题目链接:https://atcoder.jp/contests/abc127/tasks/abc127_f
题目大意
初始状态下$f(x) = 0$,现在有 2 种模式的询问,第一种以“1 a b”的形式,需要进行操作$f(x) = f(x) + |x - a| + b$;第二种以“2”的形式,求使得 f(x) 取得最小值的 x 取值和 f(x) 值,如果有多个 x,输出任意一个即可。
分析
考虑第一种询问已经出现了 k 次,现在遇到第二种询问。此时$f(x) = \sum_{i = 1}^k (|x - a_i| + b_i)$,可以发现$b_i$完全是独立于 x 的,所以只要讨论$\sum_{i = 1}^k |x - a_i|$的最小值即可。
于是问题就退化成在数轴上找一个点,使得它到给定的 k 个点的距离之和最短。
当点个数为奇数,最优解是中位数。
当点个数为偶数,最优解是位于中间的两个数中任取一个。
所以这个问题的本质就是让你动态找中位数,维护中位数可以用双堆。
代码如下
1 #include <bits/stdc++.h> 2 using namespace std; 3 4 #define INIT() ios::sync_with_stdio(false);cin.tie(0);cout.tie(0); 5 #define Rep(i,n) for (int i = 0; i < (n); ++i) 6 #define For(i,s,t) for (int i = (s); i <= (t); ++i) 7 #define rFor(i,t,s) for (int i = (t); i >= (s); --i) 8 #define ForLL(i, s, t) for (LL i = LL(s); i <= LL(t); ++i) 9 #define rForLL(i, t, s) for (LL i = LL(t); i >= LL(s); --i) 10 #define foreach(i,c) for (__typeof(c.begin()) i = c.begin(); i != c.end(); ++i) 11 #define rforeach(i,c) for (__typeof(c.rbegin()) i = c.rbegin(); i != c.rend(); ++i) 12 13 #define pr(x) cout << #x << " = " << x << " " 14 #define prln(x) cout << #x << " = " << x << endl 15 16 #define LOWBIT(x) ((x)&(-x)) 17 18 #define ALL(x) x.begin(),x.end() 19 #define INS(x) inserter(x,x.begin()) 20 21 #define ms0(a) memset(a,0,sizeof(a)) 22 #define msI(a) memset(a,inf,sizeof(a)) 23 #define msM(a) memset(a,-1,sizeof(a)) 24 25 #define MP make_pair 26 #define PB push_back 27 #define ft first 28 #define sd second 29 30 template<typename T1, typename T2> 31 istream &operator>>(istream &in, pair<T1, T2> &p) { 32 in >> p.first >> p.second; 33 return in; 34 } 35 36 template<typename T> 37 istream &operator>>(istream &in, vector<T> &v) { 38 for (auto &x: v) 39 in >> x; 40 return in; 41 } 42 43 template<typename T1, typename T2> 44 ostream &operator<<(ostream &out, const std::pair<T1, T2> &p) { 45 out << "[" << p.first << ", " << p.second << "]" << "\n"; 46 return out; 47 } 48 49 inline int gc(){ 50 static const int BUF = 1e7; 51 static char buf[BUF], *bg = buf + BUF, *ed = bg; 52 53 if(bg == ed) fread(bg = buf, 1, BUF, stdin); 54 return *bg++; 55 } 56 57 inline int ri(){ 58 int x = 0, f = 1, c = gc(); 59 for(; c<48||c>57; f = c==‘-‘?-1:f, c=gc()); 60 for(; c>47&&c<58; x = x*10 + c - 48, c=gc()); 61 return x*f; 62 } 63 64 typedef long long LL; 65 typedef unsigned long long uLL; 66 typedef pair< double, double > PDD; 67 typedef pair< int, int > PII; 68 typedef pair< string, int > PSI; 69 typedef set< int > SI; 70 typedef vector< int > VI; 71 typedef vector< PII > VPII; 72 typedef map< int, int > MII; 73 typedef pair< LL, LL > PLL; 74 typedef vector< LL > VL; 75 typedef vector< VL > VVL; 76 typedef priority_queue< int > PQIMax; 77 typedef priority_queue< int, VI, greater< int > > PQIMin; 78 const double EPS = 1e-10; 79 const LL inf = 0x7fffffff; 80 const LL infLL = 0x7fffffffffffffffLL; 81 const LL mod = 1e9 + 7; 82 const int maxN = 2e5 + 7; 83 const LL ONE = 1; 84 const LL evenBits = 0xaaaaaaaaaaaaaaaa; 85 const LL oddBits = 0x5555555555555555; 86 87 88 int Q; 89 PQIMax maxH; 90 PQIMin minH; 91 LL sumB, sumLA, sumRA, ans; 92 93 int main(){ 94 INIT(); 95 cin >> Q; 96 while(Q--) { 97 int x, a, b; 98 cin >> x; 99 if(x == 1) { 100 cin >> a >> b; 101 sumB += b; 102 if(maxH.size() == minH.size()) { 103 if(maxH.empty()) { 104 maxH.push(a); 105 sumLA += a; 106 } 107 else if(a >= maxH.top()) { 108 minH.push(a); 109 sumRA += a; 110 } 111 else { 112 maxH.push(a); 113 sumLA += a; 114 } 115 } 116 else if(maxH.size() > minH.size()) { 117 if(a >= maxH.top()) { 118 minH.push(a); 119 sumRA += a; 120 } 121 else { 122 maxH.push(a); 123 sumLA += a; 124 sumLA -= maxH.top(); 125 minH.push(maxH.top()); 126 sumRA += maxH.top(); 127 maxH.pop(); 128 } 129 } 130 else { 131 if(a < minH.top()) { 132 maxH.push(a); 133 sumLA += a; 134 } 135 else { 136 minH.push(a); 137 sumRA += a; 138 sumRA -= minH.top(); 139 maxH.push(minH.top()); 140 sumLA += minH.top(); 141 minH.pop(); 142 } 143 } 144 } 145 else { 146 int x = maxH.top(); 147 ans = sumRA - sumLA + sumB; 148 if(maxH.size() > minH.size()) ans += maxH.top(); 149 else if(maxH.size() < minH.size()) { 150 ans -= minH.top(); 151 x = minH.top(); 152 } 153 cout << x << " " << ans << endl; 154 } 155 } 156 return 0; 157 }
原文地址:https://www.cnblogs.com/zaq19970105/p/10941714.html
时间: 2024-10-28 09:12:17