**Dijkstra’s algorithm**, conceived by Dutch 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 nonnegative 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).

**Task:**

- Implement a version of Dijkstra’s algorithm that computes a shortest path from a start vertex to an end vertex in a directed graph.
- Run your program with the following directed graph to find the shortest path from vertex “a” to vertex “e.”
- Show the output of your program.

Number | Name |
---|---|

1 | a |

2 | b |

3 | c |

4 | d |

5 | e |

6 | f |

Start | End | Cost |
---|---|---|

a | b | 7 |

a | c | 9 |

a | f | 14 |

b | c | 10 |

b | d | 15 |

c | d | 11 |

c | f | 2 |

d | e | 6 |

e | f | 9 |

You can use numbers or names to identify vertices in your program.

(Modified from LiteratePrograms, which is MIT/X11 licensed.)

Solution follows Dijkstra’s algorithm as described elsewhere. Data like min-distance, previous node, neighbors, are kept in separate data structures instead of part of the vertex. We number the vertexes starting from 0, and represent the graph using an adjacency list (vector whose i’th element is the vector of neighbors that vertex i has edges to) for simplicity.

For the priority queue of vertexes, we use a self-balancing binary search tree (`std::set`

), which should bound time complexity by O(E log V). Although C++ has heaps, without knowing the index of an element it would take linear time to find it to re-order it for a changed weight. It is not easy to keep the index of vertexes in the heap because the heap operations are opaque without callbacks. On the other hand, using a self-balancing binary search tree is efficient because it has the same log(n) complexity for insertion and removal of the head element as a binary heap. In addition, a self-balancing binary search tree also allows us to find and remove any other element in log(n) time, allowing us to perform the decrease-key step in logarithmic time by removing and re-inserting.

We do not need to keep track of whether a vertex is “done” (“visited”) as in the Wikipedia description, since re-reaching such a vertex will always fail the relaxation condition (when re-reaching a “done” vertex, the new distance will never be *less* than it was originally), so it will be skipped anyway.

#include <iostream> #include <vector> #include <string> #include <list> #include <limits> // for numeric_limits #include <set> #include <utility> // for pair #include <algorithm> #include <iterator> typedef int vertex_t; typedef double weight_t; const weight_t max_weight = std::numeric_limits<double>::infinity(); struct neighbor { vertex_t target; weight_t weight; neighbor(vertex_t arg_target, weight_t arg_weight) : target(arg_target), weight(arg_weight) { } }; typedef std::vector<std::vector<neighbor> > adjacency_list_t; void DijkstraComputePaths(vertex_t source, const adjacency_list_t &adjacency_list, std::vector<weight_t> &min_distance, std::vector<vertex_t> &previous) { int n = adjacency_list.size(); min_distance.clear(); min_distance.resize(n, max_weight); min_distance[source] = 0; previous.clear(); previous.resize(n, -1); std::set<std::pair<weight_t, vertex_t> > vertex_queue; vertex_queue.insert(std::make_pair(min_distance[source], source)); while (!vertex_queue.empty()) { weight_t dist = vertex_queue.begin()->first; vertex_t u = vertex_queue.begin()->second; vertex_queue.erase(vertex_queue.begin()); // Visit each edge exiting u const std::vector<neighbor> &neighbors = adjacency_list[u]; for (std::vector<neighbor>::const_iterator neighbor_iter = neighbors.begin(); neighbor_iter != neighbors.end(); neighbor_iter++) { vertex_t v = neighbor_iter->target; weight_t weight = neighbor_iter->weight; weight_t distance_through_u = dist + weight; if (distance_through_u < min_distance[v]) { vertex_queue.erase(std::make_pair(min_distance[v], v)); min_distance[v] = distance_through_u; previous[v] = u; vertex_queue.insert(std::make_pair(min_distance[v], v)); } } } } std::list<vertex_t> DijkstraGetShortestPathTo( vertex_t vertex, const std::vector<vertex_t> &previous) { std::list<vertex_t> path; for ( ; vertex != -1; vertex = previous[vertex]) path.push_front(vertex); return path; } int main() { // remember to insert edges both ways for an undirected graph adjacency_list_t adjacency_list(6); // 0 = a adjacency_list[0].push_back(neighbor(1, 7)); adjacency_list[0].push_back(neighbor(2, 9)); adjacency_list[0].push_back(neighbor(5, 14)); // 1 = b adjacency_list[1].push_back(neighbor(0, 7)); adjacency_list[1].push_back(neighbor(2, 10)); adjacency_list[1].push_back(neighbor(3, 15)); // 2 = c adjacency_list[2].push_back(neighbor(0, 9)); adjacency_list[2].push_back(neighbor(1, 10)); adjacency_list[2].push_back(neighbor(3, 11)); adjacency_list[2].push_back(neighbor(5, 2)); // 3 = d adjacency_list[3].push_back(neighbor(1, 15)); adjacency_list[3].push_back(neighbor(2, 11)); adjacency_list[3].push_back(neighbor(4, 6)); // 4 = e adjacency_list[4].push_back(neighbor(3, 6)); adjacency_list[4].push_back(neighbor(5, 9)); // 5 = f adjacency_list[5].push_back(neighbor(0, 14)); adjacency_list[5].push_back(neighbor(2, 2)); adjacency_list[5].push_back(neighbor(4, 9)); std::vector<weight_t> min_distance; std::vector<vertex_t> previous; DijkstraComputePaths(0, adjacency_list, min_distance, previous); std::cout << "Distance from 0 to 4: " << min_distance[4] << std::endl; std::list<vertex_t> path = DijkstraGetShortestPathTo(4, previous); std::cout << "Path : "; std::copy(path.begin(), path.end(), std::ostream_iterator<vertex_t>(std::cout, " ")); std::cout << std::endl; return 0; }

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