Dijkstra's algorithm


Dijkstra's algorithm is an algorithm for finding the shortest paths between nodes in a weighted graph, which may represent, for example, a road network. It was conceived by computer scientist Edsger W. Dijkstra in 1956 and published three years later.
Dijkstra's algorithm finds the shortest path from a given source node to every other node. It can be used to find the shortest path to a specific destination node, by terminating the algorithm after determining the shortest path to that node. For example, if the nodes of the graph represent cities, and the costs of edges represent the distances between pairs of cities connected by a direct road, then Dijkstra's algorithm can be used to find the shortest route between one city and all other cities. A common application of shortest path algorithms is network routing protocols, most notably IS-IS and OSPF. It is also employed as a subroutine in algorithms such as Johnson's algorithm.
The algorithm uses a min-priority queue data structure for selecting the shortest paths known so far. Before more advanced priority queue structures were discovered, Dijkstra's original algorithm ran in time, where is the number of nodes. proposed a Fibonacci heap priority queue to optimize the running time complexity to. This is asymptotically the fastest known single-source shortest-path algorithm for arbitrary directed graphs with unbounded non-negative weights. However, specialized cases can be improved further. If preprocessing is allowed, algorithms such as contraction hierarchies can be up to seven orders of magnitude faster.
Dijkstra's algorithm is commonly used on graphs where the edge weights are positive integers or real numbers. It can be generalized to any graph where the edge weights are partially ordered, provided the subsequent labels are monotonically non-decreasing.
In many fields, particularly artificial intelligence, Dijkstra's algorithm or a variant offers a uniform cost search and is formulated as an instance of the more general idea of best-first search.

History

Dijkstra thought about the shortest path problem while working as a programmer at the Mathematical Center in Amsterdam in 1956. He wanted to demonstrate the capabilities of the new ARMAC computer. His objective was to choose a problem and a computer solution that non-computing people could understand. He designed the shortest path algorithm and later implemented it for ARMAC for a slightly simplified transportation map of 64 cities in the Netherlands. A year later, he came across another problem advanced by hardware engineers working on the institute's next computer: minimize the amount of wire needed to connect the pins on the machine's back panel. As a solution, he re-discovered Prim's minimal spanning tree algorithm. Dijkstra published the algorithm in 1959, two years after Prim and 29 years after Jarník.

Algorithm

The algorithm requires a starting node, and computes the shortest distance from that starting node to each other node. Dijkstra's algorithm starts with infinite distances and tries to improve them step by step:
  1. Create a set of all unvisited nodes: the unvisited set.
  2. Assign to every node a distance from start value: for the starting node, it is zero, and for all other nodes, it is infinity, since initially no path is known to these nodes. During execution, the distance of a node N is the length of the shortest path discovered so far between the starting node and N.
  3. From the unvisited set, select the current node to be the one with the smallest distance; initially, this is the starting node. If the unvisited set is empty, or contains only nodes with infinite distance, then the algorithm terminates by skipping to step 6. If the only concern is the path to a target node, the algorithm terminates once the current node is the target node. Otherwise, the algorithm continues.
  4. For the current node, consider all of its unvisited neighbors and update their distances through the current node; compare the newly calculated distance to the one currently assigned to the neighbor and assign the smaller one to it. For example, if the current node A is marked with a distance of 6, and the edge connecting it with its neighbor B has length 2, then the distance to B through A is 6 + 2 = 8. If B was previously marked with a distance greater than 8, then update it to 8. Otherwise, keep its current distance.
  5. After considering all of the current node's unvisited neighbors, the current node is removed from the unvisited set. Thus a visited node is never rechecked, which is correct because the distance recorded on the current node is minimal, and thus final. Repeat from step 3.
  6. Once the loop exits, every visited node contains its shortest distance from the starting node.

    Description

The shortest path between two intersections on a city map can be found by this algorithm using pencil and paper. Every intersection is listed on a separate line: one is the starting point and is labeled 0. Every other intersection is initially labeled with a distance of infinity. This is done to note that no path to these intersections has yet been established. At each iteration one intersection becomes the current intersection. For the first iteration, this is the starting point.
From the current intersection, the distance to every neighbor intersection is assessed by summing the label of the current intersection and the distance to the neighbor and then relabeling the neighbor with the lesser of that sum and the neighbor's existing label. I.e., the neighbor is relabeled if the path to it through the current intersection is shorter than previously assessed paths. If so, mark the road to the neighbor with an arrow pointing to it, and erase any other arrow that points to it. After the distances to each of the current intersection's neighbors have been assessed, the current intersection is marked as visited. The unvisited intersection with the smallest label becomes the current intersection and the process repeats until all nodes with labels less than the destination's label have been visited.
Once no unvisited nodes remain with a label smaller than the destination's label, the remaining arrows show the shortest path.

Pseudocode

In the following pseudocode, is an array that contains the current distances from the to other vertices, i.e. is the current distance from the source to the vertex. The array contains pointers to previous-hop nodes on the shortest path from source to the given vertex. The code, searches for the vertex in the vertex set that has the least value. returns the length of the edge joining the two neighbor-nodes and. The variable on line 14 is the length of the path from the node to the neighbor node if it were to go through. If this path is shorter than the current shortest path recorded for, then the distance of is updated to.
File:DijkstraDemo.gif|thumb|A demo of Dijkstra's algorithm based on Euclidean distance. Red lines are the shortest path covering, i.e., connecting u and prev. Blue lines indicate where relaxing happens, i.e., connecting v with a node u in Q, which gives a shorter path from the source to v.
1 function Dijkstra:
2
3 for each vertex v in Graph.Vertices:
4 dist ← INFINITY
5 prev ← UNDEFINED
6 add v to Q
7 dist ← 0
8
9 while Q is not empty:
10 u ← vertex in Q with minimum dist
11 Q.remove
12
13 for each edge in Graph:
14 alt ← dist + Graph.Distance
15 if alt < dist:
16 dist ← alt
17 prev ← u
18
19 return dist, prev
To find the shortest path between vertices and, the search terminates after line 10 if. The shortest path from to can be obtained by reverse iteration:
1 S ← empty sequence
2 u ← target
3 if prev is defined or u = source: // Proceed if the vertex is reachable
4 while u is defined: // Construct shortest path with stack S
5 S.push // Push the vertex onto the stack
6 u ← prev // Traverse from target to source
Now sequence is the list of vertices constituting one of the shortest paths from to, or the empty sequence if no path exists.
A more general problem is to find all the shortest paths between and . Then instead of storing only a single node in each entry of all nodes satisfying the relaxation condition can be stored. For example, if both and connect to and they lie on different shortest paths through , then both and are added to. When the algorithm completes, data structure describes a graph that is a subset of the original graph with some edges removed. Its key property is that if the algorithm was run with some starting node, then every path from that node to any other node in the new graph is the shortest path between those nodes graph, and all paths of that length from the original graph are present in the new graph. Then to actually find all these shortest paths between two given nodes, a path finding algorithm on the new graph, such as depth-first search would work.

Using a priority queue

A min-priority queue is an abstract data type that provides 3 basic operations:, and. As mentioned earlier, using such a data structure can lead to faster computing times than using a basic queue. Notably, Fibonacci heap or Brodal queue offer optimal implementations for those 3 operations. As the algorithm is slightly different in appearance, it is mentioned here, in pseudocode as well:
1 function Dijkstra:
2 Q ← Queue storing vertex priority
3
4 dist ← 0 // Initialization
5 Q.add_with_priority // associated priority equals dist
6
7 for each vertex v in Graph.Vertices:
8 if v ≠ source
9 prev ← UNDEFINED // Predecessor of v
10 dist ← INFINITY // Unknown distance from source to v
11 Q.add_with_priority
12
13
14 while Q is not empty: // The main loop
15 u ← Q.extract_min // Remove and return best vertex
16 for each arc : // Go through all v neighbors of u
17 alt ← dist + Graph.Distance
18 if alt < dist:
19 prev ← u
20 dist ← alt
21 Q.decrease_priority
22
23 return
Instead of filling the priority queue with all nodes in the initialization phase, it is possible to initialize it to contain only source; then, inside the if alt < dist block, the becomes an operation.
Yet another alternative is to add nodes unconditionally to the priority queue and to instead check after extraction that it isn't revisiting, or that no shorter connection was found yet in the if alt < dist block. This can be done by additionally extracting the associated priority p from the queue and only processing further if p dist inside the while Q is not empty loop.
These alternatives can use entirely array-based priority queues without decrease-key functionality, which have been found to achieve even faster computing times in practice. However, the difference in performance was found to be narrower for denser graphs.