迪杰斯特拉(dijkstra)算法详解
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在图的应用中,有一个很重要的需求:我们需要知道从某一个点开始,到其他所有点的最短路径。
算法描述:
算法具体步骤
复杂度分析:
再看一个例子:
步骤
S集合中
U集合中
1
选入A,此时S ={A}
此时最短路径A->A = 0
以A为中间点,从A开始找
U = {B, C, D, E, F}
A->B = 6
A->C = 3
A->U中其他顶点 = ∞
其中A->C = 3 权值为最小,路径最短
2
选入上一轮中找到的最短路径的顶点C,此时S = {A, C}
此时最短路径A->A = 0,A->C = 3
以C为中间点,从A->C=3这条最短路径开始新一轮查找
U = {B, D, E, F}
A->C->B = 5(比上面的A->B = 6要小)
替换B的权值为更小的A->C->B = 5
A->C->D = 6
A->C->E = 7
A->C->U中其他顶点 = ∞
其中A->C->B = 5 最短
3
选入B,此时S = {A, C, B}
此时最短路径 A->A = 0,A->C = 3
A->C->B = 5
以B为中间点,从A->C->B = 5这条最短路径开始新一轮查找
U = {D, E, F}
A->C->B->D = 10(比上面的A->C->D = 6大,不替换,保持D的权值为A->C->D=6)
A->C->B->U中其他顶点 = ∞
其中 A->C->D = 6 最短
4
选入D,此时 S = {A, C, B, D}
此时最短路径 A->A = 0,A->C = 3,A->C->B = 5,A->C->D = 6
以D为中间点,从A->C->D = 6这条最短路径开始新一轮查找
U = {E, F}
A->C->D->E = 8(比上面步骤2中的A->C->E = 7要长,保持E的权值为A->C->E =7)
A->C->D->F = 9
其中A->C->E = 7最短
5
选入E,此时 S = {A, C, B, D ,E}
此时最短路径 A->A = 0,A->C = 3,A->C->B = 5,A->C->D = 6,A->C->E =7,
以E为中间点,从A->C->E = 7这条最短路径开始新一轮查找
U = {F}
A->C->E->F = 12(比第4步中的A->C->D->F = 9要长,保持F的权值为A->C->D->F = 9)
其中A->C->D->F =9最短
6
选入F,此时 S = {A, C, B, D ,E, F}
此时最短路径 A->A = 0,A->C = 3,A->C->B = 5,A->C->D = 6,A->C->E =7,A->C->D->F = 9
U集合已空,查找完毕
算法实现:
伪代码
DIJSTRA(G,w,s)
1 INITIALIZE-SINGLE-SOURCE(G,s)
2 S ← Φ
3 Q ← V[G]
4 while Q≠Φ
5 do u EXTRACT-MIN(Q)
6 S ← S∪{u}
7 for each vertex v∈Adj[u]
8 do RELAX(u,v,w)
C++代码实现
template<typename vertexNameType, typename weight>
int OLGraph<vertexNameType, weight>::Dijkstra(IN const vertexNameType vertexName1)
{
int sourceIndex = getVertexIndex(vertexName1); //获取源点在容器中索引值
if (-1 == sourceIndex)
{
cerr << "There is no vertex " << endl;
return false;
}
int nVertexNo = getVertexNumber(); //获取顶点数
vector<bool> vecIncludeArray; //顶点是否已求出最短路径
vecIncludeArray.assign(nVertexNo, false); //初始化容器
vecIncludeArray[sourceIndex] = true;
vector<weight> vecDistanceArray; //路径值容器
vecDistanceArray.assign(nVertexNo, weight(INT_MAX)); //将所有顶点到源点的初始路径值为正无穷
vecDistanceArray[sourceIndex] = weight(0); //源点到自己距离置0
vector<int> vecPrevVertex; //路径中,入边弧尾顶点编号(即指向自己那个顶点的编号)
vecPrevVertex.assign(nVertexNo, sourceIndex); //指向所有顶点的弧尾都初始为源点,源点指向所有顶点
getVertexEdgeWeight(sourceIndex, vecDistanceArray); //得到源点到其余每个顶点的距离
int vFrom, vTo;
while(1)
{
weight minWeight = weight(INT_MAX);
vFrom = sourceIndex;
vTo = -1;
for (int i = 0; i < nVertexNo; i++) //找出还没求出最短距离的顶点中,距离最小的一个
{
if (!vecIncludeArray[i] && minWeight > vecDistanceArray[i])
{
minWeight = vecDistanceArray[i];
vFrom = i;
}
}
if (weight(INT_MAX) == minWeight) //若所有顶点都已求出最短路径,跳出循环
{
break;
}
vecIncludeArray[vFrom] = true; //将找出的顶点加入到已求出最短路径的顶点集合中
//更新当前最短路径,只需要更新vFrom顶点的邻接表即可,因为所有vFrom指向的边都在邻接表中
Edge<weight> *p = m_vertexArray[vFrom].firstout;
while (NULL != p)
{
weight wFT = p->edgeWeight;
vTo = p->headvex;
if (!vecIncludeArray[vTo] && vecDistanceArray[vTo] > wFT + vecDistanceArray[vFrom]) //当前顶点还未求出最短路径,并且经由新中间点得路径更短
{
vecDistanceArray[vTo] = wFT + vecDistanceArray[vFrom];
vecPrevVertex[vTo] = vFrom;
}
p = p->tlink;
}
}
for (int i = 0; i < nVertexNo; i++) //输出最短路径
{
if (weight(INT_MAX) != vecDistanceArray[i])
{
cout << getData(sourceIndex) << "->" << getData(i) << ": ";
DijkstraPrint(i, sourceIndex, vecPrevVertex);
cout << " " << vecDistanceArray[i];
cout << endl;
}
}
return 0;
}
template<typename vertexNameType, typename weight>
void OLGraph<vertexNameType, weight>::DijkstraPrint(IN int index, IN int sourceIndex, IN vector<int> vecPreVertex)
{
if (sourceIndex != index)
{
DijkstraPrint(vecPreVertex[index], sourceIndex, vecPreVertex);
}
cout << getData(index) << " ";
}
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