Dijkstra's algorithm (C++)

時間 2019-11-09

標籤 dijkstra's dijkstra algorithm c++ 欄目 C&C++ 简体版

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這是我發現的可貴的比較好文章，介紹Dijkstra 算法的c++程序部署。php

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. html

[hide] ios

[edit] 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 doubles. We define a struct representing an edge that stores its weight and target vertex (the vertex it points to):c++

<<simple graph types>>=

typedef int vertex_t;

typedef double weight_t;

struct edge {

    const vertex_t target;

    const 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:算法

<<simple graph types>>=

typedef std::map<vertex_t, std::list<edge> > adjacency_map_t;

<<definition headers>>=

#include <map>

#include <list>

[edit] Main algorithm

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

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: less

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>>=

void DijkstraComputePaths(vertex_t source,

                          const 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 u

        const std::list<edge> &edges = adjacency_map.find(u)->second;

        for (std::list<edge>::const_iterator edge_iter = edges.begin();

             edge_iter != edges.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: ide

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

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:測試

<<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

Note that the original description of Dijkstra's algorithm involves remembering that when a vertex is "done" (i.e. we have gone through an iteration of the loop with u = that vertex), we put it in a set of "visited" vertices, so that if we reach it again we can skip it. However, it is not really necessary to do this specially, as it is guaranteed that if we re-reach a "done" vertex, the new distance to it will never be lessthan its original min distance, so the relaxation condition will always fail in this case which will also cause it to be skipped.

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 iterating over the keys of adjacency_map, and then iterating over the elements in the values of adjacency_map. This is needed since this is a directed graph, and so there may be vertices without any outgoing edges (i.e. sinks) whose min distance still must be set to positive infinity.

<<initialize output parameters>>=

for (adjacency_map_t::const_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();

    for (std::list<edge>::const_iterator edge_iter = vertex_iter->second.begin();

         edge_iter != vertex_iter->second.end();

         edge_iter++)

        vertex_t v2 = edge_iter->target;

        min_distance[v2] = 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 distancemin_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.

We don't need to put all the vertices into the priority queue to start with. We only add them as they are reached.

<<visit each vertex u, always visiting vertex with smallest min_distance first>>=

std::set< std::pair<weight_t, vertex_t> > vertex_queue;

vertex_queue.insert(std::make_pair(min_distance[source], source));

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 (if we are reaching the vertex for the first time, it won't be in the queue, and erase() will do nothing):

<<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, const std::map<vertex_t, vertex_t> &previous)

    std::list<vertex_t> path;

    std::map<vertex_t, vertex_t>::const_iterator prev;

    vertex_t vertex = target;

    path.push_front(vertex);

    while((prev = previous.find(vertex)) != previous.end())

        vertex = prev->second;

        path.push_front(vertex);

    return path;

[edit] 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 function

int main()

    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 distances

    return 0;

Printing out shortest paths is just a matter of iterating over the vertices and callingDijkstraGetShortestPathTo() on each:

<<print out shortest paths and distances>>=

for (adjacency_map_t::const_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>::const_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 (note that edges are directed, so if we want an "undirected" graph, we would need to add edges going both ways):

<<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));

[edit] 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.

我修改的代碼：

 
#include <iostream> 
   
#include <vector> 
   
#include <iterator> 
   
#include <fstream> 
   
#include <sstream> 
   
#include <string> 
   
//#include <memory> 
   
#include <map> 
   
#include <list> 
   
#include <set> 
   
#include <algorithm> 
   
//#include <queue>          // priority_queue 
   
//#include <functional>     // greater 
   
   
   
using namespace std; 
   
typedef int vertex_t; 
   
typedef int weight_t; 
   
   
   
struct edge 
   
{ 
   
    const vertex_t target; 
   
    const weight_t weight; 
   
    edge (vertex_t arg_target, weight_t arg_weight) 
   
        : target(arg_target), weight( arg_weight) { }  
   
}; 
   
   
   
typedef map<vertex_t, list<edge>> adjacency_map_t; 
   
   
   
   
   
void DijkstraComputePaths(vertex_t source, 
   
                          const adjacency_map_t &adjacency_map, 
   
                          map<vertex_t, weight_t> &min_distance, 
   
                          map<vertex_t, vertex_t> &previous); 
   
istream& readData(istream &in, adjacency_map_t &adjacency_map); 
   
void showResult(const vector<int> &vec,  map<vertex_t, weight_t> &min_distance); 
   
   
   
int main(){ 
   
    //fstream fin ("data2.txt"); 
   
    fstream fin("dijkstraData.txt"); 
   
    adjacency_map_t adjacency_map; 
   
    readData(fin, adjacency_map); 
   
   
   
    map<vertex_t, weight_t> min_distance; 
   
    map<vertex_t, vertex_t> previous; 
   
   
   
    DijkstraComputePaths(1, adjacency_map, min_distance, previous); 
   
     
   
    int tmp[] = {7, 37, 59, 82, 99, 115, 133, 165, 188, 197}; 
   
    vector<int> vec_res(tmp, tmp + sizeof(tmp) / sizeof(int)  ); 
   
    showResult(vec_res, min_distance); 
   
    return 0; 
   
} 
   
   
   
   
   
  
   
   
   
void showResult(const vector<int> &vec,  map<vertex_t, weight_t> &min_distance){ 
   
    for (auto iter = vec.cbegin(); iter != vec.cend(); iter++){ 
   
        cout << min_distance[*iter] << ","; 
   
    } 
   
} 
   
   
   
   
   
istream& readData(istream &in, adjacency_map_t &adjacency_map){ 
   
    string line; 
   
    vertex_t vertex1, vertex2; 
   
    weight_t weight;     
   
    //istringstream ss; 
   
     
   
    while ( getline(in, line)){ 
   
        istringstream ss(line); 
   
        ss >> vertex1; 
   
        while ( ss >> vertex2 >> weight){ 
   
            adjacency_map[vertex1].push_back( edge(vertex2, weight)); 
   
            adjacency_map[vertex2].push_back( edge(vertex1, weight)); 
   
        }             
   
    } 
   
    cout << "Reading data completes" << endl; 
   
    return in;  
   
} 
   
   
   
   
   
void DijkstraComputePaths(vertex_t source, 
   
                          const adjacency_map_t &adjacency_map, 
   
                          map<vertex_t, weight_t> &min_distance, 
   
                          map<vertex_t, vertex_t> &previous) 
   
{ 
   
    //map<vertex_t, bool> explorerd; 
   
    for (auto vertex_iter = adjacency_map.begin(); vertex_iter != adjacency_map.end(); vertex_iter++) 
   
    { 
   
        vertex_t v = vertex_iter->first; 
   
        min_distance[v] = 1000000; 
   
        //explorerd[v] = false; 
   
        // for (auto edge_iter = vertex_iter->second.begin(); 
   
        //      edge_iter != vertex_iter->second.end(); 
   
        //      edge_iter++) 
   
        // { 
   
        //     vertex_t v2 = edge_iter->target; 
   
        //     min_distance[v2] = 1000000; 
   
        // } 
   
    } 
   
    min_distance[source] = 0; 
   
    set< pair<weight_t, vertex_t> > vertex_queue; 
   
   
   
    vertex_queue.insert(make_pair(min_distance[source], source)); 
   
    while (!vertex_queue.empty())  
   
    { 
   
        vertex_t u = vertex_queue.begin()->second; 
   
        vertex_queue.erase(vertex_queue.begin()); 
   
   
   
   
   
        // Visit each edge exiting u 
   
        const list<edge> &edges = adjacency_map.find(u)->second; 
   
        for (auto edge_iter = edges.begin(); 
   
             edge_iter != edges.end(); 
   
             edge_iter++) 
   
        { 
   
            vertex_t v = edge_iter->target; 
   
            //if (explorerd[v]) continue; 
   
            //explorerd[v] = true; 
   
            weight_t weight = edge_iter->weight; 
   
            weight_t distance_through_u = min_distance[u] + weight; 
   
            if (distance_through_u < min_distance[v]) { 
   
                vertex_queue.erase(make_pair(min_distance[v], v)); 
   
                min_distance[v] = distance_through_u; 
   
                previous[v] = u; 
   
                vertex_queue.insert(make_pair(min_distance[v], v)); 
   
            } 
   
        } 
   
    } 
   
}