# Dijkstra's algorithm (C Plus Plus)

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Dijkstra's algorithm is a graph algorithm that simultaneously finds the **shortest path** from a single vertex in a weighted graph to all other vertices in the graph, called the single-source shortest path problem. It works for directed and undirected graphs, but unlike the Bellman-Ford algorithm, requires nonnegative edge weights.

## Contents |

## Simple graph representation

The data structures supplied by C++'s Standard Template Library facilitate a straightforward implementation of Dijkstra's algorithm. For simplicity, we start with a simple graph representation where the vertices are numbered by sequential integers (0, 1, 2, ...) and the edge weights are `double`

s. We define a `struct`

representing an edge that stores its weight and target vertex (the vertex it points to):

<<simple graph types>>=typedefintvertex_t;typedefdoubleweight_t;structedge{vertex_t target; weight_t weight; edge(vertex_t arg_target, weight_t arg_weight) : target(arg_target), weight(arg_weight){}};

Because we'll need to iterate over the successors of each vertex, we will find an adjacency list representation most convenient. We will represent this as an *adjacency map*, a mapping from each vertex to the list of edges exiting that vertex, as defined by the following `typedef`

:

<<simple graph types>>=typedefstd::map<vertex_t, std::list<edge> > adjacency_map_t;<<definition headers>>=#include <map>#include <list>

## Main algorithm

We're now prepared to define our method. We separate the computation into two stages:

- Compute the minimum distance from the source to each vertex in the graph. Simultaneously, keep track of the
`previous`

map, which for each vertex*v*gives the previous vertex on the shortest path from the source vertex to*v*. This is the expensive step. - Later, any time we want to find a particular shortest path between the source vertex and a given vertex, we use the previous array to quickly construct it.

For the first part, we write `DijkstraComputePaths`

, which takes two input parameters and two output parameters:

- source (in): The source vertex from which all shortest paths are found
- adjacency_map (in): The adjacency map specifying the connection between vertices and edge weights.
- min_distance (out): Will receive the shortest distance from
`source`

to each vertex in the graph. - previous (out): Receives a map that can be passed back into
`DijkstraGetShortestPathTo()`

later on to quickly get a shortest path from the source vertex to any desired vertex.

<<simple compute paths function>>=voidDijkstraComputePaths(vertex_t source, adjacency_map_t& adjacency_map, std::map<vertex_t, weight_t>& min_distance, std::map<vertex_t, vertex_t>& previous){initialize output parameters min_distance[source] = 0; visit each vertex u, always visiting vertex with smallest min_distance first // Visit each edge exiting ufor(std::list<edge>::iterator edge_iter = adjacency_map[u].begin(); edge_iter != adjacency_map[u].end(); edge_iter++){vertex_t v = edge_iter->target; weight_t weight = edge_iter->weight; relax the edge (u,v)}}}

The outline of how the function works is shown above: we visit each vertex, looping over its out-edges and adjusting `min_distance`

as necessary. The critical operation is relaxing the edges, which is based on the following formula:

- if (
*u*,*v*) is an edge and*u*is on the shortest path to*v*,*d*(*u*) +*w*(*u*,*v*) =*d*(*v*).

In other words, we can reach *v* by going from the source to *u*, then following the edge (*u*,*v*). Eventually, we will visit every predecessor of *v* reachable from the source. The shortest path goes through one of these. We keep track of the shortest distance seen so far by setting `min_distance`

and the vertex it went through by setting `previous`

:

<<relax the edge (u,v)>>=weight_t distance_through_u = min_distance[u] + weight;if(distance_through_u < min_distance[v]){remove v from queue min_distance[v] = distance_through_u; previous[v] = u; re-add v to queue}

We also need to initialize the output parameters so that at first all min distances are positive infinity (as large as possible). We can visit all vertices by just iterating over the keys of `adjacency_map`

:

<<initialize output parameters>>=for(adjacency_map_t::iterator vertex_iter = adjacency_map.begin(); vertex_iter != adjacency_map.end(); vertex_iter++){vertex_t v = vertex_iter->first; min_distance[v] = std::numeric_limits<double>::infinity();}<<definition headers>>=#include <limits> // for numeric_limits

Finally, we need a way to visit the vertices in order of their minimum distance. We could do this using a heap data structure, but later on we will also need to decrease the distance of particular vertices, which will involve re-ordering that vertex in the heap. However, finding a particular element in a heap is a linear-time operation. To avoid this, we instead use a self-balancing binary search tree, which will also allow us to find the vertex of minimum distance, as well as find any particular vertex, quickly. We will use a `std::set`

of pairs, where each pair contains a vertex *v* along with its minimum distance `min_distance[v]`

. Fortunately, C++'s `std::pair`

class already implements an ordering, which orders lexicographically (by first element, then by second element), which will work fine for us, as long as we use the distance as the first element of the pair.

<<visit each vertex u, always visiting vertex with smallest min_distance first>>=std::set< std::pair<weight_t, vertex_t> > vertex_queue;for(adjacency_map_t::iterator vertex_iter = adjacency_map.begin(); vertex_iter != adjacency_map.end(); vertex_iter++){vertex_t v = vertex_iter->first; vertex_queue.insert(std::make_pair(min_distance[v], v));}while(!vertex_queue.empty()){vertex_t u = vertex_queue.begin()->second; vertex_queue.erase(vertex_queue.begin());<<definition headers>>=#include <set>#include <utility> // for pair

We access and remove the smallest element using `begin()`

, which works because the set is ordered by minimum distance. If we change a vertex's minimum distance, we must update its key in the map as well:

<<remove v from queue>>=vertex_queue.erase(std::make_pair(min_distance[v], v));<<re-add v to queue>>=vertex_queue.insert(std::make_pair(min_distance[v], v));

This completes `DijkstraComputePaths()`

. `DijkstraGetShortestPathTo()`

is much simpler, just following the linked list in the `previous`

map from the target back to the source:

<<get shortest path function>>=std::list<vertex_t> DijkstraGetShortestPathTo( vertex_t target, std::map<vertex_t, vertex_t>& previous){std::list<vertex_t> path; std::map<vertex_t, vertex_t>::iterator prev; vertex_t vertex = target; path.push_front(vertex);while((prev = previous.find(vertex)) != previous.end()){vertex = prev->second; path.push_front(vertex);}returnpath;}

## Sample code

Here's some code demonstrating how we use the above functions:

<<dijkstra_example.cpp>>=#include <iostream>#include <vector>#include <string>definition headers simple graph types simple compute paths function get shortest path functionintmain(){adjacency_map_t adjacency_map; std::vector<std::string> vertex_names; initialize adjacency map std::map<vertex_t, weight_t> min_distance; std::map<vertex_t, vertex_t> previous; DijkstraComputePaths(0, adjacency_map, min_distance, previous); print out shortest paths and distancesreturn0;}

Printing out shortest paths is just a matter of iterating over the vertices and calling `DijkstraGetShortestPathTo()`

on each:

<<print out shortest paths and distances>>=for(adjacency_map_t::iterator vertex_iter = adjacency_map.begin(); vertex_iter != adjacency_map.end(); vertex_iter++){vertex_t v = vertex_iter->first; std::cout << "Distance to " << vertex_names[v] << ": " << min_distance[v] << std::endl; std::list<vertex_t> path = DijkstraGetShortestPathTo(v, previous); std::list<vertex_t>::iterator path_iter = path.begin(); std::cout << "Path: ";for( ; path_iter != path.end(); path_iter++){std::cout << vertex_names[*path_iter] << " ";}std::cout << std::endl;}

For this example, we choose vertices corresponding to some East Coast U.S. cities. We add edges corresponding to interstate highways, with the edge weight set to the driving distance between the cities in miles as determined by Mapquest:

<<initialize adjacency map>>=vertex_names.push_back("Harrisburg"); // 0 vertex_names.push_back("Baltimore"); // 1 vertex_names.push_back("Washington"); // 2 vertex_names.push_back("Philadelphia"); // 3 vertex_names.push_back("Binghamton"); // 4 vertex_names.push_back("Allentown"); // 5 vertex_names.push_back("New York"); // 6 adjacency_map[0].push_back(edge(1, 79.83)); adjacency_map[0].push_back(edge(5, 81.15)); adjacency_map[1].push_back(edge(0, 79.75)); adjacency_map[1].push_back(edge(2, 39.42)); adjacency_map[1].push_back(edge(3, 103.00)); adjacency_map[2].push_back(edge(1, 38.65)); adjacency_map[3].push_back(edge(1, 102.53)); adjacency_map[3].push_back(edge(5, 61.44)); adjacency_map[3].push_back(edge(6, 96.79)); adjacency_map[4].push_back(edge(5, 133.04)); adjacency_map[5].push_back(edge(0, 81.77)); adjacency_map[5].push_back(edge(3, 62.05)); adjacency_map[5].push_back(edge(4, 134.47)); adjacency_map[5].push_back(edge(6, 91.63)); adjacency_map[6].push_back(edge(3, 97.24)); adjacency_map[6].push_back(edge(5, 87.94));

## Alternatives

In an application that does a lot of graph manipulation, a good option is the Boost graph library, which includes support for Dijkstra's algorithm.

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