Brief description:
。。给定一个 N 个顶点的无向图（稠密）、回答 Q 次询问。
问一条边的权值增加为 ci’ 后，新图的 MST。
(. .. N <= 3000, Q <= 10^6.. .) ..
Analysis:
下面介绍 O(mlogm) 算法、设原图的最小生成树为 T，若修改一条非树边、那么对结果没有影响（修改操作单调递增）.. .
如果是树边、那么这条边可能会因为这次修改操作而被一条非树边替换掉、不难发现用以替换的边是唯一的。
于是得到算法，对每一条非树边、松弛生成树上这两点之间路径的所有树边。
则对于每一询问 （a, b, w’) .. .
if (!树边) return mst; else return mst - w[a][b] + min(w', s[a][b]);
其中 s[a][b] 是嗯嗯嗯嗯嗯的意思。
（如果使用 Kruskal() 算法求解最小生成树。。那么松弛的过程可以用并查集优化到 O(m) 。。。
次小生成树的算法有很多，对于这题、今年北京现场赛区 Problem A 的算法可能并不适用？。。）
/** ` Micro Mezzo Macro Flation -- Overheated Economy ., **/
#include <algorithm>
#include <iostream>
#include <iomanip>
#include <sstream>
#include <cstring>
#include <complex>
#include <cstdio>
#include <string>
#include <vector>
#include <bitset>
#include <queue>
#include <stack>
#include <cmath>
#include <ctime>
#include <list>
#include <set>
#include <map>
using namespace std;
#define REP(i, n) for (int i=0;i<int(n);++i)
#define FOR(i, a, b) for (int i=int(a);i<int(b);++i)
#define DWN(i, b, a) for (int i=int(b-1);i>=int(a);--i)
#define REP_1(i, n) for (int i=1;i<=int(n);++i)
#define FOR_1(i, a, b) for (int i=int(a);i<=int(b);++i)
#define DWN_1(i, b, a) for (int i=int(b);i>=int(a);--i)
#define REP_C(i, n) for (int n____=int(n),i=0;i<n____;++i)
#define FOR_C(i, a, b) for (int b____=int(b),i=a;i<b____;++i)
#define DWN_C(i, b, a) for (int a____=int(a),i=b-1;i>=a____;--i)
#define REP_N(i, n) for (i=0;i<int(n);++i)
#define FOR_N(i, a, b) for (i=int(a);i<int(b);++i)
#define DWN_N(i, b, a) for (i=int(b-1);i>=int(a);--i)
#define REP_1_C(i, n) for (int n____=int(n),i=1;i<=n____;++i)
#define FOR_1_C(i, a, b) for (int b____=int(b),i=a;i<=b____;++i)
#define DWN_1_C(i, b, a) for (int a____=int(a),i=b;i>=a____;--i)
#define REP_1_N(i, n) for (i=1;i<=int(n);++i)
#define FOR_1_N(i, a, b) for (i=int(a);i<=int(b);++i)
#define DWN_1_N(i, b, a) for (i=int(b);i>=int(a);--i)
#define REP_C_N(i, n) for (n____=int(n),i=0;i<n____;++i)
#define FOR_C_N(i, a, b) for (b____=int(b),i=a;i<b____;++i)
#define DWN_C_N(i, b, a) for (a____=int(a),i=b-1;i>=a____;--i)
#define REP_1_C_N(i, n) for (n____=int(n),i=1;i<=n____;++i)
#define FOR_1_C_N(i, a, b) for (b____=int(b),i=a;i<=b____;++i)
#define DWN_1_C_N(i, b, a) for (a____=int(a),i=b;i>=a____;--i)
#define DO(n) while(n--)
#define DO_C(n) int n____ = n; while(n____--)
#define TO(i, a, b) int s_=a<b?1:-1,b_=b+s_;for(int i=a;i!=b_;i+=s_)
#define TO_1(i, a, b) int s_=a<b?1:-1,b_=b;for(int i=a;i!=b_;i+=s_)
#define SQZ(i, j, a, b) for (int i=int(a),j=int(b)-1;i<j;++i,--j)
#define SQZ_1(i, j, a, b) for (int i=int(a),j=int(b);i<=j;++i,--j)
#define REP_2(i, j, n, m) REP(i, n) REP(j, m)
#define REP_2_1(i, j, n, m) REP_1(i, n) REP_1(j, m)
#define ALL(A) A.begin(), A.end()
#define LLA(A) A.rbegin(), A.rend()
#define CPY(A, B) memcpy(A, B, sizeof(A))
#define INS(A, P, B) A.insert(A.begin() + P, B)
#define ERS(A, P) A.erase(A.begin() + P)
#define BSC(A, X) find(ALL(A), X) // != A.end()
#define CTN(T, x) (T.find(x) != T.end())
#define SZ(A) int(A.size())
#define PB push_back
#define MP(A, B) make_pair(A, B)
#define Rush int T____; RD(T____); DO(T____)
#pragma comment(linker, "/STACK:36777216")
#pragma GCC optimize ("O2")
#define Ruby system("ruby main.rb")
#define Haskell system("runghc main.hs")
#define Pascal system("fpc main.pas")
typedef long long LL;
typedef double DB;
typedef unsigned UINT;
typedef unsigned long long ULL;
typedef vector<int> VI;
typedef vector<char> VC;
typedef vector<string> VS;
typedef vector<LL> VL;
typedef vector<DB> VD;
typedef set<int> SI;
typedef set<string> SS;
typedef set<LL> SL;
typedef set<DB> SD;
typedef map<int, int> MII;
typedef map<string, int> MSI;
typedef map<LL, int> MLI;
typedef map<DB, int> MDI;
typedef map<int, bool> MIB;
typedef map<string, bool> MSB;
typedef map<LL, bool> MLB;
typedef map<DB, bool> MDB;
typedef pair<int, int> PII;
typedef pair<int, bool> PIB;
typedef vector<PII> VII;
typedef vector<VI> VVI;
typedef vector<VII> VVII;
typedef set<PII> SII;
typedef map<PII, int> MPIII;
typedef map<PII, bool> MPIIB;
/** I/O Accelerator **/
/* ... :" We are I/O Accelerator ... Use us at your own risk ;) ... " .. */
template<class T> inline void RD(T &);
template<class T> inline void OT(const T &);
inline int RD(){ int x; RD(x); return x;}
template<class T> inline T& _RD(T &x){ RD(x); return x;}
inline void RC(char &c){scanf(" %c", &c);}
inline void RS(char *s){scanf("%s", s);}
template<class T0, class T1> inline void RD(T0 &x0, T1 &x1){RD(x0), RD(x1);}
template<class T0, class T1, class T2> inline void RD(T0 &x0, T1 &x1, T2 &x2){RD(x0), RD(x1), RD(x2);}
template<class T0, class T1, class T2, class T3> inline void RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3){RD(x0), RD(x1), RD(x2), RD(x3);}
template<class T0, class T1, class T2, class T3, class T4> inline void RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4){RD(x0), RD(x1), RD(x2), RD(x3), RD(x4);}
template<class T0, class T1, class T2, class T3, class T4, class T5> inline void RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5){RD(x0), RD(x1), RD(x2), RD(x3), RD(x4), RD(x5);}
template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5, T6 &x6){RD(x0), RD(x1), RD(x2), RD(x3), RD(x4), RD(x5), RD(x6);}
template<class T0, class T1> inline void OT(T0 &x0, T1 &x1){OT(x0), OT(x1);}
template<class T0, class T1, class T2> inline void OT(T0 &x0, T1 &x1, T2 &x2){OT(x0), OT(x1), OT(x2);}
template<class T0, class T1, class T2, class T3> inline void OT(T0 &x0, T1 &x1, T2 &x2, T3 &x3){OT(x0), OT(x1), OT(x2), OT(x3);}
template<class T0, class T1, class T2, class T3, class T4> inline void OT(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4){OT(x0), OT(x1), OT(x2), OT(x3), OT(x4);}
template<class T0, class T1, class T2, class T3, class T4, class T5> inline void OT(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5){OT(x0), OT(x1), OT(x2), OT(x3), OT(x4), OT(x5);}
template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void OT(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5, T6 &x6){OT(x0), OT(x1), OT(x2), OT(x3), OT(x4), OT(x5), OT(x6);}
template<class T> inline void RST(T &A){memset(A, 0, sizeof(A));}
template<class T0, class T1> inline void RST(T0 &A0, T1 &A1){RST(A0), RST(A1);}
template<class T0, class T1, class T2> inline void RST(T0 &A0, T1 &A1, T2 &A2){RST(A0), RST(A1), RST(A2);}
template<class T0, class T1, class T2, class T3> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3){RST(A0), RST(A1), RST(A2), RST(A3);}
template<class T0, class T1, class T2, class T3, class T4> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4){RST(A0), RST(A1), RST(A2), RST(A3), RST(A4);}
template<class T0, class T1, class T2, class T3, class T4, class T5> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5){RST(A0), RST(A1), RST(A2), RST(A3), RST(A4), RST(A5);}
template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5, T6 &A6){RST(A0), RST(A1), RST(A2), RST(A3), RST(A4), RST(A5), RST(A6);}
template<class T> inline void CLR(T &A){A.clear();}
template<class T0, class T1> inline void CLR(T0 &A0, T1 &A1){CLR(A0), CLR(A1);}
template<class T0, class T1, class T2> inline void CLR(T0 &A0, T1 &A1, T2 &A2){CLR(A0), CLR(A1), CLR(A2);}
template<class T0, class T1, class T2, class T3> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3){CLR(A0), CLR(A1), CLR(A2), CLR(A3);}
template<class T0, class T1, class T2, class T3, class T4> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4){CLR(A0), CLR(A1), CLR(A2), CLR(A3), CLR(A4);}
template<class T0, class T1, class T2, class T3, class T4, class T5> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5){CLR(A0), CLR(A1), CLR(A2), CLR(A3), CLR(A4), CLR(A5);}
template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5, T6 &A6){CLR(A0), CLR(A1), CLR(A2), CLR(A3), CLR(A4), CLR(A5), CLR(A6);}
template<class T> inline void CLR(T &A, int n){REP(i, n) CLR(A[i]);}
template<class T> inline void FLC(T &A, int x){memset(A, x, sizeof(A));}
template<class T0, class T1> inline void FLC(T0 &A0, T1 &A1, int x){FLC(A0, x), FLC(A1, x);}
template<class T0, class T1, class T2> inline void FLC(T0 &A0, T1 &A1, T2 &A2){FLC(A0), FLC(A1), FLC(A2);}
template<class T0, class T1, class T2, class T3> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3){FLC(A0), FLC(A1), FLC(A2), FLC(A3);}
template<class T0, class T1, class T2, class T3, class T4> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4){FLC(A0), FLC(A1), FLC(A2), FLC(A3), FLC(A4);}
template<class T0, class T1, class T2, class T3, class T4, class T5> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5){FLC(A0), FLC(A1), FLC(A2), FLC(A3), FLC(A4), FLC(A5);}
template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5, T6 &A6){FLC(A0), FLC(A1), FLC(A2), FLC(A3), FLC(A4), FLC(A5), FLC(A6);}
template<class T> inline void SRT(T &A){sort(ALL(A));}
template<class T, class C> inline void SRT(T &A, C B){sort(ALL(A), B);}
/** Add - On **/
const int MOD = 1000000007;
const int INF = 0x7f7f7f7f;
const DB EPS = 1e-6;
const DB OO = 1e15;
const DB PI = acos(-1.0);
// <<= ` 0. Daily Use .,
template<class T> inline void checkMin(T &a,const T b){if (b<a) a=b;}
template<class T> inline void checkMax(T &a,const T b){if (b>a) a=b;}
template <class T, class C> inline void checkMin(T& a, const T b, C c){if (c(b,a)) a = b;}
template <class T, class C> inline void checkMax(T& a, const T b, C c){if (c(a,b)) a = b;}
template<class T> inline T min(T a, T b, T c){return min(min(a, b), c);}
template<class T> inline T max(T a, T b, T c){return max(max(a, b), c);}
template<class T> inline T sqr(T a){return a*a;}
template<class T> inline T cub(T a){return a*a*a;}
int Ceil(int x, int y){return (x - 1) / y + 1;}
// <<= ` 1. Bitwise Operation .,
inline bool _1(int x, int i){return x & 1<<i;}
inline int _1(int i){return 1<<i;}
inline int _U(int i){return _1(i) - 1;};
inline int count_bits(int x){
x = (x & 0x55555555) + ((x & 0xaaaaaaaa) >> 1);
x = (x & 0x33333333) + ((x & 0xcccccccc) >> 2);
x = (x & 0x0f0f0f0f) + ((x & 0xf0f0f0f0) >> 4);
x = (x & 0x00ff00ff) + ((x & 0xff00ff00) >> 8);
x = (x & 0x0000ffff) + ((x & 0xffff0000) >> 16);
return x;
}
template<class T> inline T low_bit(T x) {
return x & -x;
}
template<class T> inline T high_bit(T x) {
T p = low_bit(x);
while (p != x) x -= p, p = low_bit(x);
return p;
}
// <<= ` 2. Modular Arithmetic Basic .,
inline void INC(int &a, int b){a += b; if (a >= MOD) a -= MOD;}
inline int sum(int a, int b){a += b; if (a >= MOD) a -= MOD; return a;}
inline void DEC(int &a, int b){a -= b; if (a < 0) a += MOD;}
inline int dff(int a, int b){a -= b; if (a < 0) a += MOD; return a;}
inline void MUL(int &a, int b){a = int((LL)a * b % MOD);}
inline int pdt(int a, int b){return int((LL)a * b % MOD);}
// <<= ' 0. I/O Accelerator interface .,
template<class T> inline void RD(T &x){
//cin >> x;
//scanf("%d", &x);
char c; for (c = getchar(); c < '0'; c = getchar()); x = c - '0'; for (c = getchar(); c >= '0'; c = getchar()) x = x * 10 + c - '0';
//char c; c = getchar(); x = c - '0'; for (c = getchar(); c >= '0'; c = getchar()) x = x * 10 + c - '0';
}
int ____Case;
template<class T> inline void OT(const T &x){
//cout << x << endl;
printf("%.4lf\n", x);
//printf("%.2lf\n", x);
//printf("Case %d: %d\n", ++____Case, x);
}
#define For_each(it, A) for (SII::iterator it = A.begin(); it != A.end(); ++it)
/* .................................................................................................................................. */
const int N = 3009, M = N * N;
int w[N][N], s[N][N]; bool InMst[N][N]; int P[N], R[N];
PII E[M]; VI adj[N]; bool c[N]; int d[N]; int prd[N], vis[N], m, _i;
int n, q, mst;
void Make(int x){
P[x] = x, R[x] = 0;
}
int Find(int x){
if (P[x] != x) P[x] = Find(P[x]);
return P[x];
}
void Union(int x, int y){
if (R[x] < R[y]) P[y] = x;
else {
if (R[x] == R[y]) R[y]++;
P[x] = y;
}
}
void Prim1(){
m = 0, RST(c), FLC(d, 0x7f), CPY(s, w), d[0] = 0;
REP(i, n){
int u = 0; while (c[u]) ++u; FOR(i, u+1, n) if (!c[i] && d[i] < d[u]) u = i;
if (d[u]) s[prd[u]][u] = INF - 1, E[m++] = MP(prd[u], u); c[u] = true;
REP(v, n) if (!c[v] && w[u][v] < d[v]) d[v] = w[u][v], prd[v] = u;
}
}
void Prim2(){
RST(c), FLC(d, 0x7f), d[0] = 0;
REP(i, n){
int u = 0; while(c[u]) ++u; FOR(i, u+1, n) if (!c[i] && d[i] < d[u]) u = i;
if (d[u]) s[prd[u]][u] = INF - 1, E[m++] = MP(prd[u], u); c[u] = true;
REP(v, n) if (!c[v] && s[u][v] < d[v]) d[v] = s[u][v], prd[v] = u;
}
}
#define a first
#define b second
bool comp(PII x, PII y){
return w[x.a][x.b] < w[y.a][y.b];
}
void Kruskal(){
sort(E, E+m, comp); REP(i, n) Make(i); RST(InMst); mst = 0;
REP(i, n) CLR(adj[i]);
REP(i, m){
if (Find(E[i].a) == Find(E[i].b)) continue; Union(Find(E[i].a), Find(E[i].b));
InMst[E[i].a][E[i].b] = true, mst += w[E[i].a][E[i].b], adj[E[i].a].PB(E[i].b), adj[E[i].b].PB(E[i].a);
}
}
#define v adj[u][i]
void dfs(int u = 0, int p = -1){
prd[u] = p; REP(i, SZ(adj[u])) if (v != p){
dfs(v, u);
}
}
#undef v
int lca(int a, int b){
while (a != 0){
vis[a] = _i;
a = prd[a];
}
vis[0] = _i;
while (vis[b] != _i){
b = prd[b];
}
return b;
}
void link(int a, int b, int c){
int p = lca(a, b), t;
#define Cloze(a, b) if (!s[a][b]) s[a][b] = s[b][a] = c
while (a != p){
Cloze(a, prd[a]);
t = prd[a], prd[a] = p, a = t;
}
while (b != p){
Cloze(b, prd[b]);
t = prd[b], prd[b] = p, b = t;
}
}
void Relax(){
RST(vis, prd, s); dfs(); REP(i, m){
if (InMst[E[i].a][E[i].b]) continue; _i = i + 1;
link(E[i].a, E[i].b, w[E[i].a][E[i].b]);
}
REP(i, n) FOR(j, i+1, n) if (!s[i][j]) s[i][j] = INF;
}
#undef a
#undef b
int main(){
//freopen("in.txt", "r", stdin);
while (scanf("%d %d", &n, &m) != EOF && n){
int a, b; FLC(w, 0x7f); REP(i, m){
RD(a, b); if (a > b) swap(a, b); RD(w[a][b]);
}
if (false && m > 2 * n) Prim1(), Prim2();
else {m = 0; REP(i, n) FOR(j, i+1, n) if (w[a][b] != INF) E[m++] = MP(i, j);}
Kruskal(); Relax();
int sum = 0; int ww; DO_C(_RD(q)){
RD(a, b, ww); if (a > b) swap(a, b);
if (!InMst[a][b]) sum += mst;
else sum += mst - w[a][b] + min(ww, s[a][b]);
}
OT(DB(sum) / q);
}
}
Further discussion:
求 O(n2) 算法！！！
External link:
http://acm.hdu.edu.cn/showproblem.php?pid=4126
http://www.cppblog.com/MatoNo1/archive/2011/11/16/149812.html
https://www.shuizilong.com/house/wp-admin/post.php?post=2498&action=edit&message=1
