C++ Dijkstra堆优化算法

时间复杂度为：O((n+m)logn)

算法特点：非负边权、单源最短路、顶点数、边数<1000000，数据结构前置：领接表、哈希表、二叉堆

算法：

第一步，建图，任何算法我们都要去思考，用什么数据结构来存储，这个算法我们采用邻接表来存储，有时候输入数据，并不是我们期待的那样，所以需要对图进行一些处理，这就是建图的过程

第二步，辅助数组，对于图G = <V, E>，源点为s，dist $i$ 表示s到i的最短路，visited $i$ 表示dist $i$ 是否已经确定（布尔值），即s到i的最短路，是否已经确定。

第三步，辅助堆，利用一个小顶堆heap存放二元组（v, dist $v$ ），小顶堆扮演的是优先队列作用，dist $v$ 值越小的，会从优先队列中优先出列。

第四步，初始化，初始化所有顶点的数据见下，

dist $i$ = 无穷大（0 =< i =< n）

visited $i$ = false

dist $s$ = 0

heap.push(s, dist $s$ )

第五步，找距离最小的点，从小顶堆中不断弹出元素u，并且判断visited $u$ 是否为true，如果为true，则继续弹出；否则标记为true，并且从u出发，进行松弛操作，如果堆为空，算法结束。

第六步，松弛操作，更新从u出发，到达顶点v的最短路dist $v$ ，

dist $v$ = min(dist $v$ , dist $u$ + w(u, v))

代码分析：第一步，定义距离二元组 Dist(v, w)

第二步，初始化邻接表

function initEdges(n, edges $maxn$ )

for i -> (0, n-1)

edges $i$ = {}

第三步，邻接表加边，

function addEdge(edges $maxn$ , u, v, w)

edges $u$ .append(Dist(v, w))

第四步，建图

addEdge(edges, u1, v1, w1)

addEdge(edges, u2, v2, w2)

...

第五步，框架代码

function DijkstraHeap(n, st, edges $maxn$ , d $maxn$ )

heap = Heap()

visited $maxn$ = false

dijkstraInit(n, st, heap, visited, d)

while(not heap.empty())

u = dijkstraFindMin(heap)

dijkstraUpdate(u, edges, heap, visited, d)

第六步，初始化

function dijkstrainit(n, st, heap, visited $maxn$ $maxn$ )

for i->(0, n-1)

d $i$ = inf

visited $i$ = false

dist $st$ = 0

heap.push(Dist(st, d $st$ ))

第七步，获取最小值

function dijkstraFindMin(heap)

s = heap.top()

heap.pop()

return s.v

第八步，松弛操作

function dijkstraUpdate(u, edges $maxn$ , heap, visited $maxn$ , d $maxn$ )

if not visited $u$

visited $u$ = true

for i-> (0, edges $u$ .size() - 1)

v = edges $u$ $i$ .v

w = edges $u$ $i$ .w

if (d $u$ + w < d $v$ )

d $v$ = d $u$ + w

heap.push(Dist(v, d $v$ ))

代码练习，对应蓝桥云课 Dijkstra求最短路2 代码见下

cpp 复制代码

#include <iostream>
#include <vector>
#include <queue>

using namespace std;

#define inf 1000000001
#define maxn 100001
#define ValueType int

struct Dist{
  int v;
  ValueType w;
  Dist() {}
  Dist(int _v, ValueType _w): v(_v), w(_w) {}
  bool operator < (const Dist& d) const{
    return w > d.w;
  }
};

typedef priority_queue<Dist> Heap;

void addEdge(vector<Dist>* edges, int u, int v, ValueType w){
  edges[u].push_back(Dist(v, w));
}

void dijkstraInit(int n, int st, Heap& heap, bool *visited, ValueType* d){
  for(int i=0; i<n; ++i){
    d[i] = inf;
    visited[i] = false;
  }
  d[st] = 0;
  heap.push(Dist(st, d[st]));
}

int dijkstraFindMin(Heap& heap){
  Dist s = heap.top();
  heap.pop();
  return s.v;
}

void dijkstraUpdate(int u, vector<Dist>* edges, Heap& heap, bool *visited, ValueType* d){
  if(visited[u]){
    return;
  }
  visited[u] = true;
  for(int i=0; i<edges[u].size(); ++i){
    int v = edges[u][i].v;
    ValueType w = edges[u][i].w;
    if(d[u] + w < d[v]){
      d[v] = d[u] + w;
      heap.push(Dist(v, d[v]));
    }
  }
}

void DijkstraHeap(int n, int st, vector<Dist>* edges, ValueType* d){
  Heap heap;
  bool visited[maxn] = {false};
  dijkstraInit(n, st, heap, visited, d);
  while(!heap.empty()){
    int u = dijkstraFindMin(heap);
    dijkstraUpdate(u, edges, heap, visited, d);
  }

}

vector<Dist> edges[maxn];
ValueType d[maxn];

int main()
{
  int n, m;
  cin >> n >> m;
  while(m--){
    int u, v, w;
    cin >> u >> v >> w;
    --u, --v;
    addEdge(edges, u, v, w);
  }
  DijkstraHeap(n, 0, edges, d);
  if(d[n-1] == inf){
    cout << -1 << endl;
  }else{
    cout << d[n-1] << endl;
  }

  // 请在此输入您的代码
  return 0;
}

代码练习 2 对应蓝桥云课蓝桥王国代码见下

cpp 复制代码

#include <iostream>
#include <vector>
#include <queue>

using namespace std;

#define inf 1000000001000000001
#define maxn 300001
#define ValueType long long 

struct Dist{
  int v;
  ValueType w;
  Dist() {}
  Dist(int _v, ValueType _w): v(_v), w(_w) {}
  bool operator < (const Dist& d) const{
    return w > d.w;
  }
};

typedef priority_queue<Dist> Heap;

void addEdge(vector<Dist>* edges, int u, int v, ValueType w){
  edges[u].push_back(Dist(v, w));
}

void dijkstraInit(int n, int st, Heap& heap, bool *visited, ValueType* d){
  for(int i=0; i<n; ++i){
    d[i] = inf;
    visited[i] = false;
  }
  d[st] = 0;
  heap.push(Dist(st, d[st]));
}

int dijkstraFindMin(Heap& heap){
  Dist s = heap.top();
  heap.pop();
  return s.v;
}

void dijkstraUpdate(int u, vector<Dist>* edges, Heap& heap, bool *visited, ValueType* d){
  if(visited[u]){
    return;
  }
  visited[u] = true;
  for(int i=0; i<edges[u].size(); ++i){
    int v = edges[u][i].v;
    ValueType w = edges[u][i].w;
    if(d[u] + w < d[v]){
      d[v] = d[u] + w;
      heap.push(Dist(v, d[v]));
    }
  }
}

void DijkstraHeap(int n, int st, vector<Dist>* edges, ValueType* d){
  Heap heap;
  bool visited[maxn] = {false};
  dijkstraInit(n, st, heap, visited, d);
  while(!heap.empty()){
    int u = dijkstraFindMin(heap);
    dijkstraUpdate(u, edges, heap, visited, d);
  }

}

vector<Dist> edges[maxn];
ValueType d[maxn];

int main(){
  int n, m;
  cin >> n >> m;
  for(int i=0; i<m; ++i){
    int u, v, w;
    cin >> u >> v >> w;
    --u, --v;
    addEdge(edges, u, v, w);
  }
  DijkstraHeap(n, 0, edges, d);
  for(int i=0; i<n; ++i){
    if(i){
      cout << " ";
    }
    if(d[i] == inf){
      cout << "-1";
    }else{
      cout << d[i];
    }
  }
  cout << endl;
  return 0;
}