — 12 October 2007. Austin, Texas: The University of Texas at Austin, Department of Computer Sciences.
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- Dijkstra 1959, p. 270
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- Fredman, Michael Lawrence; Tarjan, Robert E. (1984). "Fibonacci heaps and their uses in improved network optimization algorithms". 25th Annual Symposium on Foundations of Computer Science. IEEE. pp. 338–346. doi:10.1109/SFCS.1984.715934.
- Fredman, Michael Lawrence; Tarjan, Robert E. (1987). "Fibonacci heaps and their uses in improved network optimization algorithms". Journal of the Association for Computing Machinery 34 (3): 596–615. doi:10.1145/28869.28874.
- Zhan, F. Benjamin; Noon, Charles E. (February 1998). "Shortest Path Algorithms: An Evaluation Using Real Road Networks". Transportation Science 32 (1): 65–73. doi:10.1287/trsc.32.1.65.
- Leyzorek, M.; Gray, R. S.; Johnson, A. A.; Ladew, W. C.; Meaker, Jr., S. R.; Petry, R. M.; Seitz, R. N. (1957). Investigation of Model Techniques — First Annual Report — 6 June 1956 — 1 July 1957 — A Study of Model Techniques for Communication Systems. Cleveland, Ohio: Case Institute of Technology.
- Knuth, D.E. (1977). "A Generalization of Dijkstra's Algorithm". Information Processing Letters 6 (1): 1–5.
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