四叉树空间索引原理及其实现

今天依然在放假中，在此将以前在学校写的四叉树的东西拿出来和大家分享。

四叉树索引的基本思想是将地理空间递归划分为不同层次的树结构。它将已知范围的空间等分成四个相等的子空间，如此递归下去，直至树的层次达到一定深度或者满足某种要求后停止分割。四叉树的结构比较简单，并且当空间数据对象分布比较均匀时，具有比较高的空间数据插入和查询效率，因此四叉树是GIS中常用的空间索引之一。常规四叉树的结构如图所示，地理空间对象都存储在叶子节点上，中间节点以及根节点不存储地理空间对象。

 

四叉树示意图

 

四叉树对于区域查询，效率比较高。但如果空间对象分布不均匀，随着地理空间对象的不断插入，四叉树的层次会不断地加深，将形成一棵严重不平衡的四叉树，那么每次查询的深度将大大的增多，从而导致查询效率的急剧下降。

 

本节将介绍一种改进的四叉树索引结构。四叉树结构是自顶向下逐步划分的一种树状的层次结构。传统的四叉树索引存在着以下几个缺点：

(1)空间实体只能存储在叶子节点中，中间节点以及根节点不能存储空间实体信息，随着空间对象的不断插入，最终会导致四叉树树的层次比较深，在进行空间数据窗口查询的时候效率会比较低下。

(2)同一个地理实体在四叉树的分裂过程中极有可能存储在多个节点中，这样就导致了索引存储空间的浪费。

(3)由于地理空间对象可能分布不均衡，这样会导致常规四叉树生成一棵极为不平衡的树，这样也会造成树结构的不平衡以及存储空间的浪费。

相应的改进方法，将地理实体信息存储在完全包含它的最小矩形节点中，不存储在它的父节点中，每个地理实体只在树中存储一次，避免存储空间的浪费。首先生成满四叉树，避免在地理实体插入时需要重新分配内存，加快插入的速度，最后将空的节点所占内存空间释放掉。改进后的四叉树结构如下图所示。四叉树的深度一般取经验值4-7之间为最佳。

 

图改进的四叉树结构

 

为了维护空间索引与对存储在文件或数据库中的空间数据的一致性，作者设计了如下的数据结构支持四叉树的操作。

(1)四分区域标识

分别定义了一个平面区域的四个子区域索引号，右上为第一象限0，左上为第二象限1，左下为第三象限2，右下为第四象限3。

typedef enum

{

      UR = 0,// UR第一象限

      UL = 1, // UL为第二象限

      LL = 2, // LL为第三象限

      LR = 3  // LR为第四象限

}QuadrantEnum;

(2)空间对象数据结构

空间对象数据结构是对地理空间对象的近似，在空间索引中，相当一部分都是采用MBR作为近似。

/*空间对象MBR信息*/

typedef struct SHPMBRInfo

{

      int nID;       //空间对象ID号

      MapRect Box;    //空间对象MBR范围坐标

}SHPMBRInfo;

nID是空间对象的标识号，Box是空间对象的最小外包矩形（MBR）。

(3)四叉树节点数据结构

四叉树节点是四叉树结构的主要组成部分，主要用于存储空间对象的标识号和MBR，也是四叉树算法操作的主要部分。

/*四叉树节点类型结构*/

typedef struct QuadNode

{

      MapRect            Box;                   //节点所代表的矩形区域

      int                nShpCount;        //节点所包含的所有空间对象个数

      SHPMBRInfo* pShapeObj;          //空间对象指针数组

      int         nChildCount;            //子节点个数

      QuadNode *children[4];             //指向节点的四个孩子

}QuadNode;

Box是代表四叉树对应区域的最小外包矩形，上一层的节点的最小外包矩形包含下一层最小外包矩形区域；nShpCount代表本节点包含的空间对象的个数；pShapeObj代表指向空间对象存储地址的首地址，同一个节点的空间对象在内存中连续存储；nChildCount代表节点拥有的子节点的数目；children是指向孩子节点指针的数组。

上述理论部分都都讲的差不多了，下面就贴上我的C语言实现版本代码。

头文件如下：

#ifndef __QUADTREE_H_59CAE94A_E937_42AD_AA27_794E467715BB__ #define __QUADTREE_H_59CAE94A_E937_42AD_AA27_794E467715BB__ /* 一个矩形区域的象限划分：: UL(1) | UR(0) ----------|----------- LL(2) | LR(3) 以下对该象限类型的枚举 */ typedef enum { UR = 0, UL = 1, LL = 2, LR = 3 }QuadrantEnum; /*空间对象MBR信息*/ typedef struct SHPMBRInfo { int nID; //空间对象ID号 MapRect Box; //空间对象MBR范围坐标 }SHPMBRInfo; /* 四叉树节点类型结构 */ typedef struct QuadNode { MapRect Box; //节点所代表的矩形区域 int nShpCount; //节点所包含的所有空间对象个数 SHPMBRInfo* pShapeObj; //空间对象指针数组 int nChildCount; //子节点个数 QuadNode *children[4]; //指向节点的四个孩子 }QuadNode; /* 四叉树类型结构 */ typedef struct quadtree_t { QuadNode *root; int depth; // 四叉树的深度 }QuadTree; //初始化四叉树节点 QuadNode *InitQuadNode(); //层次创建四叉树方法（满四叉树） void CreateQuadTree(int depth,GeoLayer *poLayer,QuadTree* pQuadTree); //创建各个分支 void CreateQuadBranch(int depth,MapRect &rect,QuadNode** node); //构建四叉树空间索引 void BuildQuadTree(GeoLayer*poLayer,QuadTree* pQuadTree); //四叉树索引查询(矩形查询) void SearchQuadTree(QuadNode* node,MapRect &queryRect,vector<int>& ItemSearched); //四叉树索引查询(矩形查询)并行查询 void SearchQuadTreePara(vector<QuadNode*> resNodes,MapRect &queryRect,vector<int>& ItemSearched); //四叉树的查询（点查询） void PtSearchQTree(QuadNode* node,double cx,double cy,vector<int>& ItemSearched); //将指定的空间对象插入到四叉树中 void Insert(long key,MapRect &itemRect,QuadNode* pNode); //将指定的空间对象插入到四叉树中 void InsertQuad(long key,MapRect &itemRect,QuadNode* pNode); //将指定的空间对象插入到四叉树中 void InsertQuad2(long key,MapRect &itemRect,QuadNode* pNode); //判断一个节点是否是叶子节点 bool IsQuadLeaf(QuadNode* node); //删除多余的节点 bool DelFalseNode(QuadNode* node); //四叉树遍历(所有要素) void TraversalQuadTree(QuadNode* quadTree,vector<int>& resVec); //四叉树遍历（所有节点） void TraversalQuadTree(QuadNode* quadTree,vector<QuadNode*>& arrNode); //释放树的内存空间 void ReleaseQuadTree(QuadNode** quadTree); //计算四叉树所占的字节的大小 long CalByteQuadTree(QuadNode* quadTree,long& nSize); #endif

源文件如下：

#include 'QuadTree.h' QuadNode *InitQuadNode() { QuadNode *node = new QuadNode; node->Box.maxX = 0; node->Box.maxY = 0; node->Box.minX = 0; node->Box.minY = 0; for (int i = 0; i < 4; i ) { node->children[i] = NULL; } node->nChildCount = 0; node->nShpCount = 0; node->pShapeObj = NULL; return node; } void CreateQuadTree(int depth,GeoLayer *poLayer,QuadTree* pQuadTree) { pQuadTree->depth = depth; GeoEnvelope env; //整个图层的MBR poLayer->GetExtent(&env); MapRect rect; rect.minX = env.MinX; rect.minY = env.MinY; rect.maxX = env.MaxX; rect.maxY = env.MaxY; //创建各个分支 CreateQuadBranch(depth,rect,&(pQuadTree->root)); int nCount = poLayer->GetFeatureCount(); GeoFeature **pFeatureClass = new GeoFeature*[nCount]; for (int i = 0; i < poLayer->GetFeatureCount(); i ) { pFeatureClass[i] = poLayer->GetFeature(i); } //插入各个要素 GeoEnvelope envObj; //空间对象的MBR //#pragma omp parallel for for (int i = 0; i < nCount; i ) { pFeatureClass[i]->GetGeometry()->getEnvelope(&envObj); rect.minX = envObj.MinX; rect.minY = envObj.MinY; rect.maxX = envObj.MaxX; rect.maxY = envObj.MaxY; InsertQuad(i,rect,pQuadTree->root); } //DelFalseNode(pQuadTree->root); } void CreateQuadBranch(int depth,MapRect &rect,QuadNode** node) { if (depth != 0) { *node = InitQuadNode(); //创建树根 QuadNode *pNode = *node; pNode->Box = rect; pNode->nChildCount = 4; MapRect boxs[4]; pNode->Box.Split(boxs,boxs 1,boxs 2,boxs 3); for (int i = 0; i < 4; i ) { //创建四个节点并插入相应的MBR pNode->children[i] = InitQuadNode(); pNode->children[i]->Box = boxs[i]; CreateQuadBranch(depth-1,boxs[i],&(pNode->children[i])); } } } void BuildQuadTree(GeoLayer *poLayer,QuadTree* pQuadTree) { assert(poLayer); GeoEnvelope env; //整个图层的MBR poLayer->GetExtent(&env); pQuadTree->root = InitQuadNode(); QuadNode* rootNode = pQuadTree->root; rootNode->Box.minX = env.MinX; rootNode->Box.minY = env.MinY; rootNode->Box.maxX = env.MaxX; rootNode->Box.maxY = env.MaxY; //设置树的深度( 根据等比数列的求和公式) //pQuadTree->depth = log(poLayer->GetFeatureCount()*3/8.0 1)/log(4.0); int nCount = poLayer->GetFeatureCount(); MapRect rect; GeoEnvelope envObj; //空间对象的MBR for (int i = 0; i < nCount; i ) { poLayer->GetFeature(i)->GetGeometry()->getEnvelope(&envObj); rect.minX = envObj.MinX; rect.minY = envObj.MinY; rect.maxX = envObj.MaxX; rect.maxY = envObj.MaxY; InsertQuad2(i,rect,rootNode); } DelFalseNode(pQuadTree->root); } void SearchQuadTree(QuadNode* node,MapRect &queryRect,vector<int>& ItemSearched) { assert(node); //int coreNum = omp_get_num_procs(); //vector<int> * pResArr = new vector<int>[coreNum]; if (NULL != node) { for (int i = 0; i < node->nShpCount; i ) { if (queryRect.Contains(node->pShapeObj[i].Box) || queryRect.Intersects(node->pShapeObj[i].Box)) { ItemSearched.push_back(node->pShapeObj[i].nID); } } //并行搜索四个孩子节点 /*#pragma omp parallel sections { #pragma omp section if ((node->children[0] != NULL) && (node->children[0]->Box.Contains(queryRect) || node->children[0]->Box.Intersects(queryRect))) { int tid = omp_get_thread_num(); SearchQuadTree(node->children[0],queryRect,pResArr[tid]); } #pragma omp section if ((node->children[1] != NULL) && (node->children[1]->Box.Contains(queryRect) || node->children[1]->Box.Intersects(queryRect))) { int tid = omp_get_thread_num(); SearchQuadTree(node->children[1],queryRect,pResArr[tid]); } #pragma omp section if ((node->children[2] != NULL) && (node->children[2]->Box.Contains(queryRect) || node->children[2]->Box.Intersects(queryRect))) { int tid = omp_get_thread_num(); SearchQuadTree(node->children[2],queryRect,pResArr[tid]); } #pragma omp section if ((node->children[3] != NULL) && (node->children[3]->Box.Contains(queryRect) || node->children[3]->Box.Intersects(queryRect))) { int tid = omp_get_thread_num(); SearchQuadTree(node->children[3],queryRect,pResArr[tid]); } }*/ for (int i = 0; i < 4; i ) { if ((node->children[i] != NULL) && (node->children[i]->Box.Contains(queryRect) || node->children[i]->Box.Intersects(queryRect))) { SearchQuadTree(node->children[i],queryRect,ItemSearched); //node = node->children[i]; //非递归 } } } /*for (int i = 0 ; i < coreNum; i ) { ItemSearched.insert(ItemSearched.end(),pResArr[i].begin(),pResArr[i].end()); }*/ } void SearchQuadTreePara(vector<QuadNode*> resNodes,MapRect &queryRect,vector<int>& ItemSearched) { int coreNum = omp_get_num_procs(); omp_set_num_threads(coreNum); vector<int>* searchArrs = new vector<int>[coreNum]; for (int i = 0; i < coreNum; i ) { searchArrs[i].clear(); } #pragma omp parallel for for (int i = 0; i < resNodes.size(); i ) { int tid = omp_get_thread_num(); for (int j = 0; j < resNodes[i]->nShpCount; j ) { if (queryRect.Contains(resNodes[i]->pShapeObj[j].Box) || queryRect.Intersects(resNodes[i]->pShapeObj[j].Box)) { searchArrs[tid].push_back(resNodes[i]->pShapeObj[j].nID); } } } for (int i = 0; i < coreNum; i ) { ItemSearched.insert(ItemSearched.end(), searchArrs[i].begin(),searchArrs[i].end()); } delete [] searchArrs; searchArrs = NULL; } void PtSearchQTree(QuadNode* node,double cx,double cy,vector<int>& ItemSearched) { assert(node); if (node->nShpCount >0) //节点 { for (int i = 0; i < node->nShpCount; i ) { if (node->pShapeObj[i].Box.IsPointInRect(cx,cy)) { ItemSearched.push_back(node->pShapeObj[i].nID); } } } else if (node->nChildCount >0) //节点 { for (int i = 0; i < 4; i ) { if (node->children[i]->Box.IsPointInRect(cx,cy)) { PtSearchQTree(node->children[i],cx,cy,ItemSearched); } } } //找出重复元素的位置 sort(ItemSearched.begin(),ItemSearched.end()); //先排序,默认升序 vector<int>::iterator unique_iter = unique(ItemSearched.begin(),ItemSearched.end()); ItemSearched.erase(unique_iter,ItemSearched.end()); } void Insert(long key, MapRect &itemRect,QuadNode* pNode) { QuadNode *node = pNode; //保留根节点副本 SHPMBRInfo pShpInfo; //节点有孩子 if (0 < node->nChildCount) { for (int i = 0; i < 4; i ) { //如果包含或相交，则将节点插入到此节点 if (node->children[i]->Box.Contains(itemRect) || node->children[i]->Box.Intersects(itemRect)) { //node = node->children[i]; Insert(key,itemRect,node->children[i]); } } } //如果当前节点存在一个子节点时 else if (1 == node->nShpCount) { MapRect boxs[4]; node->Box.Split(boxs,boxs 1,boxs 2,boxs 3); //创建四个节点并插入相应的MBR node->children[UR] = InitQuadNode(); node->children[UL] = InitQuadNode(); node->children[LL] = InitQuadNode(); node->children[LR] = InitQuadNode(); node->children[UR]->Box = boxs[0]; node->children[UL]->Box = boxs[1]; node->children[LL]->Box = boxs[2]; node->children[LR]->Box = boxs[3]; node->nChildCount = 4; for (int i = 0; i < 4; i ) { //将当前节点中的要素移动到相应的子节点中 for (int j = 0; j < node->nShpCount; j ) { if (node->children[i]->Box.Contains(node->pShapeObj[j].Box) || node->children[i]->Box.Intersects(node->pShapeObj[j].Box)) { node->children[i]->nShpCount = 1; node->children[i]->pShapeObj = (SHPMBRInfo*)malloc(node->children[i]->nShpCount*sizeof(SHPMBRInfo)); memcpy(node->children[i]->pShapeObj,&(node->pShapeObj[j]),sizeof(SHPMBRInfo)); free(node->pShapeObj); node->pShapeObj = NULL; node->nShpCount = 0; } } } for (int i = 0; i < 4; i ) { //如果包含或相交，则将节点插入到此节点 if (node->children[i]->Box.Contains(itemRect) || node->children[i]->Box.Intersects(itemRect)) { if (node->children[i]->nShpCount == 0) //如果之前没有节点 { node->children[i]->nShpCount = 1; node->pShapeObj = (SHPMBRInfo*)malloc(sizeof(SHPMBRInfo)*node->children[i]->nShpCount); } else if (node->children[i]->nShpCount > 0) { node->children[i]->nShpCount = 1; node->children[i]->pShapeObj = (SHPMBRInfo *)realloc(node->children[i]->pShapeObj, sizeof(SHPMBRInfo)*node->children[i]->nShpCount); } pShpInfo.Box = itemRect; pShpInfo.nID = key; memcpy(node->children[i]->pShapeObj, &pShpInfo,sizeof(SHPMBRInfo)); } } } //当前节点没有空间对象 else if (0 == node->nShpCount) { node->nShpCount = 1; node->pShapeObj = (SHPMBRInfo*)malloc(sizeof(SHPMBRInfo)*node->nShpCount); pShpInfo.Box = itemRect; pShpInfo.nID = key; memcpy(node->pShapeObj,&pShpInfo,sizeof(SHPMBRInfo)); } } void InsertQuad(long key,MapRect &itemRect,QuadNode* pNode) { assert(pNode != NULL); if (!IsQuadLeaf(pNode)) //非叶子节点 { int nCorver = 0; //跨越的子节点个数 int iIndex = -1; //被哪个子节点完全包含的索引号 for (int i = 0; i < 4; i ) { if (pNode->children[i]->Box.Contains(itemRect) && pNode->Box.Contains(itemRect)) { nCorver = 1; iIndex = i; } } //如果被某一个子节点包含，则进入该子节点 if (/*pNode->Box.Contains(itemRect) || pNode->Box.Intersects(itemRect)*/1 <= nCorver) { InsertQuad(key,itemRect,pNode->children[iIndex]); } //如果跨越了多个子节点，直接放在这个节点中 else if (nCorver == 0) { if (pNode->nShpCount == 0) //如果之前没有节点 { pNode->nShpCount = 1; pNode->pShapeObj = (SHPMBRInfo*)malloc(sizeof(SHPMBRInfo)*pNode->nShpCount); } else { pNode->nShpCount = 1; pNode->pShapeObj = (SHPMBRInfo *)realloc(pNode->pShapeObj,sizeof(SHPMBRInfo)*pNode->nShpCount); } SHPMBRInfo pShpInfo; pShpInfo.Box = itemRect; pShpInfo.nID = key; memcpy(pNode->pShapeObj pNode->nShpCount-1,&pShpInfo,sizeof(SHPMBRInfo)); } } //如果是叶子节点，直接放进去 else if (IsQuadLeaf(pNode)) { if (pNode->nShpCount == 0) //如果之前没有节点 { pNode->nShpCount = 1; pNode->pShapeObj = (SHPMBRInfo*)malloc(sizeof(SHPMBRInfo)*pNode->nShpCount); } else { pNode->nShpCount = 1; pNode->pShapeObj = (SHPMBRInfo *)realloc(pNode->pShapeObj,sizeof(SHPMBRInfo)*pNode->nShpCount); } SHPMBRInfo pShpInfo; pShpInfo.Box = itemRect; pShpInfo.nID = key; memcpy(pNode->pShapeObj pNode->nShpCount-1,&pShpInfo,sizeof(SHPMBRInfo)); } } void InsertQuad2(long key,MapRect &itemRect,QuadNode* pNode) { QuadNode *node = pNode; //保留根节点副本 SHPMBRInfo pShpInfo; //节点有孩子 if (0 < node->nChildCount) { for (int i = 0; i < 4; i ) { //如果包含或相交，则将节点插入到此节点 if (node->children[i]->Box.Contains(itemRect) || node->children[i]->Box.Intersects(itemRect)) { //node = node->children[i]; Insert(key,itemRect,node->children[i]); } } } //如果当前节点存在一个子节点时 else if (0 == node->nChildCount) { MapRect boxs[4]; node->Box.Split(boxs,boxs 1,boxs 2,boxs 3); int cnt = -1; for (int i = 0; i < 4; i ) { //如果包含或相交，则将节点插入到此节点 if (boxs[i].Contains(itemRect)) { cnt = i; } } //如果有一个矩形包含此对象，则创建四个孩子节点 if (cnt > -1) { for (int i = 0; i < 4; i ) { //创建四个节点并插入相应的MBR node->children[i] = InitQuadNode(); node->children[i]->Box = boxs[i]; } node->nChildCount = 4; InsertQuad2(key,itemRect,node->children[cnt]); //递归 } //如果都不包含，则直接将对象插入此节点 if (cnt == -1) { if (node->nShpCount == 0) //如果之前没有节点 { node->nShpCount = 1; node->pShapeObj = (SHPMBRInfo*)malloc(sizeof(SHPMBRInfo)*node->nShpCount); } else if (node->nShpCount > 0) { node->nShpCount = 1; node->pShapeObj = (SHPMBRInfo *)realloc(node->pShapeObj, sizeof(SHPMBRInfo)*node->nShpCount); } pShpInfo.Box = itemRect; pShpInfo.nID = key; memcpy(node->pShapeObj, &pShpInfo,sizeof(SHPMBRInfo)); } } //当前节点没有空间对象 /*else if (0 == node->nShpCount) { node->nShpCount = 1; node->pShapeObj = (SHPMBRInfo*)malloc(sizeof(SHPMBRInfo)*node->nShpCount); pShpInfo.Box = itemRect; pShpInfo.nID = key; memcpy(node->pShapeObj,&pShpInfo,sizeof(SHPMBRInfo)); }*/ } bool IsQuadLeaf(QuadNode* node) { if (NULL == node) { return 1; } for (int i = 0; i < 4; i ) { if (node->children[i] != NULL) { return 0; } } return 1; } bool DelFalseNode(QuadNode* node) { //如果没有子节点且没有要素 if (node->nChildCount ==0 && node->nShpCount == 0) { ReleaseQuadTree(&node); } //如果有子节点 else if (node->nChildCount > 0) { for (int i = 0; i < 4; i ) { DelFalseNode(node->children[i]); } } return 1; } void TraversalQuadTree(QuadNode* quadTree,vector<int>& resVec) { QuadNode *node = quadTree; int i = 0; if (NULL != node) { //将本节点中的空间对象存储数组中 for (i = 0; i < node->nShpCount; i ) { resVec.push_back((node->pShapeObj i)->nID); } //遍历孩子节点 for (i = 0; i < node->nChildCount; i ) { if (node->children[i] != NULL) { TraversalQuadTree(node->children[i],resVec); } } } } void TraversalQuadTree(QuadNode* quadTree,vector<QuadNode*>& arrNode) { deque<QuadNode*> nodeQueue; if (quadTree != NULL) { nodeQueue.push_back(quadTree); while (!nodeQueue.empty()) { QuadNode* queueHead = nodeQueue.at(0); //取队列头结点 arrNode.push_back(queueHead); nodeQueue.pop_front(); for (int i = 0; i < 4; i ) { if (queueHead->children[i] != NULL) { nodeQueue.push_back(queueHead->children[i]); } } } } } void ReleaseQuadTree(QuadNode** quadTree) { int i = 0; QuadNode* node = *quadTree; if (NULL == node) { return; } else { for (i = 0; i < 4; i ) { ReleaseQuadTree(&node->children[i]); } free(node); node = NULL; } node = NULL; } long CalByteQuadTree(QuadNode* quadTree,long& nSize) { if (quadTree != NULL) { nSize = sizeof(QuadNode) quadTree->nChildCount*sizeof(SHPMBRInfo); for (int i = 0; i < 4; i ) { if (quadTree->children[i] != NULL) { nSize = CalByteQuadTree(quadTree->children[i],nSize); } } } return 1; }

代码有点长，有需要的朋友可以借鉴并自己优化。