洛谷 P4998 信号站 题解

首先，尝试枚举建立信号站的位置是 \(1e16\) 的复杂度， 现在就要 \(O(1)\) 求出每个信号站的不合理值。

设当前枚举的信号站的位置为 \(i\), 在 \(i\) 左边的信号站有 \(cnt_i\) 个.

对于位于信号站左边的信号站。

\[\Sigma_{j=1}^{cnt_i }|i - pos_j| \]

\[i*cnt_i - \Sigma_{j=1}^{cnt_i} pos_j \]

对于右边的信号站同理， 使用二分查找 \(cnt_i\)

复杂度 \(O(1e6*log_2N)\)

但是有可能存在拜访位置为负数的情况，那么让每个位置向右偏离 \(+10^6\)

//+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-
//
//      By txp2024 www.luogu.com.cn  TXP2023 www.github.com
// 
//+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-

#pragma once
#include <vector>
#include <stdio.h>
#include <string.h>
#include <algorithm>
#include <numeric>
#include <ctype.h>
#include <cstdarg>
#include <climits>
#include <time.h>
#include <iostream>
#include <stdint.h>

#define _FREAD        true
#define MAX_INF       1e18
#define MAX_NUM_SIZE  35
#define MAXN          (size_t)(1e6+5)

typedef long long int ll;
typedef unsigned long long int ull;

template<typename _T>
inline _T fpow(_T a, _T n, _T mod) {
    _T base = a, ret = 1;
    while (n) {
        if (n & 1) {
            ret = ret * base;
            ret %= mod;
        }
        base = base * base;
        base %= mod;
        n >>= 1;
    }
    return ret % mod;
}

//快读函数声明
template< typename Type >
inline Type fread(Type* p = nullptr);

//快速输出函数
template<typename Type>
inline void fwrite(Type x);

ll pos[MAXN], sum[MAXN], val[(ll)2e6 + 5];
ll n, k, ans;

int main() {

#ifdef _FREOPEN
    freopen("input.txt", "r", stdin);
#endif // _FREOPEN

#ifdef _RUN_TIME
    clock_t start = clock();
#endif // _RUN_TIME

    fread(&n), fread(&k);

    for (size_t i = 1; i <= n; i++) {
        pos[i] = fread<ll>() + 1e6;
    }

    std::sort(pos + 1, pos + 1 + n);

    for (size_t i = 1; i <= n; i++) {
        sum[i] = sum[i - 1] + pos[i];
    }

    for (size_t i = 0; i <= 2e6; i++) {
        ll temp = (std::lower_bound(pos + 1, pos + 1 + n, i) - pos - 1);
        if (temp != n + 1 && temp != 0) {
            val[i] += i * temp - sum[temp];
        }
        temp = (std::upper_bound(pos + 1, pos + 1 + n, i) - pos);
        if (temp != n + 1) {
            val[i] += sum[n] - sum[temp - 1] - i * (n - temp + 1);
        }
    }

    std::sort(val, val + (ll)2e6);

    for (size_t i = 0; i < k; i++) {
        ans += val[i];
    }

    printf("%lld\n", ans);






#ifdef _RUN_TIME
    printf("The running duration is not less than %ld ms\n", clock() - start);
#endif // _RUN_TIME
    return 0;
}

template< typename Type >
inline Type fread(Type* p) {
#if _FREAD
    Type ret = 0, sgn = 0, ch = getchar();
    while (!isdigit(ch)) {
        sgn |= ch == '-', ch = getchar();
    }
    while (isdigit(ch)) ret = ret * 10 + ch - '0', ch = getchar();
    if (p != nullptr) {
        *p = Type(sgn ? -ret : ret);
    }
    return sgn ? -ret : ret;
#else
    if (p == nullptr) {
        Type temp;
        p = &temp;
    }
    scanf("%lld", p);
    return *p;


#endif // _FREAD
}


template<typename Type>
inline void fwrite(Type x) {
    int sta[MAX_NUM_SIZE];
    int top = 0;
    do {
        sta[top++] = x % 10, x /= 10;
    } while (x);
    while (top) {
        putchar(sta[--top] + '0');
    }  // 48 是 '0'
    return;
}



/**
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 *       ,"                      ,"|        ,"        ,"  |
 *      +-----------------------+  |      ,"        ,"    |
 *      |  .-----------------.  |  |     +---------+      |
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 *      |  |                 |  |  |     |         |      |
 *      |  |  C:\>rp++       |  |  |     |`---=    |      |
 *      |  |                 |  |  |     |==== ooo |      ;
 *      |  |                 |  |  |     |(((( [33]|    ,"
 *      |  `-----------------'  | /      |((((     |  ,"
 *      +-----------------------+/       |         |,"
 *         /_)______________(_/          +---------+
 *    _______________________________
 *   /  oooooooooooooooo  .o.  oooo /,   /-----------
 *  / ==ooooooooooooooo==.o.  ooo= //   /\--{)B     ,"
 * /_==__==========__==_ooo__ooo=_/'   /___________,"
 *
 */