最短路径算法——Dijkstra算法
Dijkstra算法在刚开始在学数据结构的时候，完全没弄明白，主要是也不怎么想去弄明白。而从学校出来到现在，最短路径算法都没有实际运用过，最近在一个GIS项目中总算用到了，于是乎把教材重温了下，同时查阅了网上一些资料，借鉴了一些别人的东西，并顺利用写进了项目中，文中的主要代码来自于园子里的一位大哥，这位大哥对通用框架的研究很深入，他的链接为：http://zhuweisky.cnblogs.com/archive/2005/09/29/246677.html（最短路径）。另外，文章的最后面的一些链接是我找资料的时候用到过的，有兴趣
的朋友可以去看看。
最短路径分析在事故抢修、交通指挥、GPS导航等行业应用中使用的非常广泛， 以至于大多数GIS平台都会把这个分析功能作为一个最基础的功能集成进去，如ARCGIS，SuperMap等。个人感觉想要了解这个算法的来龙去脉，一方面是参与相关书籍仔细理解，另外一个最重要的是要去调试代码。由于历史原因，对于书上的伪C代码，我是完全不感兴趣的，而且由于有严格的数学证明，所以看起来相对较难，而对于面向对象实现的算法，我很感兴趣，也感觉很容易理解，所以一边针对C#实现的面向对象代码再一边对照书籍，感觉理解的层次就加深了。
Dijkstra算法又称为单源最短路径，所谓单源是在一个有向图中，从一个顶点出发，求该顶点至所有可到达顶点的最短路径问题。要顺利实现算法，要求理解Dijstra的算法，同时还要理解图的一些基本概念，图由节点和边构成，将节点和边看成对象，每个对象有自己的特有属性，如在GIS中，一个节点必须都有ID，横坐标，纵坐标等基本属性，边有起点节点，终点节点，长度等属性，而最短路径分析，就是根据边的长度（权值）进行分析的。
边的定义如下：
public
class
Edge
{
public string StartNodeID;
public string EndNodeID;
public double Weight; // 权值，代价
}
{
public string StartNodeID;
public string EndNodeID;
public double Weight; // 权值，代价
}
节点的定义：
public
class
Node
{
private string iD;
private List < Edge > edgeList; // Edge的集合－－出边表
public Node( string id)
{
this .iD = id;
this .edgeList = new List < Edge > ();
}
#region property
public string ID
{
get
{
return this .iD;
}
}
public List < Edge > EdgeList
{
get
{
return this .edgeList;
}
}
#endregion
}
{
private string iD;
private List < Edge > edgeList; // Edge的集合－－出边表
public Node( string id)
{
this .iD = id;
this .edgeList = new List < Edge > ();
}
#region property
public string ID
{
get
{
return this .iD;
}
}
public List < Edge > EdgeList
{
get
{
return this .edgeList;
}
}
#endregion
}
本次用于分析的拓扑图如下：(A为起点，D为终点，边上的数字为权值)
利用上述的边与节点的定义，可以通过代码简单的构成如下图：
public
class
Graph
{
public List < Node > m_nodeList = null ;
public Graph()
{
m_nodeList = new List < Node > ();
}
/// <summary>
/// 获取图的节点集合
/// </summary>
public List < Node > NodeList
{
get { return this .m_nodeList;}
}
/// <summary>
/// 初始化拓扑图
/// </summary>
public void Init()
{
// *****************BNode*******************
NodeaNode = new Node( " A " );
m_nodeList.Add(aNode);
// A->B
EdgeaEdge1 = new Edge();
aEdge1.StartNodeID = aNode.ID;
aEdge1.EndNodeID = " B " ;
aEdge1.Weight = 10 ;
aNode.EdgeList.Add(aEdge1);
// A->C
EdgeaEdge2 = new Edge();
aEdge2.StartNodeID = aNode.ID;
aEdge2.EndNodeID = " C " ;
aEdge2.Weight = 20 ;
aNode.EdgeList.Add(aEdge2);
// A->E
EdgeaEdge3 = new Edge();
aEdge3.StartNodeID = aNode.ID;
aEdge3.EndNodeID = " E " ;
aEdge3.Weight = 30 ;
aNode.EdgeList.Add(aEdge3);
// *****************BNode*******************
NodebNode = new Node( " B " );
m_nodeList.Add(bNode);
// B->C
EdgebEdge1 = new Edge();
bEdge1.StartNodeID = bNode.ID;
bEdge1.EndNodeID = " C " ;
bEdge1.Weight = 5 ;
bNode.EdgeList.Add(bEdge1);
// B->E
EdgebEdge2 = new Edge();
bEdge2.StartNodeID = bNode.ID;
bEdge2.EndNodeID = " E " ;
bEdge2.Weight = 10 ;
bNode.EdgeList.Add(bEdge2);
// *****************CNode*******************
NodecNode = new Node( " C " );
m_nodeList.Add(cNode);
// C->D
EdgecEdge1 = new Edge();
cEdge1.StartNodeID = cNode.ID;
cEdge1.EndNodeID = " D " ;
cEdge1.Weight = 30 ;
cNode.EdgeList.Add(cEdge1);
// *****************DNode*******************
NodedNode = new Node( " D " );
m_nodeList.Add(dNode);
// *****************ENode*******************
NodeeNode = new Node( " E " );
m_nodeList.Add(eNode);
// E->D
EdgeeEdge1 = new Edge();
eEdge1.StartNodeID = eNode.ID;
eEdge1.EndNodeID = " D " ;
eEdge1.Weight = 20 ;
eNode.EdgeList.Add(eEdge1);
}
}
{
public List < Node > m_nodeList = null ;
public Graph()
{
m_nodeList = new List < Node > ();
}
/// <summary>
/// 获取图的节点集合
/// </summary>
public List < Node > NodeList
{
get { return this .m_nodeList;}
}
/// <summary>
/// 初始化拓扑图
/// </summary>
public void Init()
{
// *****************BNode*******************
NodeaNode = new Node( " A " );
m_nodeList.Add(aNode);
// A->B
EdgeaEdge1 = new Edge();
aEdge1.StartNodeID = aNode.ID;
aEdge1.EndNodeID = " B " ;
aEdge1.Weight = 10 ;
aNode.EdgeList.Add(aEdge1);
// A->C
EdgeaEdge2 = new Edge();
aEdge2.StartNodeID = aNode.ID;
aEdge2.EndNodeID = " C " ;
aEdge2.Weight = 20 ;
aNode.EdgeList.Add(aEdge2);
// A->E
EdgeaEdge3 = new Edge();
aEdge3.StartNodeID = aNode.ID;
aEdge3.EndNodeID = " E " ;
aEdge3.Weight = 30 ;
aNode.EdgeList.Add(aEdge3);
// *****************BNode*******************
NodebNode = new Node( " B " );
m_nodeList.Add(bNode);
// B->C
EdgebEdge1 = new Edge();
bEdge1.StartNodeID = bNode.ID;
bEdge1.EndNodeID = " C " ;
bEdge1.Weight = 5 ;
bNode.EdgeList.Add(bEdge1);
// B->E
EdgebEdge2 = new Edge();
bEdge2.StartNodeID = bNode.ID;
bEdge2.EndNodeID = " E " ;
bEdge2.Weight = 10 ;
bNode.EdgeList.Add(bEdge2);
// *****************CNode*******************
NodecNode = new Node( " C " );
m_nodeList.Add(cNode);
// C->D
EdgecEdge1 = new Edge();
cEdge1.StartNodeID = cNode.ID;
cEdge1.EndNodeID = " D " ;
cEdge1.Weight = 30 ;
cNode.EdgeList.Add(cEdge1);
// *****************DNode*******************
NodedNode = new Node( " D " );
m_nodeList.Add(dNode);
// *****************ENode*******************
NodeeNode = new Node( " E " );
m_nodeList.Add(eNode);
// E->D
EdgeeEdge1 = new Edge();
eEdge1.StartNodeID = eNode.ID;
eEdge1.EndNodeID = " D " ;
eEdge1.Weight = 20 ;
eNode.EdgeList.Add(eEdge1);
}
}
有了拓扑节点和边，就可以根据算法构造其他最短路径分析的对象了，主要步骤如下：
Ø初始化图中的从A出发的路径集合：
///
<summary>
/// PlanCourse缓存从源节点到其它任一节点的最小权值路径（路径表）
/// </summary>
public class PlanCourse
{
private HashtablehtPassedPath;
#region ctor
public PlanCourse(List < Node > nodeList, string originID)
{
this .htPassedPath = new Hashtable();
NodeoriginNode = null ;
foreach (Nodenode in nodeList)
{
if (node.ID == originID)
{
originNode = node;
}
else
{
PassedPathpPath = new PassedPath(node.ID);
this .htPassedPath.Add(node.ID,pPath);
}
}
if (originNode == null )
{
throw new Exception( " Theoriginnodeisnotexist! " );
}
this .InitializeWeight(originNode);
}
/// <summary>
/// 通过指定节点的边的权值初始化路径表
/// </summary>
/// <paramname="originNode"></param>
private void InitializeWeight(NodeoriginNode)
{
if ((originNode.EdgeList == null ) || (originNode.EdgeList.Count == 0 ))
{
return ;
}
foreach (Edgeedge in originNode.EdgeList)
{
PassedPathpPath = this [edge.EndNodeID];
if (pPath == null )
{
continue ;
}
pPath.PassedIDList.Add(originNode.ID);
pPath.Weight = edge.Weight;
}
}
#endregion
/// <summary>
/// 获取指定点的路径表
/// </summary>
/// <paramname="nodeID"></param>
/// <returns></returns>
public PassedPath this [ string nodeID]
{
get
{
return (PassedPath) this .htPassedPath[nodeID];
}
}
}
/// PlanCourse缓存从源节点到其它任一节点的最小权值路径（路径表）
/// </summary>
public class PlanCourse
{
private HashtablehtPassedPath;
#region ctor
public PlanCourse(List < Node > nodeList, string originID)
{
this .htPassedPath = new Hashtable();
NodeoriginNode = null ;
foreach (Nodenode in nodeList)
{
if (node.ID == originID)
{
originNode = node;
}
else
{
PassedPathpPath = new PassedPath(node.ID);
this .htPassedPath.Add(node.ID,pPath);
}
}
if (originNode == null )
{
throw new Exception( " Theoriginnodeisnotexist! " );
}
this .InitializeWeight(originNode);
}
/// <summary>
/// 通过指定节点的边的权值初始化路径表
/// </summary>
/// <paramname="originNode"></param>
private void InitializeWeight(NodeoriginNode)
{
if ((originNode.EdgeList == null ) || (originNode.EdgeList.Count == 0 ))
{
return ;
}
foreach (Edgeedge in originNode.EdgeList)
{
PassedPathpPath = this [edge.EndNodeID];
if (pPath == null )
{
continue ;
}
pPath.PassedIDList.Add(originNode.ID);
pPath.Weight = edge.Weight;
}
}
#endregion
/// <summary>
/// 获取指定点的路径表
/// </summary>
/// <paramname="nodeID"></param>
/// <returns></returns>
public PassedPath this [ string nodeID]
{
get
{
return (PassedPath) this .htPassedPath[nodeID];
}
}
}
Ø从A中最短路径集合中找到一个最短的路径点Vi开始分析
///
<summary>
/// 从PlanCourse取出一个当前累积权值最小，并且没有被处理过的节点
/// </summary>
/// <returns></returns>
private NodeGetMinWeightRudeNode(PlanCourseplanCourse,List < Node > nodeList, string originID)
{
double weight = double .MaxValue;
NodedestNode = null ;
foreach (Nodenode in nodeList)
{
if (node.ID == originID)
{
continue ;
}
PassedPathpPath = planCourse[node.ID];
if (pPath.BeProcessed)
{
continue ;
}
if (pPath.Weight < weight)
{
weight = pPath.Weight;
destNode = node;
}
}
return destNode;
}
/// 从PlanCourse取出一个当前累积权值最小，并且没有被处理过的节点
/// </summary>
/// <returns></returns>
private NodeGetMinWeightRudeNode(PlanCourseplanCourse,List < Node > nodeList, string originID)
{
double weight = double .MaxValue;
NodedestNode = null ;
foreach (Nodenode in nodeList)
{
if (node.ID == originID)
{
continue ;
}
PassedPathpPath = planCourse[node.ID];
if (pPath.BeProcessed)
{
continue ;
}
if (pPath.Weight < weight)
{
weight = pPath.Weight;
destNode = node;
}
}
return destNode;
}
Ø修正从A出发至Vi最短路径，并重新选择另一个最短路径点Vj点开始分析，重新执行上述步骤的路径分析
while
(curNode
!=
null
)
{
PassedPathcurPath = planCourse[curNode.ID];
foreach (Edgeedge in curNode.EdgeList)
{
PassedPathtargetPath = planCourse[edge.EndNodeID];
double tempWeight = curPath.Weight + edge.Weight;
if (tempWeight < targetPath.Weight)
{
targetPath.Weight = tempWeight;
targetPath.PassedIDList.Clear();
for ( int i = 0 ;i < curPath.PassedIDList.Count;i ++ )
{
targetPath.PassedIDList.Add(curPath.PassedIDList[i].ToString());
}
targetPath.PassedIDList.Add(curNode.ID);
}
}
// 标志为已处理
planCourse[curNode.ID].BeProcessed = true ;
// 获取下一个未处理节点
curNode = this .GetMinWeightRudeNode(planCourse,nodeList,originID);
}
{
PassedPathcurPath = planCourse[curNode.ID];
foreach (Edgeedge in curNode.EdgeList)
{
PassedPathtargetPath = planCourse[edge.EndNodeID];
double tempWeight = curPath.Weight + edge.Weight;
if (tempWeight < targetPath.Weight)
{
targetPath.Weight = tempWeight;
targetPath.PassedIDList.Clear();
for ( int i = 0 ;i < curPath.PassedIDList.Count;i ++ )
{
targetPath.PassedIDList.Add(curPath.PassedIDList[i].ToString());
}
targetPath.PassedIDList.Add(curNode.ID);
}
}
// 标志为已处理
planCourse[curNode.ID].BeProcessed = true ;
// 获取下一个未处理节点
curNode = this .GetMinWeightRudeNode(planCourse,nodeList,originID);
}
Ø重复上述两个步骤，一直到所有的对象都分析完为止。
Ø这个时候的路径集合表中已经保存了从A到任意一点的最短路径集合了。
///
<summary>
/// 从PlanCourse表中取出目标节点的PassedPath，这个PassedPath即是规划结果
/// </summary>
/// <returns></returns>
private RoutePlanResultGetResult(PlanCourseplanCourse, string destID)
{
PassedPathpPath = planCourse[destID];
if (pPath.Weight == int .MaxValue)
{
RoutePlanResultresult1 = new RoutePlanResult( null , int .MaxValue);
return result1;
}
string []passedNodeIDs = new string [pPath.PassedIDList.Count];
for ( int i = 0 ;i < passedNodeIDs.Length;i ++ )
{
passedNodeIDs[i] = pPath.PassedIDList[i].ToString();
}
RoutePlanResultresult = new RoutePlanResult(passedNodeIDs,pPath.Weight);
return result;
}
/// 从PlanCourse表中取出目标节点的PassedPath，这个PassedPath即是规划结果
/// </summary>
/// <returns></returns>
private RoutePlanResultGetResult(PlanCourseplanCourse, string destID)
{
PassedPathpPath = planCourse[destID];
if (pPath.Weight == int .MaxValue)
{
RoutePlanResultresult1 = new RoutePlanResult( null , int .MaxValue);
return result1;
}
string []passedNodeIDs = new string [pPath.PassedIDList.Count];
for ( int i = 0 ;i < passedNodeIDs.Length;i ++ )
{
passedNodeIDs[i] = pPath.PassedIDList[i].ToString();
}
RoutePlanResultresult = new RoutePlanResult(passedNodeIDs,pPath.Weight);
return result;
}
最短路径的结果类定义如下：
public
class
RoutePlanResult
{
public RoutePlanResult( string []passedNodes, double value)
{
m_resultNodes = passedNodes;
m_value = value;
}
private string []m_resultNodes;
/// <summary>
/// 最短路径经过的节点
/// </summary>
public string []ResultNodes
{
get { return m_resultNodes;}
}
private double m_value;
/// <summary>
/// 最短路径的值
/// </summary>
private double Value
{
get { return m_value;}
}
}
{
public RoutePlanResult( string []passedNodes, double value)
{
m_resultNodes = passedNodes;
m_value = value;
}
private string []m_resultNodes;
/// <summary>
/// 最短路径经过的节点
/// </summary>
public string []ResultNodes
{
get { return m_resultNodes;}
}
private double m_value;
/// <summary>
/// 最短路径的值
/// </summary>
private double Value
{
get { return m_value;}
}
}
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