「洛谷 P4275」萃香的请柬

给定一个只含 B, L 的字符串序列

定义一次变换为:B 扩展为 BL,L 变成 B,例如 BBLBL -> BLBLBBLB

求经过无数次变换后,i=lr[si=B]\sum_{i=l}^r [s_i=\text{B}] ( lr263)(~l\le r\le 2^{63})


这是一道很有意思的题

首先无数次变换表示只需考虑从单字符 B 开始的变换

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a[1] = B
a[2] = BL
a[3] = BLB
a[4] = BLBBL
a[5] = BLBBLBLB
a[6] = BLBBLBLBBLBBL

发现变换非常相似

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BLBBLBLB 看成 a[5] = a[4] + a[3]

a[4] -> a[5]
a[3] -> a[4]

a[5] -> a[5] + a[4] = a[6]

发现 a 的长度、a 中 B 的数量和斐波那契 不可告人的 关系,具体请您自行打表发现

可以保证 s1ns_{1\cdots n} 一定由若干个 a 由长到短依次拼接而成(归纳可证),例如

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BLBBLBLBBLBB
BLBBLBLB | BLB | B
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#include <algorithm>
#include <bitset>
#include <cassert>
#include <cmath>
#include <cstdio>
#include <cstring>
#include <deque>
#include <iostream>
#include <map>
#include <queue>
#include <random>
#include <set>
#include <stack>
#include <string>
#include <utility>
#include <vector>
// #pragma GCC optimize("Ofast")
// #pragma GCC optimize("inline")
// #pragma GCC optimize("-fgcse")
// #pragma GCC optimize("-fgcse-lm")
// #pragma GCC optimize("-fipa-sra")
// #pragma GCC optimize("-ftree-pre")
// #pragma GCC optimize("-ftree-vrp")
// #pragma GCC optimize("-fpeephole2")
// #pragma GCC optimize("-ffast-math")
// #pragma GCC optimize("-fsched-spec")
// #pragma GCC optimize("unroll-loops")
// #pragma GCC optimize("-falign-jumps")
// #pragma GCC optimize("-falign-loops")
// #pragma GCC optimize("-falign-labels")
// #pragma GCC optimize("-fdevirtualize")
// #pragma GCC optimize("-fcaller-saves")
// #pragma GCC optimize("-fcrossjumping")
// #pragma GCC optimize("-fthread-jumps")
// #pragma GCC optimize("-funroll-loops")
// #pragma GCC optimize("-fwhole-program")
// #pragma GCC optimize("-freorder-blocks")
// #pragma GCC optimize("-fschedule-insns")
// #pragma GCC optimize("inline-functions")
// #pragma GCC optimize("-ftree-tail-merge")
// #pragma GCC optimize("-fschedule-insns2")
// #pragma GCC optimize("-fstrict-aliasing")
// #pragma GCC optimize("-fstrict-overflow")
// #pragma GCC optimize("-falign-functions")
// #pragma GCC optimize("-fcse-skip-blocks")
// #pragma GCC optimize("-fcse-follow-jumps")
// #pragma GCC optimize("-fsched-interblock")
// #pragma GCC optimize("-fpartial-inlining")
// #pragma GCC optimize("no-stack-protector")
// #pragma GCC optimize("-freorder-functions")
// #pragma GCC optimize("-findirect-inlining")
// #pragma GCC optimize("-fhoist-adjacent-loads")
// #pragma GCC optimize("-frerun-cse-after-loop")
// #pragma GCC optimize("inline-small-functions")
// #pragma GCC optimize("-finline-small-functions")
// #pragma GCC optimize("-ftree-switch-conversion")
// #pragma GCC optimize("-foptimize-sibling-calls")
// #pragma GCC optimize("-fexpensive-optimizations")
// #pragma GCC optimize("-funsafe-loop-optimizations")
// #pragma GCC optimize("inline-functions-called-once")
// #pragma GCC optimize("-fdelete-null-pointer-checks")
#define rep(i, l, r) for (int i = (l); i <= (r); ++i)
#define per(i, l, r) for (int i = (l); i >= (r); --i)
using std::cerr;
using std::endl;
using std::make_pair;
using std::pair;
typedef pair<int, int> pii;
typedef long long ll;
typedef unsigned int ui;
typedef unsigned long long ull;

// #define DEBUG 1 //调试开关
struct IO {
#define MAXSIZE (1 << 20)
#define isdigit(x) (x >= '0' && x <= '9')
char buf[MAXSIZE], *p1, *p2;
char pbuf[MAXSIZE], *pp;
#if DEBUG
#else
IO() : p1(buf), p2(buf), pp(pbuf) {}
~IO() { fwrite(pbuf, 1, pp - pbuf, stdout); }
#endif
inline char gc() {
#if DEBUG //调试,可显示字符
return getchar();
#endif
if (p1 == p2) p2 = (p1 = buf) + fread(buf, 1, MAXSIZE, stdin);
return p1 == p2 ? -1 : *p1++;
}
inline bool blank(char ch) { return ch == ' ' || ch == '\n' || ch == '\r' || ch == '\t'; }
template <class T>
inline void read(T &x) {
register double tmp = 1;
register bool sign = 0;
x = 0;
register char ch = gc();
for (; !isdigit(ch); ch = gc())
if (ch == '-') sign = 1;
for (; isdigit(ch); ch = gc()) x = x * 10 + (ch - '0');
if (ch == '.')
for (ch = gc(); isdigit(ch); ch = gc()) tmp /= 10.0, x += tmp * (ch - '0');
if (sign) x = -x;
}
inline void read(char *s) {
register char ch = gc();
for (; blank(ch); ch = gc())
;
for (; !blank(ch); ch = gc()) *s++ = ch;
*s = 0;
}
inline void read(char &c) {
for (c = gc(); blank(c); c = gc())
;
}
inline void push(const char &c) {
#if DEBUG //调试,可显示字符
putchar(c);
#else
if (pp - pbuf == MAXSIZE) fwrite(pbuf, 1, MAXSIZE, stdout), pp = pbuf;
*pp++ = c;
#endif
}
template <class T>
inline void write(T x) {
if (x < 0) x = -x, push('-'); // 负数输出
static T sta[35];
T top = 0;
do {
sta[top++] = x % 10, x /= 10;
} while (x);
while (top) push(sta[--top] + '0');
}
inline void write(const char *s) {
while (*s != '\0') push(*(s++));
}
template <class T>
inline void write(T x, char lastChar) {
write(x), push(lastChar);
}
} io;

const int N = 94;
ull fib[100];

int get(ll n) { return (std::upper_bound(fib, fib + 1 + N, n) - 1 - fib); }

ll s(ll n) {
if (n <= 0) return 0;
ll ret = 0;
while (n) {
int x = get(n);
ret += fib[x - 1];
n -= fib[x];
}
return ret;
}

int main() {
#ifdef LOCAL
freopen("input", "r", stdin);
#endif
char c;
io.read(c);
int q;
io.read(q);
fib[0] = 0, fib[1] = 1;
rep(i, 2, 93) fib[i] = fib[i - 1] + fib[i - 2];
fib[94] = (1ULL << 63);
while (q--) {
ll l, r;
io.read(l), io.read(r);
io.write(s(r) - s(l - 1), '\n');
}
return 0;
}
作者

Gesrua

发布于

2019-08-13

更新于

2020-11-21

许可协议