[算法整理]树上求LCA算法合集

Posted 2020-08-28 阿波罗2003

1#树上倍增

　　以前写的博客:http://www.cnblogs.com/yyf0309/p/5972701.html

　　预处理时间复杂度O(nlog₂n)，查询O(log₂n),也不算难写。

2#st表(RMQ)

　　首先对一棵树进行dfs,得到欧拉序列，记录下每个节点的第一次出现位置。

　　（先序遍历这棵树，访问到的节点(无论是从深的一层返回还是父节点访问)就加入到序列中，序列长度为2 * n - 1）

　　根据欧拉序列神奇的特性，两个点第一次出现的位置之间，深度最小的一个点，是这两个点LCA（反正我是不会证明）。于是可以干什么呢？就建st表就行了。

　　预处理时间复杂度O(2nlog₂(2n) + n)，查询时间复杂度O(log₂n)。

codevs 商务旅行:

  1 /**
  2  * codevs
  3  * Problem#1036
  4  * Accepted
  5  * Time:52ms
  6  * Memory:2736k
  7  */
  8 #include<iostream>
  9 #include<sstream>
 10 #include<cstdio>
 11 #include<cmath>
 12 #include<cstdlib>
 13 #include<cstring>
 14 #include<cctype>
 15 #include<queue>
 16 #include<set>
 17 #include<map>
 18 #include<stack>
 19 #include<vector>
 20 #include<algorithm>
 21 #ifndef    WIN32
 22 #define    AUTO "%I64d"
 23 #else
 24 #define AUTO "%lld"
 25 #endif
 26 using namespace std;
 27 typedef bool boolean;
 28 #define smin(a, b) (a) = min((a), (b))
 29 #define smax(a, b) (a) = max((a), (b))
 30 template<typename T>
 31 inline void readInteger(T& u){
 32     char x;
 33     int aFlag = 1;
 34     while(!isdigit((x = getchar())) && x != \'-\');
 35     if(x == \'-\'){
 36         aFlag = -1;
 37         x = getchar();
 38     }
 39     for(u = x - \'0\'; isdigit((x = getchar())); u = u * 10 + x - \'0\');
 40     ungetc(x, stdin);
 41     u *= aFlag;
 42 }
 43 
 44 template<typename T>class Matrix{
 45     public:
 46         T *p;
 47         int lines;
 48         int rows;
 49         Matrix():p(NULL){    }
 50         Matrix(int rows, int lines):lines(lines), rows(rows){
 51             p = new T[(lines * rows)];
 52         }
 53         T* operator [](int pos){
 54             return (p + pos * lines);
 55         }
 56 };
 57 #define matset(m, i, s) memset((m).p, (i), (s) * (m).lines * (m).rows)
 58 
 59 ///map template starts
 60 typedef class Edge{
 61     public:
 62         int end;
 63         int next;
 64         Edge(const int end = 0, const int next = 0):end(end), next(next){}
 65 }Edge;
 66 typedef class MapManager{
 67     public:
 68         int ce;
 69         int *h;
 70         Edge *edge;
 71         MapManager(){}
 72         MapManager(int points, int limit):ce(0){
 73             h = new int[(const int)(points + 1)];
 74             edge = new Edge[(const int)(limit + 1)];
 75             memset(h, 0, sizeof(int) * (points + 1));
 76         }
 77         inline void addEdge(int from, int end){
 78             edge[++ce] = Edge(end, h[from]);
 79             h[from] = ce;
 80         }
 81         inline void addDoubleEdge(int from, int end){
 82             addEdge(from, end);
 83             addEdge(end, from);
 84         }
 85         Edge& operator[] (int pos) {
 86             return edge[pos];
 87         }
 88 }MapManager;
 89 #define m_begin(g, i) (g).h[(i)]
 90 ///map template ends
 91 
 92 int n, m;
 93 int cnt = 0;
 94 Matrix<int> st;
 95 int* seq;
 96 int *dep, *app;
 97 MapManager g;
 98 const int P = 15;
 99 int *mlog2;
100 
101 inline void init() {
102     readInteger(n);
103     g = MapManager(n, 2  * n);
104     seq = new int[(const int)(2 * n + 1)];
105     dep = new int[(const int)(n + 1)];
106     app = new int[(const int)(n + 1)];
107     for(int i = 1, a, b; i < n; i++){
108         readInteger(a);
109         readInteger(b);
110         g.addDoubleEdge(a, b);
111     }
112     dep[0] = 0;
113 }
114 
115 void dfs(int node, int f) {
116     seq[++cnt] = node;
117     dep[node] = dep[f] + 1;
118     app[node] = cnt;
119     for(int i = m_begin(g, node); i != 0; i = g[i].next) {
120         int& e = g[i].end;
121         if(e == f)    continue;
122         dfs(e, node);
123         seq[++cnt] = node;
124     }
125 }
126 
127 inline void init_log() {
128     mlog2 = new int[(const int)(2 * n + 1)];
129     mlog2[1] = 0;
130     for(int i = 2; i <= 2 * n; i++)
131         mlog2[i] = mlog2[i / 2] + 1; 
132 }
133 
134 inline void init_st() {
135     init_log();
136     st = Matrix<int>(cnt, mlog2[cnt] + 1);
137     for(int i = 1; i <= cnt; i++)
138         st[i][0] = seq[i];//,cout << i << " " << 0 << ":" << st[i][0] << endl;
139     for(int j = 1; j <= P; j++)
140         for(int i = 1; i + (1 << j) - 1 <= cnt; i++)
141             st[i][j] = (dep[st[i][j - 1]] < dep[st[i + (1 << (j - 1))][j - 1]]) ? (st[i][j - 1]) : (st[i + (1 << (j - 1))][j - 1]);
142 //            cout << i << " " << j << ":" << st[i][j] << endl;
143 }
144 
145 inline int lca(int a, int b) {
146     if(app[a] > app[b])    swap(a, b);
147     int pos = mlog2[app[b] - app[a] + 1];
148     int u = st[app[a]][pos];
149     int v = st[app[b] - (1 << pos) + 1][pos];
150     return (dep[u] > dep[v]) ? (v) : (u);
151 }
152 
153 int last;
154 int dist;
155 inline void solve() {
156     readInteger(m);
157     readInteger(last);
158     for(int i = 1, a; i < m; i++){
159         readInteger(a);
160         int l = lca(a, last);
161         dist += dep[a] + dep[last] - 2 * dep[l];
162         last = a;
163     }
164     printf("%d", dist);
165 }
166 
167 int main() {
168     init();
169     dfs(1, 0);
170     init_st();
171     solve();
172     return 0;
173 }

商务旅行(st表)

3#Tarjan算法（离线算法）

　　Tarjan需要一个并查集来辅助。

　　算法思路大概是这样:

　　init:初始化并查集，对查询建一个邻接链表，还是按照存边的方式，要双向的,再用一个数组记录第i个询问是否解决了。

　　1.访问子树

　　2.每次访问结束后将子树的并查集中的父节点指向当前节点。

　　3.子树都访问完了后，开始处理以该节点为起点的询问，如果终点已经被访问过了并且没有被解决，那么这个点和终点的LCA是终点在并查集中的父节点，然后再记录是否解决的那个数组中把对应位置设上true。

　　（画画图，还是很容易理解的）

  1 #include<iostream>
  2 #include<sstream>
  3 #include<cstdio>
  4 #include<cmath>
  5 #include<cstdlib>
  6 #include<cstring>
  7 #include<cctype>
  8 #include<queue>
  9 #include<set>
 10 #include<map>
 11 #include<stack>
 12 #include<vector>
 13 #include<algorithm>
 14 #ifndef    WIN32
 15 #define    AUTO "%I64d"
 16 #else
 17 #define AUTO "%lld"
 18 #endif
 19 using namespace std;
 20 typedef bool boolean;
 21 #define smin(a, b) (a) = min((a), (b))
 22 #define smax(a, b) (a) = max((a), (b))
 23 template<typename T>
 24 inline void readInteger(T& u){
 25     char x;
 26     int aFlag = 1;
 27     while(!isdigit((x = getchar())) && x != \'-\');
 28     if(x == \'-\'){
 29         aFlag = -1;
 30         x = getchar();
 31     }
 32     for(u = x - \'0\'; isdigit((x = getchar())); u = u * 10 + x - \'0\');
 33     ungetc(x, stdin);
 34     u *= aFlag;
 35 }
 36 
 37 typedef class Edge {
 38     public:
 39         int end;
 40         int next;
 41         int w;
 42         Edge(const int end = 0, const int next = 0, const int w = 0):end(end), next(next), w(w){        }
 43 }Edge;
 44 
 45 typedef class MapManager{
 46     public:
 47         int ce;
 48         Edge* edges;
 49         int* h;
 50         MapManager():ce(0), edges(NULL), h(NULL){        }
 51         MapManager(int points, int limit):ce(0){
 52             edges = new Edge[(const int)(limit + 1)];
 53             h = new int[(const int)(points + 1)];
 54             memset(h, 0, sizeof(int) * (points + 1));
 55         }
 56         inline void addEdge(int from, int end, int w){
 57             edges[++ce] = Edge(end, h[from], w);
 58             h[from] = ce;
 59         }
 60         inline void addDoubleEdge(int from, int end, int w){
 61             addEdge(from, end, w);
 62             addEdge(end, from, w);
 63         }
 64         Edge& operator [](int pos){
 65             return edges[pos];
 66         }
 67 }MapManager;
 68 #define m_begin(g, i) (g).h[(i)]
 69 
 70 typedef class union_found{
 71     public:
 72         int *f;
 73         union_found():f(NULL) {}
 74         union_found(int points) {
 75             f = new int[(const int)(points + 1)];
 76         }
 77         int find(int x) {
 78             if(f[x] != x)    return f[x] = find(f[x]);
 79             return f[x];
 80         }
 81         void unit(int fa, int so) {
 82             int ffa = find(fa);
 83             int fso = find(so);
 84             f[fso] = ffa;
 85         }
 86         int& operator [](int pos){
 87             return f[pos];
 88         }
 89 }union_found;
 90 
 91 int n, m;
 92 MapManager g;
 93 MapManager q;
 94 int *results;
 95 boolean* enable;
 96 int *querya, *queryb;
 97 union_found uf;
 98 boolean* visited;
 99 int* dist;
100 
101 inline void init(){
102     readInteger(n);
103     g = MapManager(n, 2 * n);
104     for(int i = 1, a, b, c; i < n; i++){
105         readInteger(a);
106         readInteger(b);
107         readInteger(c);
108         g.addDoubleEdge(a, b, c);
109     }
110     readInteger(m);
111     q = MapManager(n, 2 * m);
112     querya = new int[(const int)(m + 1)];
113     queryb = new int[(const int)(m + 1)];
114     results = new int[(const int)(m + 1)];
115     enable = new boolean[(const int)(m + 1)];
116     dist = new int[(const int)(n + 1)];
117     uf = union_found(n);
118     visited = new boolean[(const int)(n + 1)];
119     memset(visited, false, sizeof(boolean) * (n + 1));
120     memset(enable, true, sizeof(boolean) * (m + 1));
121     for(int i = 1; i <= m; i++){
122         readInteger(querya[i]);
123         readInteger(queryb[i]);
124         q.addDoubleEdge(querya[i], queryb[i], i);
125     }
126     dist[1] = 0;
127 }
128 
129 void tarjan(int node, int f){
130     uf[node] = node;
131     visited[node] = true;
132     for(int i = m_begin(g, node); i != 0; i = g[i].next){
133         int& e = g[i].end;
134         if(e == f)    continue;
135         dist[e] = dist[node] + g[i].w;
136         tarjan(e, node);
137         uf[e] = node;
138     }
139     for(int i = m_begin(q, node); i != 0; i = q[i].next) {
140         int& e = q[i].end;
141         if(visited[e] && enable[q[i].w]){
142             int lca = uf.find(e);
143             results[q[i].w] = lca;
144             enable[q[i].w] = false;
145         }
146     }
147 }
148 
149 inline void solve(){
150     tarjan(1, 0);
151     for(int i = 1; i <= m; i++){
152         int dis = dist[querya[i]] + dist[queryb[i]] - 2 * dist[results[i]];
153         printf("%d\\n", dis);
154     }
155 }
156 
157 int main(){
158     init();
159     solve();
160     return 0;
161 }

Tajan

4#树链剖分

　　详见:http://www.cnblogs.com/yyf0309/p/6344982.html

　　树链剖分的空间复杂度比上面任何一种方法都要优，所以当空间很紧的时候(当然，我相信出题人不会那么坑人)是不错的选择。