Radix Heap ---Dijkstra算法的优化 BY Gremount
Radix Heap 算法是在Dijkstra的Dial实现的基础上，通过减少对桶的使用，来优化算法的时间复杂度:
Dial 时间复杂度是O(m+nC) -------C是最长的链路
Radix Heap 时间复杂度是O(m+n*log(nC))
本质上来说Radix Heap是Dial和Dijkstra的折中，因为Dial用NC个桶，Dijkstra用1个桶
下面是本程序中使用的一些函数介绍：
void radix_heap():
（1）对所有桶的range初始化（2）所有节点的初始化(这一步可以在建图的时候完成)；
（2）除开始点，其他全部放入无穷桶里，注意这里的无穷桶的id=初始d值对2求对数后上去整；
（3）更新源节点
（4）后续是找最小值和更新的操作循环（类似于dijkstra算法）
int FindMin():
（1）移动指针到第一个不为空的桶
（2）如果桶的编号是0或者1，则任取元素出来返回；
如果编号大于1且桶内元素个数为1，则将该元素取出来返回；
如果编号大于1且桶内的元素个数大于1，则先找出最小值，再执行redistribute()
重置桶里的元素
int Redistribute():
（1）根据最小值，更改桶的range，修改range和桶的映射；
（2）将该桶内的元素重置位置，返回最小值；
void Update(int i):
根据i点的d值和邻接边的值来更改与i相连的其他节点的d值；
下面是我写的程序，分为三部分，第一部分是主程序，第二部分是资源程序，第三部分是头文件
第一部分：创建如下图：
//radix_heap.cpp 主程序：完成图的创建和最短路函数调用
#include "resources.h"
CEdge::CEdge(int a, int b, int c, int d){
tail=a;
head=b;
weight=c;
capacity=d;
}
CEdge::CEdge(int a, int b, int c){
head=b;
tail=a;
weight=c;
}
CEdge::CEdge(CEdge & x){
tail=x.getTail();
head=x.getHead();
weight=x.getWeight();
capacity=x.getCap();
}
CGraph::CGraph(list<CEdge*> listEdge){
IncidentList=listEdge;
numVertex=N;
numEdge=listEdge.size();
}
int main()
{
list<CEdge*> listEdge;
CEdge* e1=new CEdge(1,2,1,10);
CEdge* e2=new CEdge(2,1,1,10);
CEdge* e3=new CEdge(1,7,1,10);
CEdge* e4=new CEdge(7,1,1,10);
CEdge* e5=new CEdge(7,10,1,10);
CEdge* e6=new CEdge(10,7,1,10);
CEdge* e7=new CEdge(3,2,1,10);
CEdge* e8=new CEdge(2,3,1,10);
CEdge* e9=new CEdge(3,4,1,10);
CEdge* e10=new CEdge(4,3,1,10);
CEdge* e11=new CEdge(4,10,1,10);
CEdge* e12=new CEdge(10,4,1,10);
CEdge* e13=new CEdge(1,5,1,10);
CEdge* e14=new CEdge(5,1,1,10);
CEdge* e15=new CEdge(6,5,1,10);
CEdge* e16=new CEdge(5,6,1,10);
CEdge* e17=new CEdge(6,10,1,10);
CEdge* e18=new CEdge(10,6,1,10);
CEdge* e19=new CEdge(8,10,1,10);
CEdge* e20=new CEdge(10,8,1,10);
CEdge* e21=new CEdge(9,10,1,10);
CEdge* e22=new CEdge(10,9,1,10);
listEdge.push_back(e1);
listEdge.push_back(e2);
listEdge.push_back(e3);
listEdge.push_back(e4);
listEdge.push_back(e5);
listEdge.push_back(e6);
listEdge.push_back(e7);
listEdge.push_back(e8);
listEdge.push_back(e9);
listEdge.push_back(e10);
listEdge.push_back(e11);
listEdge.push_back(e12);
listEdge.push_back(e13);
listEdge.push_back(e14);
listEdge.push_back(e15);
listEdge.push_back(e16);
listEdge.push_back(e17);
listEdge.push_back(e18);
listEdge.push_back(e19);
listEdge.push_back(e20);
listEdge.push_back(e21);
listEdge.push_back(e22);
CGraph g(listEdge);
g.p1();
g.p2();
g.p3();
g.p4();
g.radix_heap(g,1);
getchar();
return 0;
}第二部分：
//resources.h 所用的类：点，边，图
//图中包含对图的测量函数（p1`、p2、p3、p4）
//radix_heap() FindMin() Update() Redistribute()
#include "common.h"
class CVertex{
public:
int d;
int p;
int ID;
CVertex(int x){ID=x;d=INF;p=NULL;}
~CVertex(){;}
};
class CEdge{
private:
int tail, head;
int weight, capacity;
public:
CEdge(int a, int b, int c, int d);
CEdge(int a, int b, int c);
CEdge(CEdge &x);
int getHead(){return head;}
int getTail(){return tail;}
int getWeight(){return weight;}
int getCap(){return capacity;}
bool operator<(CEdge& x){
if(weight<x.weight)
return 1;
else
return 0;
}
};
class CGraph{
private:
int numVertex;
int numEdge;
list<CEdge*> IncidentList;
public:
set<int> S;
set<int> V;
int d[N+10];
int p[N+10];
map<int,int> degree_vertex;
multimap<int,int> degree_vertex2;
map<int,list<int>> adjlist;
vector<vector<CEdge*>> adjmatrix;
map<int,list<CEdge*>> nelist;
map<int,map<int, CVertex*>> mapBuckets;
map<int, CVertex*> mapVID_Vertex;
map<int,map<int, CVertex*>>::iterator itBucket;
map<int, CVertex*>::iterator itcv;
int range[2024];
int rbegin[20];
int rend[20];
int zhishu;
int db[20];
CGraph(char* inputFile);
CGraph(list<CEdge*> listEdge);
CGraph(CGraph &);
void init()
{
//range表初始化
zhishu=2;
rbegin[0]=1;
rend[0]=1;
rbegin[1]=2;
rend[1]=2;
for(int i=2;i<=10;i++)
{
zhishu=zhishu*2;
rbegin[i]=rend[i-1]+1;
rend[i]=zhishu;
}
//等比数列初始化
db[1]=1;
for(int i=2;i<=10;i++)
db[i]=db[i-1]*2;
}
int getNumVertex(){
return numVertex;
}
int getNumEdge(){
return numEdge;
}
int cmp(const pair<int,int> &x, const pair<int, int> &y){
return x.second > y.second;
}
void p1(){
list<CEdge*>::iterator it,iend;
multimap<int,int>::iterator it2;
iend=IncidentList.end();
vector<pair<int,int>> dv_vec;
for(int i=1;i<=N;i++)
degree_vertex[i]=0;
for(it=IncidentList.begin();it!=iend;it++)
degree_vertex[(*it)->getTail()]++;
for(int i=1;i<=N;i++)
degree_vertex2.insert(pair<int,int>(degree_vertex[i],i));
it2=--degree_vertex2.end();
printf("project 1:\n");
for(;it2!=degree_vertex2.begin();it2--)
printf("%d,%d\n",it2->second,it2->first);
}
void p2(){
list<CEdge*>::iterator it,iend;
list<int>::iterator it2,iend2;
iend=IncidentList.end();
//printf("incidentList:\n");
for(it=IncidentList.begin();it!=iend;it++){
adjlist[(*it)->getTail()].push_back((*it)->getHead());
//printf("%d,%d\n",(*it)->getTail(),(*it)->getHead());
}
it2=adjlist[3].begin();
iend2=adjlist[3].end();
printf("\n");
printf("project 2:\n");
printf("3:");
for(;it2!=iend2;it2++)
printf("%d ",*it2);
}
void p3(){
list<CEdge*>::iterator it,iend;
iend=IncidentList.end();
CEdge* emptyedge=new CEdge(-1,-1,-1,-1);
for(int i=0;i<=numVertex;i++)
{
vector<CEdge*> vec;
for(int j=0;j<=numVertex;j++)
{
vec.push_back(emptyedge);
}
adjmatrix.push_back(vec);
}
for(it=IncidentList.begin();it!=iend;it++){
//CEdge* e = new CEdge((*it)->getTail(),(*it)->getHead(),(*it)->getWeight(),(*it)->getCap());
adjmatrix[(*it)->getTail()][(*it)->getHead()] = *it ;
}
printf("\n");
printf("project 3:\n");
printf("%d,%d",adjmatrix[2][3]->getTail(),adjmatrix[2][3]->getHead());
}
void p4(){
list<CEdge*>::iterator it,iend;
iend=IncidentList.end();
for(it=IncidentList.begin();it!=iend;it++){
nelist[(*it)->getTail()].push_back(*it);
}
printf("\n");
printf("project 4:\n");
list<CEdge*>::iterator it2,iend2;
iend2=nelist[3].end();
for(it2=nelist[3].begin();it2!=iend2;it2++)
printf("%d %d\n",(*it2)->getTail(),(*it2)->getHead());
}
void Update(int i){
list<CEdge*>::iterator it,iend;
it=nelist[i].begin();
iend=nelist[i].end();
for(;it!=iend;it++)
if((*it)->getWeight()+mapVID_Vertex[i]->d < mapVID_Vertex[(*it)->getHead()]->d){
int d;
d=(*it)->getWeight()+mapVID_Vertex[i]->d;
//增加新的点，删除旧的点，改动点信息
int k;
k = mapVID_Vertex[(*it)->getHead()]->d;
//pair<int,CVertex*> p((*it)->getHead(),mapBuckets[int(ceil((log((float)k))/(log((float)2))))][(*it)->getHead()]);
pair<int,CVertex*> p((*it)->getHead(),mapVID_Vertex[(*it)->getHead()]);
mapBuckets[int(ceil((log((float)d))/(log((float)2))))].insert(p);
mapBuckets[int(ceil((log((float)k))/(log((float)2))))].erase((*it)->getHead());
mapVID_Vertex[(*it)->getHead()]->d = (*it)->getWeight()+mapVID_Vertex[i]->d;
mapVID_Vertex[(*it)->getHead()]->p = i;
}
//printf("update\n");
}
int Redistribute(int dmin, int mini)
{
rbegin[0]=dmin;
rend[0]=dmin;
rbegin[1]=dmin+1;
rend[1]=dmin+1;
range[dmin]=0;
range[dmin+1]=1;
for(int i=2;i<=10;i++)
{
rbegin[i]=rend[i-1]+1;
rend[i]=rend[i-1]+db[i];
for(int j=rbegin[i];j<=rend[i];j++)
range[j]=i;
}
cout<<"middle"<<endl;
//重置该桶内的所有节点
itcv=itBucket->second.begin();
for(;itcv!=itBucket->second.end();)
{
//增加新的点，删除旧的点
cout<<"middle2"<<endl;
pair<int,CVertex*> p(itcv->first,itcv->second);
//mapBuckets[range[mapVID_Vertex[itcv->first]->d]].insert(p);
mapBuckets[range[itcv->second->d]].insert(p);
cout<<"middle3"<<endl;
mapBuckets[itBucket->first].erase(itcv++);
cout<<"middle4"<<endl;
}
itBucket=mapBuckets.begin();
cout<<"Before_FindMin"<<endl;
int reward;
reward=FindMin();
return reward;
}
int FindMin(){
set<int>::iterator vi,vend;
//移动指针到第一个不为空的桶
while(1)
{
if(itBucket->second.empty()==true)
itBucket++;
else
{itcv=itBucket->second.begin();break;}
}
if(itBucket->first==0 || itBucket->first==1)
{
int mini;
printf("1 if\n");
mini=itcv->first;
mapBuckets[itBucket->first].erase(itcv++);
printf("min is %d\n",mini);
return mini;
}
else if(itBucket->second.size()==1)
{
int mini;
printf("2 if\n");
mini=itcv->first;
mapBuckets[itBucket->first].erase(itcv++);
//printf("min is %d\n",mini);
return mini;
}
else
{
int dmin=10000;
int mini;
printf("3 if\n");
itcv=itBucket->second.begin();
cout<<mapVID_Vertex[itcv->first]->d<<endl;
for(;itcv!=itBucket->second.end();itcv++)
{
if(mapVID_Vertex[itcv->first]->d < dmin)
{dmin=mapVID_Vertex[itcv->first]->d;mini=itcv->first;}
}
cout<<dmin<<mini<<endl;
itcv=itBucket->second.begin();
return Redistribute(dmin,mini);
}
}
void radix_heap(CGraph &g,int s){
int i,j;
init();
for(i=1;i<=10;i++)
V.insert(i);
for(i=1;i<=N;i++)//所有节点的初始化
{
CVertex* vertex=new CVertex(i);
pair<int, CVertex*> p(i,vertex);
mapVID_Vertex.insert(p);
}
for(i=2;i<=N;i++)//除开始点，其他全部放入无穷桶里
{
pair<int,CVertex*> p2(i,mapVID_Vertex[i]);
mapBuckets[ceil(log(float(INF))/log(2.0))].insert(p2);//10=log1024,对应无穷桶，方便在删除时操作
}
S.insert(s);
V.erase(s);
mapVID_Vertex[1]->d=0;
Update(s);
itBucket=mapBuckets.begin();
while (V.size()!=0)
{
j=FindMin();
S.insert(j);
V.erase(j);
cout<<"V :"<<j<<" "<<V.size()<<endl;
Update(j);
}
printf("\n 1->9:%d\n",mapVID_Vertex[9]->d);
}
};
第三部分：
//一些需要用到的头文件 #ifndef _COMMON_H_ #define _COMMON_H_ #include <map> #include <vector> #include <list> #include <set> #include <math.h> using namespace std; #include <iostream> #include <stdio.h> #include <algorithm> #define INF 1024 #define N 10 #define C 1 #endif
