最短路徑

從圖中的某個頂點出發到達另外一個頂點的所經過的邊的權重和最小的一條路徑，稱為最短路徑? ? ? ? ? ? ? ? ? ? ? ? ? ??

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Dijkstra算法

算法來源

Dijkstra算法是由一個叫Dijkstra的荷蘭人發明的，故稱此算法為Dijkstra算法

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算法思想

• 將圖上的初始點看作一個集合S，其它點看作另一個集合
• 根據初始點，求出其它點到初始點的距離d[i] （若相鄰，則d[i]為邊權值；若不相鄰，則d[i]為無限大）
• 選取最小的d[i]（記為d[x]），并將此d[i]邊對應的點（記為x）加入集合S（實際上，加入集合的這個點的d[x]值就是它到初始點的最短距離）
• 再根據x，更新跟 x 相鄰點 y 的d[y]值：d[y] = min{ d[y], d[x] + 邊權值w[x][y] }，因為可能把距離調小，所以這個更新操作叫做松弛操作。
• 重復3，4兩步，直到目標點也加入了集合，此時目標點所對應的d[i]即為最短路徑長度。

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算法缺陷

只能處理有向無環，且權重為正的圖

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代碼實現

/*! *@file Dijkstra.h *@brief 狄克斯特拉算法,適用于有向無環圖，且權重非負 */#ifndef DIJKSTRA_H #define DIJKSTRA_H#include <iostream> #include <string> #include <set> #include <list> #include <map>const double g_Infinity = 1E100;struct CNode;struct CEdge {CNode* m_neighborNode;double m_dist;CEdge(CNode* pNeighborNode, double dist) : m_neighborNode(pNeighborNode), m_dist(dist) {}~CEdge() {} };struct CNode {std::string m_name;std::set<CEdge*> m_edges;CNode(std::string name) : m_name(name) {}~CNode() {for (auto iter = m_edges.begin(); iter != m_edges.end(); ++iter){delete* iter;}m_edges.clear();}void AddEdge(CNode* pNode, double dist){m_edges.insert(new CEdge(pNode, dist));}void AddEdge(CEdge* pEdge){m_edges.insert(pEdge);}bool operator<(const CNode& node) const{return this->m_name.compare(node.m_name) < 0;} };class CGraph { public:CGraph() {}~CGraph() {}bool Dijkstra(CNode* pNodeStart, CNode* pNodeEnd, std::list<CNode*>& path, double& dist){if (!pNodeStart || !pNodeEnd)return false;Init(pNodeStart);bool bSuc(false);CNode* pNodeLowest = FindLowestNode(dist);while (pNodeLowest){if (pNodeLowest == pNodeEnd){bSuc = true;break;}std::set<CEdge*> edges = pNodeLowest->m_edges;for (auto iter = edges.begin(); iter != edges.end(); ++iter){double dCostNew = dist + (*iter)->m_dist;UpdateCostMap((*iter)->m_neighborNode, pNodeLowest, dCostNew);}m_processedNodes.insert(pNodeLowest);pNodeLowest = FindLowestNode(dist);}if (bSuc){bSuc = CreatePath(pNodeLowest, pNodeStart, path);}return bSuc;} private:void Init(CNode* pNodeStart){m_costMap.clear();m_processedNodes.clear();std::set<CEdge*> edges = pNodeStart->m_edges;for (auto iter = edges.begin(); iter != edges.end(); ++iter){CNode* pNode = (*iter)->m_neighborNode;double dCost = (*iter)->m_dist;m_costMap.insert(std::make_pair(pNode, std::make_pair(pNodeStart, dCost)));}}void UpdateCostMap(CNode* pNode, CNode* pParentNode, double cost){auto iter = m_costMap.find(pNode);if (iter != m_costMap.cend()){double oldCost = iter->second.second;if (oldCost > cost){m_costMap[pNode] = std::make_pair(pParentNode, cost);}}else{m_costMap.insert(std::make_pair(pNode, std::make_pair(pParentNode, cost)));}}CNode* FindLowestNode(double& dist){dist = g_Infinity;CNode* pResult(nullptr);for (auto iter = m_costMap.begin(); iter != m_costMap.end(); ++iter){CNode* pNode = iter->first;double distNode = iter->second.second;if (distNode < dist && m_processedNodes.find(pNode) == m_processedNodes.end()){pResult = pNode;dist = distNode;}}return pResult;}bool CreatePath(CNode* pNode, CNode* pStartNode, std::list<CNode*>& path){path.push_front(pNode);while (pNode != pStartNode){if (!pNode){path.clear();return false;}auto iter = m_costMap.find(pNode);if (iter == m_costMap.end()){path.clear();return false;}pNode = iter->second.first;path.push_front(pNode);}return true;}private:std::map<CNode*, std::pair<CNode*, double>> m_costMap;std::set<CNode*> m_processedNodes; };void Print(const std::list<CNode*>& path) {std::cout << "path: ";for (auto iter = path.begin(); iter != path.end(); ++iter){std::cout << " --> " << (*iter)->m_name;}//std::cout << std::endl; }void DijkstraTest() {CNode* pNodeS = new CNode("start");CNode* pNodeF = new CNode("end");CNode* pNodeA = new CNode("A");CNode* pNodeB = new CNode("B");pNodeS->AddEdge(pNodeA, 6);pNodeS->AddEdge(pNodeB, 2);pNodeB->AddEdge(pNodeA, 3);pNodeB->AddEdge(pNodeF, 5);pNodeA->AddEdge(pNodeF, 1);CGraph cg;std::list<CNode*> path;double dist = g_Infinity;bool bSuc = cg.Dijkstra(pNodeS, pNodeF, path, dist);if (bSuc){Print(path);std::cout << " dist: " << dist << std::endl;}else{std::cout << "failed!!!" << std::endl;}delete pNodeS;delete pNodeF;delete pNodeA;delete pNodeB;pNodeS = nullptr;pNodeF = nullptr;pNodeA = nullptr;pNodeB = nullptr; }#endif // !DIJKSTRA_H

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