Dijkstra's algorithm

Last updated

Dijkstra's algorithm
Dijkstra Animation.gif
Dijkstra's algorithm to find the shortest path between a and b. 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.
Class Search algorithm
Data structure Graph
Worst-case performance

Dijkstra's algorithm (or Dijkstra's Shortest Path First algorithm, SPF algorithm) [1] is an algorithm for finding the shortest paths between nodes in a graph, which may represent, for example, road networks. It was conceived by computer scientist Edsger W. Dijkstra in 1956 and published three years later. [2] [3] [4]

Contents

The algorithm exists in many variants. Dijkstra's original algorithm found the shortest path between two given nodes, [4] 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 node in the graph, the algorithm finds the shortest path between that node and every other. [5] :196–206 It can also be used for finding the shortest paths from a single node to a single destination node by stopping the algorithm once the shortest path to the destination node has been determined. For example, if the nodes of the graph represent cities and edge path costs represent driving distances between pairs of cities connected by a direct road (for simplicity, ignore red lights, stop signs, toll roads and other obstructions), Dijkstra's algorithm can be used to find the shortest route between one city and all other cities. A widely used application of shortest path algorithm is network routing protocols, most notably IS-IS (Intermediate System to Intermediate System) and Open Shortest Path First (OSPF). It is also employed as a subroutine in other algorithms such as Johnson's.

The Dijkstra algorithm uses labels that are positive integers or real numbers, which are totally ordered. It can be generalized to use any labels that are partially ordered, provided the subsequent labels (a subsequent label is produced when traversing an edge) are monotonically non-decreasing. This generalization is called the generic Dijkstra shortest-path algorithm. [6]

Dijkstra's algorithm uses a data structure for storing and querying partial solutions sorted by distance from the start. The original algorithm uses a min-priority queue and runs in time (where is the number of nodes). The idea of this algorithm is also given in Leyzorek et al. 1957. Fredman & Tarjan 1984 propose using a Fibonacci heap min-priority queue to optimize the running time complexity to (where is the number of edges). This is asymptotically the fastest known single-source shortest-path algorithm for arbitrary directed graphs with unbounded non-negative weights. However, specialized cases (such as bounded/integer weights, directed acyclic graphs etc.) can indeed be improved further as detailed in Specialized variants.

In some fields, artificial intelligence in particular, 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. [7]

History

What is the shortest way to travel from Rotterdam to Groningen, in general: from given city to given city. It is the algorithm for the shortest path, which I designed in about twenty minutes. One morning I was shopping in Amsterdam 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. As I said, it was a twenty-minute invention. In fact, it was published in '59, three years later. The publication is still readable, it is, in fact, quite nice. One of the reasons that it is so nice was that I designed it without pencil and paper. I learned later that one of the advantages of designing without pencil and paper is that you are almost forced to avoid all avoidable complexities. Eventually that algorithm became, to my great amazement, one of the cornerstones of my fame.

Edsger Dijkstra, in an interview with Philip L. Frana, Communications of the ACM, 2001 [3]

Dijkstra thought about the shortest path problem when working at the Mathematical Center in Amsterdam in 1956 as a programmer to demonstrate the capabilities of a new computer called ARMAC. [8] His objective was to choose both a problem and a solution (that would be produced by computer) 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 (64, so that 6 bits would be sufficient to encode the city number). [3] 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 (known earlier to Jarník, and also rediscovered by Prim). [9] [10] Dijkstra published the algorithm in 1959, two years after Prim and 29 years after Jarník. [11] [12]

Algorithm

Illustration of Dijkstra's algorithm finding a path from a start node (lower left, red) to a goal node (upper right, green) in a robot motion planning problem. Open nodes represent the "tentative" set (aka set of "unvisited" nodes). Filled nodes are visited ones, with color representing the distance: the greener, the closer. Nodes in all the different directions are explored uniformly, appearing more-or-less as a circular wavefront as Dijkstra's algorithm uses a heuristic identically equal to 0. Dijkstras progress animation.gif
Illustration of Dijkstra's algorithm finding a path from a start node (lower left, red) to a goal node (upper right, green) in a robot motion planning problem. Open nodes represent the "tentative" set (aka set of "unvisited" nodes). Filled nodes are visited ones, with color representing the distance: the greener, the closer. Nodes in all the different directions are explored uniformly, appearing more-or-less as a circular wavefront as Dijkstra's algorithm uses a heuristic identically equal to 0.

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.

  1. Mark all nodes unvisited. Create a set of all the unvisited nodes called the unvisited set.
  2. 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. [13]
  3. For the current node, consider all of its unvisited neighbours and calculate their tentative distances through the current node. 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 neighbour 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, the current value will be kept.
  4. When we are done considering all of the unvisited neighbours 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.
  5. 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.
  6. Otherwise, select the unvisited node that is marked with the smallest tentative distance, set it as the new "current node", and go back to step 3.

When planning a route, it is actually not necessary to wait until the destination node is "visited" as above: the algorithm can stop once the destination node has the smallest tentative distance among all "unvisited" nodes (and thus could be selected as the next "current").

Description

Suppose you would like to find the shortest path between two intersections on a city map: a starting point and a destination. Dijkstra's algorithm initially marks the distance (from the starting point) to every other intersection on the map with infinity. This is done not to imply that there is an infinite distance, but to note that those intersections have not been visited yet. Some variants of this method leave the intersections' distances unlabeled. Now select the current intersection at each iteration. 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 a 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 then relabeling the unvisited intersection with this value (the sum) if it is less than the unvisited intersection's 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 an unvisited intersection with minimal distance (from the starting point) – or the lowest label—as the current intersection. Intersections 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, marking the current intersection as visited, and moving onto a 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. In the algorithm's implementations, this is usually done (after the algorithm has reached the destination node) by following the nodes' parents from the destination node up to the starting node; that's why we also keep track of each node's parent.

This algorithm makes no attempt of 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.

Pseudocode

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 dist[u] value. length(u, v) returns the length of the edge joining (i.e. the distance between) the two neighbor-nodes u and v. The variable alt on line 18 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 prev array is populated with a pointer to the "next-hop" node on the source graph to get the shortest route to the source.

A demo of Dijkstra's algorithm based on Euclidean distance. Red lines are the shortest path covering, i.e., connecting u and prev[u]. 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. DijkstraDemo.gif
A demo of Dijkstra's algorithm based on Euclidean distance. Red lines are the shortest path covering, i.e., connecting u and prev[u]. 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(Graph, source):  2  3      create vertex set Q  4  5      for each vertex v in Graph:               6          dist[v] ← INFINITY                    7          prev[v] ← UNDEFINED                   8          add v to Q                       10      dist[source] ← 0                         11       12      whileQ is not empty: 13          u ← vertex in Q with min dist[u]     14                                               15          remove u from Q  16           17          for each neighbor v of u:           // only v that are still in Q 18              alt ← dist[u] + length(u, v) 19              ifalt < dist[v]:                20                  dist[v] ← alt  21                  prev[v] ← u  22 23      return dist[], prev[]

If we are only interested in a shortest path between vertices source and target, we can terminate the search after line 15 if u = target. Now we can read the shortest path from source to target by reverse iteration:

1  S ← empty sequence 2  utarget 3  if prev[u] is defined oru = source:          // Do something only if the vertex is reachable 4      whileu is defined:                       // Construct the shortest path with a stack S 5          insert u at the beginning of S// Push the vertex onto the stack 6          u ← prev[u]                           // Traverse from target to source


Now sequence S is the list of vertices constituting one of the shortest paths from source to target, or the empty sequence if no path exists.

A more general problem would be to find all the shortest paths between source and target (there might be several different ones of the same length). Then instead of storing only a single node in each entry of prev[] we would store all nodes satisfying the relaxation condition. For example, if both r and 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 r and source to prev[target]. When the algorithm completes, prev[] 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 type that provides 3 basic operations : add_with_priority(), decrease_priority() and 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 4      create vertex priority queue Q 5 6      for each vertex v in Graph:            7          ifvsource 8              dist[v] ← INFINITY                 // Unknown distance from source to v 9              prev[v] ← UNDEFINED                // Predecessor of v 10 11         Q.add_with_priority(v, dist[v]) 12 13 14     whileQ is not empty:                      // The main loop 15         uQ.extract_min()                    // Remove and return best vertex 16         for each neighbor v of u:              // only v that are still in Q 17             alt ← dist[u] + length(u, v)  18             ifalt < dist[v] 19                 dist[v] ← alt 20                 prev[v] ← u 21                 Q.decrease_priority(v, alt) 22 23     return dist, 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 ifalt < dist[v] block, the node must be inserted if not already in the queue (instead of performing a decrease_priority operation). [5] :198

Other data structures can be used to achieve even faster computing times in practice. [14]

Proof of correctness

Proof of Dijkstra's algorithm is constructed by induction on the number of visited nodes.

Invariant hypothesis: For each visited node v, dist[v] is considered the shortest distance from source to v; and for each unvisited node u, dist[u] is assumed the shortest distance when traveling via visited nodes only, from source to u. This assumption is only considered if a path exists, otherwise the distance is set to infinity. (Note : we do not assume dist[u] is the actual shortest distance for unvisited nodes)

The base case is when there is just one visited node, namely the initial node source, in which case the hypothesis is trivial.

Otherwise, assume the hypothesis for n-1 visited nodes. In which case, we choose an edge vu where u has the least dist[u] of any unvisited nodes and the edge vu is such that dist[u] = dist[v] + length[v,u]. dist[u] is considered to be the shortest distance from source to u because if there were a shorter path, and if w was the first unvisited node on that path then by the original hypothesis dist[w] > dist[u] which creates a contradiction. Similarly if there were a shorter path to u without using unvisited nodes, and if the last but one node on that path were w, then we would have had dist[u] = dist[w] + length[w,u], also a contradiction.

After processing u it will still be true that for each unvisited nodes w, dist[w] will be the shortest distance from source to w using visited nodes only, because if there were a shorter path that doesn't go by u we would have found it previously, and if there were a shorter path using u we would have updated it when processing u.

Running time

Bounds of the running time of Dijkstra's algorithm on a graph with edges E and vertices V can be expressed as a function of the number of edges, denoted , and the number of vertices, denoted , using big-O notation. The complexity bound depends mainly on the data structure used to represent the set Q. In the following, upper bounds can be simplified because is for any graph, but that simplification disregards the fact that in some problems, other upper bounds on may hold.

For any data structure for the vertex set Q, the running time is in

where and are the complexities of the decrease-key and extract-minimum operations in Q, respectively. The simplest version of Dijkstra's algorithm stores the vertex set Q as an ordinary linked list or array, and extract-minimum is simply a linear search through all vertices in Q. In this case, the running time is .

If the graph is stored as an adjacency list, the running time for a dense graph (i.e., where ) 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 Q changes. With a self-balancing binary search tree or binary heap, the algorithm requires

time in the worst case (where denotes the binary logarithm ); 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 [5] :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). [5] :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. [7]

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 [7] [15] [16] and can be expressed in pseudocode as

procedure uniform_cost_search(Graph, start, goal) is     node ← start     cost ← 0     frontier ← priority queue containing node only     explored ← empty set     doif frontier is empty thenreturn failure         node ← frontier.pop()         if node is goal thenreturn solution         explored.add(node)         for each of node's neighbors ndoifn is not in explored then                 frontier.add(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, each edge has cost at least ε, and the number of neighbors per node is bounded by b, then the algorithm's worst-case time and space complexity are both in O(b1+⌊C*ε). [15]

Further optimizations of Dijkstra's algorithm for the single-target case include bidirectional variants, goal-directed variants such as the A* algorithm (see § Related problems and algorithms), graph pruning to determine which nodes are likely to form the middle segment of shortest paths (reach-based routing), and hierarchical decompositions of the input graph that reduce st routing to connecting s and t to their respective "transit nodes" followed by shortest-path computation between these transit nodes using a "highway". [17] Combinations of such techniques may be needed for optimal practical performance on specific problems. [18]

Specialized variants

When arc weights are small integers (bounded by a parameter C), a monotone priority queue can be used to speed up Dijkstra's algorithm. The first algorithm of this type was Dial's algorithm, which used a bucket queue to obtain a running time that depends on the weighted diameter of a graph with integer edge weights ( Dial 1969 ). The use of a Van Emde Boas tree as the priority queue brings the complexity to ( Ahuja et al. 1990 ). Another interesting variant 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. [19] [20]

In the special case of integer weights and undirected connected graphs, Dijkstra's algorithm can be completely countered with a linear complexity algorithm, given by ( Thorup 1999 ).

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.

Dijkstra's algorithm is usually the working principle behind link-state routing protocols, OSPF and IS-IS being the most common ones.

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. (This statement assumes that a "path" is allowed to repeat vertices. In graph theory that is normally not allowed. In theoretical computer science it often is allowed.) 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 edges.

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.

The 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. [21] [22] [23]

In fact, Dijkstra's explanation of the logic behind the algorithm, [24] 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 .

is a paraphrasing of Bellman's famous Principle of Optimality in the context of the shortest path problem.

See also

Notes

  1. "OSPF Incremental SPF". Cisco.
  2. Richards, Hamilton. "Edsger Wybe Dijkstra". A.M. Turing Award. Association for Computing Machinery. Retrieved 16 October 2017. At the Mathematical Centre a major project was building the ARMAC computer. For its official inauguration in 1956, Dijkstra devised a program to solve a problem interesting to a nontechnical audience: Given a network of roads connecting cities, what is the shortest route between two designated cities?
  3. 1 2 3 Frana, Phil (August 2010). "An Interview with Edsger W. Dijkstra". Communications of the ACM. 53 (8): 41–47. doi:10.1145/1787234.1787249.
  4. 1 2 Dijkstra, E. W. (1959). "A note on two problems in connexion with graphs" (PDF). Numerische Mathematik. 1: 269–271. doi:10.1007/BF01386390.
  5. 1 2 3 4 Mehlhorn, Kurt; Sanders, Peter (2008). "Chapter 10. Shortest Paths" (PDF). Algorithms and Data Structures: The Basic Toolbox. Springer. doi:10.1007/978-3-540-77978-0. ISBN   978-3-540-77977-3.
  6. Szcześniak, Ireneusz; Jajszczyk, Andrzej; Woźna-Szcześniak, Bożena (2019). "Generic Dijkstra for optical networks". Journal of Optical Communications and Networking. 11 (11): 568–577. arXiv: 1810.04481 . doi:10.1364/JOCN.11.000568.
  7. 1 2 3 Felner, Ariel (2011). Position Paper: Dijkstra's Algorithm versus Uniform Cost Search or a Case Against Dijkstra's Algorithm. Proc. 4th Int'l Symp. on Combinatorial Search. In a route-finding problem, Felner finds that the queue can be a factor 500–600 smaller, taking some 40% of the running time.
  8. "ARMAC". Unsung Heroes in Dutch Computing History. 2007. Archived from the original on 13 November 2013.
  9. Dijkstra, Edsger W., Reflections on "A note on two problems in connexion with graphs (PDF)
  10. Tarjan, Robert Endre (1983), Data Structures and Network Algorithms, CBMS_NSF Regional Conference Series in Applied Mathematics, 44, Society for Industrial and Applied Mathematics, p. 75, The third classical minimum spanning tree algorithm was discovered by Jarník and rediscovered by Prim and Dikstra; it is commonly known as Prim's algorithm.
  11. Prim, R.C. (1957). "Shortest connection networks and some generalizations" (PDF). Bell System Technical Journal. 36 (6): 1389–1401. Bibcode:1957BSTJ...36.1389P. doi:10.1002/j.1538-7305.1957.tb01515.x. Archived from the original (PDF) on 18 July 2017. Retrieved 18 July 2017.
  12. V. Jarník: O jistém problému minimálním [About a certain minimal problem], Práce Moravské Přírodovědecké Společnosti, 6, 1930, pp. 57–63. (in Czech)
  13. Gass, Saul; Fu, Michael (2013). "Dijkstra's Algorithm". Encyclopedia of Operations Research and Management Science. Springer. 1. doi:10.1007/978-1-4419-1153-7. ISBN   978-1-4419-1137-7 via Springer Link.
  14. 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 (PDF). Austin, Texas: The University of Texas at Austin, Department of Computer Sciences.
  15. 1 2 Russell, Stuart; Norvig, Peter (2009) [1995]. Artificial Intelligence: A Modern Approach (3rd ed.). Prentice Hall. pp. 75, 81. ISBN   978-0-13-604259-4.
  16. Sometimes also least-cost-first search: Nau, Dana S. (1983). "Expert computer systems" (PDF). Computer. IEEE. 16 (2): 63–85. doi:10.1109/mc.1983.1654302.
  17. Wagner, Dorothea; Willhalm, Thomas (2007). Speed-up techniques for shortest-path computations. STACS. pp. 23–36.
  18. Bauer, Reinhard; Delling, Daniel; Sanders, Peter; Schieferdecker, Dennis; Schultes, Dominik; Wagner, Dorothea (2010). "Combining hierarchical and goal-directed speed-up techniques for Dijkstra's algorithm". J. Experimental Algorithmics. 15: 2.1. doi:10.1145/1671970.1671976.
  19. "Boost Graph Library: Directed Acyclic Graph Shortest Paths – 1.44.0". www.boost.org.
  20. Cormen et al. 2001 , p. 655
  21. Sniedovich, M. (2006). "Dijkstra's algorithm revisited: the dynamic programming connexion" (PDF). Journal of Control and Cybernetics. 35 (3): 599–620. Online version of the paper with interactive computational modules.
  22. Denardo, E.V. (2003). Dynamic Programming: Models and Applications. Mineola, NY: Dover Publications. ISBN   978-0-486-42810-9.
  23. Sniedovich, M. (2010). Dynamic Programming: Foundations and Principles. Francis & Taylor. ISBN   978-0-8247-4099-3.
  24. Dijkstra 1959 , p. 270

Related Research Articles

Shortest path problem computational problem

In graph theory, the shortest path problem is the problem of finding a path between two vertices in a graph such that the sum of the weights of its constituent edges is minimized.

Prims algorithm algorithm

In computer science, Prim'salgorithm is a greedy algorithm that finds a minimum spanning tree for a weighted undirected graph. This means it finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized. The algorithm operates by building this tree one vertex at a time, from an arbitrary starting vertex, at each step adding the cheapest possible connection from the tree to another vertex.

Breadth-first search algorithm for searching the nodes of a graph in order by their hop count from a starting node

Breadth-first search (BFS) is an algorithm for traversing or searching tree or graph data structures. It starts at the tree root, and explores all of the neighbor nodes at the present depth prior to moving on to the nodes at the next depth level.

The Bellman–Ford algorithm is an algorithm that computes shortest paths from a single source vertex to all of the other vertices in a weighted digraph. It is slower than Dijkstra's algorithm for the same problem, but more versatile, as it is capable of handling graphs in which some of the edge weights are negative numbers. The algorithm was first proposed by Alfonso Shimbel (1955), but is instead named after Richard Bellman and Lester Ford Jr., who published it in 1958 and 1956, respectively. Edward F. Moore also published the same algorithm in 1957, and for this reason it is also sometimes called the Bellman–Ford–Moore algorithm.

In computer science, the Floyd–Warshall algorithm is an algorithm for finding shortest paths in a weighted graph with positive or negative edge weights. A single execution of the algorithm will find the lengths of shortest paths between all pairs of vertices. Although it does not return details of the paths themselves, it is possible to reconstruct the paths with simple modifications to the algorithm. Versions of the algorithm can also be used for finding the transitive closure of a relation , or widest paths between all pairs of vertices in a weighted graph.

In computer science, the Edmonds–Karp algorithm is an implementation of the Ford–Fulkerson method for computing the maximum flow in a flow network in time. The algorithm was first published by Yefim Dinitz in 1970 and independently published by Jack Edmonds and Richard Karp in 1972. Dinic's algorithm includes additional techniques that reduce the running time to .

Johnson's algorithm is a way to find the shortest paths between all pairs of vertices in an edge-weighted, directed graph. It allows some of the edge weights to be negative numbers, but no negative-weight cycles may exist. It works by using the Bellman–Ford algorithm to compute a transformation of the input graph that removes all negative weights, allowing Dijkstra's algorithm to be used on the transformed graph. It is named after Donald B. Johnson, who first published the technique in 1977.

Pathfinding Plotting by a computer application

Pathfinding or pathing is the plotting, by a computer application, of the shortest route between two points. It is a more practical variant on solving mazes. This field of research is based heavily on Dijkstra's algorithm for finding the shortest path on a weighted graph.

In graph theory, reachability refers to the ability to get from one vertex to another within a graph. A vertex can reach a vertex if there exists a sequence of adjacent vertices which starts with and ends with .

In computer science, the Hopcroft–Karp algorithm is an algorithm that takes as input a bipartite graph and produces as output a maximum cardinality matching – a set of as many edges as possible with the property that no two edges share an endpoint. It runs in time in the worst case, where is set of edges in the graph, is set of vertices of the graph, and it is assumed that . In the case of dense graphs the time bound becomes , and for sparse random graphs it runs in near-linear time.

In computer science, graph traversal refers to the process of visiting each vertex in a graph. Such traversals are classified by the order in which the vertices are visited. Tree traversal is a special case of graph traversal.

Given a connected, undirected graph G, a shortest-path tree rooted at vertex v is a spanning tree T of G, such that the path distance from root v to any other vertex u in T is the shortest path distance from v to u in G.

The d-ary heap or d-heap is a priority queue data structure, a generalization of the binary heap in which the nodes have d children instead of 2. Thus, a binary heap is a 2-heap, and a ternary heap is a 3-heap. According to Tarjan and Jensen et al., d-ary heaps were invented by Donald B. Johnson in 1975.

In theoretical computer science and network routing, Suurballe's algorithm is an algorithm for finding two disjoint paths in a nonnegatively-weighted directed graph, so that both paths connect the same pair of vertices and have minimum total length. The algorithm was conceived by John W. Suurballe and published in 1974. The main idea of Suurballe's algorithm is to use Dijkstra's algorithm to find one path, to modify the weights of the graph edges, and then to run Dijkstra's algorithm a second time. The output of the algorithm is formed by combining these two paths, discarding edges that are traversed in opposite directions by the paths, and using the remaining edges to form the two paths to return as the output. The modification to the weights is similar to the weight modification in Johnson's algorithm, and preserves the non-negativity of the weights while allowing the second instance of Dijkstra's algorithm to find the correct second path.

In computer science, the method of contraction hierarchies is a speed-up technique for finding the shortest-path in a graph. The most intuitive applications are car-navigation systems: A user wants to drive from to using the quickest possible route. The metric optimized here is the travel time. Intersections are represented by vertices, the street sections connecting them by edges. The edge weights represent the time it takes to drive along this segment of the street. A path from to is a sequence of edges (streets); the shortest path is the one with the minimal sum of edge weights among all possible paths. The shortest path in a graph can be computed using Dijkstra's algorithm; but given that road networks consist of tens of millions of vertices, this is impractical. Contraction hierarchies is a speed-up method optimized to exploit properties of graphs representing road networks. The speed-up is achieved by creating shortcuts in a preprocessing phase which are then used during a shortest-path query to skip over "unimportant" vertices. This is based on the observation that road networks are highly hierarchical. Some intersections, for example highway junctions, are "more important" and higher up in the hierarchy than for example a junction leading into a dead end. Shortcuts can be used to save the precomputed distance between two important junctions such that the algorithm doesn't have to consider the full path between these junctions at query time. Contraction hierarchies do not know about which roads humans consider "important", but they are provided with the graph as input and are able to assign importance to vertices using heuristics.

The Shortest Path Faster Algorithm (SPFA) is an improvement of the Bellman–Ford algorithm which computes single-source shortest paths in a weighted directed graph. The algorithm is believed to work well on random sparse graphs and is particularly suitable for graphs that contain negative-weight edges. However, the worst-case complexity of SPFA is the same as that of Bellman–Ford, so for graphs with nonnegative edge weights Dijkstra's algorithm is preferred. The SPFA algorithm was first published by Edward F. Moore in 1959, as a generalization of breadth first search; the same algorithm was rediscovered in 1994 by Fanding Duan.

Yen's algorithm computes single-source K-shortest loopless paths for a graph with non-negative edge cost. The algorithm was published by Jin Y. Yen in 1971 and employs any shortest path algorithm to find the best path, then proceeds to find K − 1 deviations of the best path.

In computer science, a monotone priority queue is a variant of the priority queue abstract data type in which the priorities of extracted items are required to form a monotonic sequence. That is, for a priority queue in which each successively extracted item is the one with the minimum priority, the minimum priority should be monotonically increasing. Conversely for a max-heap the maximum priority should be monotonically decreasing. The assumption of monotonicity arises naturally in several applications of priority queues, and can be used as a simplifying assumption to speed up certain types of priority queues.

Stoer–Wagner algorithm

In graph theory, the Stoer–Wagner algorithm is a recursive algorithm to solve the minimum cut problem in undirected weighted graphs with non-negative weights. It was proposed by Mechthild Stoer and Frank Wagner in 1995. The essential idea of this algorithm is to shrink the graph by merging the most intensive vertices, until the graph only contains two combined vertex sets. At each phase, the algorithm finds the minimum - cut for two vertices and chosen at its will. Then the algorithm shrinks the edge between and to search for non - cuts. The minimum cut found in all phases will be the minimum weighted cut of the graph.

External memory graph traversal is a type of graph traversal optimized for accessing externally stored memory.

References