「模拟测试」颜色-分块/莫队

给出一个序列,每个位置有同一个颜色,每个颜色有一个价值,要求支持单点修改,查询区间 $[l, r]$ 的权值和,相同颜色只算一次。

题解

这不是数颜色加强版吗?
直接带修莫队不就完了,特判一些特殊数据开块的大小会比较快,否则可能会 TLE,时间复杂度 $O(n ^ {\frac 5 3})$。

但这么做是不优雅的,这题还可以分块/树套树(分块真的比树套树快),用一个 set 维护每个位置前一个相同颜色的位置,记为 pre 数组,每次查询就是在 pre 里查 $[l, r]$ 中小于 $l$ 的权值和,于是我们可以分块维护,维护一个原数组,和一个排序后的数组,以及前缀和。

时间复杂度 $O(n \sqrt{n} \log n)$,然而奇快无比

代码

分块

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/**
* Copyright (c) 2017, xehoth
* All rights reserved.
* 「SuperOJ 2022」颜色 30-10-2017
* 分块
* @author xehoth
*/
#include <bits/stdc++.h>
namespace IO {
inline char read() {
static const int IN_LEN = 1000000;
static char buf[IN_LEN], *s, *t;
s == t ? t = (s = buf) + fread(buf, 1, IN_LEN, stdin) : 0;
return s == t ? -1 : *s++;
}
template <typename T>
inline bool read(T &x) {
static char c;
static bool iosig;
for (c = read(), iosig = false; !isdigit(c); c = read()) {
if (c == -1) return false;
c == '-' ? iosig = true : 0;
}
for (x = 0; isdigit(c); c = read()) x = x * 10 + (c ^ '0');
iosig ? x = -x : 0;
return true;
}
inline void read(char &c) {
while (c = read(), isspace(c) && c != -1)
;
}
inline int read(char *buf) {
register int s = 0;
register char c;
while (c = read(), isspace(c) && c != -1)
;
if (c == -1) {
*buf = 0;
return -1;
}
do
buf[s++] = c;
while (c = read(), !isspace(c) && c != -1);
buf[s] = 0;
return s;
}
const int OUT_LEN = 1000000;
char obuf[OUT_LEN], *oh = obuf;
inline void print(char c) {
oh == obuf + OUT_LEN ? (fwrite(obuf, 1, OUT_LEN, stdout), oh = obuf) : 0;
*oh++ = c;
}
template <typename T>
inline void print(T x) {
static int buf[30], cnt;
if (x == 0) {
print('0');
} else {
x < 0 ? (print('-'), x = -x) : 0;
for (cnt = 0; x; x /= 10) buf[++cnt] = x % 10 | 48;
while (cnt) print((char)buf[cnt--]);
}
}
inline void print(const char *s) {
for (; *s; s++) print(*s);
}
inline void flush() { fwrite(obuf, 1, oh - obuf, stdout); }
struct InputOutputStream {
template <typename T>
inline InputOutputStream &operator>>(T &x) {
read(x);
return *this;
}
template <typename T>
inline InputOutputStream &operator<<(const T &x) {
print(x);
return *this;
}
~InputOutputStream() { flush(); }
} io;
}
namespace {
using IO::io;
const int MAXN = 100000;
const int MAXC = 100000;
const int MAX_BLOCK = 245;
const int BLOCK_SIZE = 412;
int blockSize;
int n, m, a[MAXN + 1], w[MAXC + 1], pre[MAXC + 1];
struct Block {
std::pair<int, int> a[MAX_BLOCK + 1][BLOCK_SIZE + 1];
std::pair<int, int> sorted[MAX_BLOCK + 1][BLOCK_SIZE + 1];
int sum[MAX_BLOCK][BLOCK_SIZE + 1];
int n, cnt, size[MAX_BLOCK + 1];
inline void init(int n, int *color) {
this->n = n, cnt = 0, blockSize = sqrt(n) * 1.3;
register int r = 0, c = 0;
for (register int i = 0; i < n; i++) {
a[r][c].first = color[i], a[r][c].second = w[::a[i]];
sorted[r][c] = a[r][c], size[r]++, c++;
if (c == blockSize) c = 0, r++;
}
cnt = r + 1;
for (register int i = 0; i < cnt; i++) {
std::sort(sorted[i], sorted[i] + size[i]);
sum[i][0] = sorted[i][0].second;
for (register int j = 1; j < size[i]; j++)
sum[i][j] += sum[i][j - 1] + sorted[i][j].second;
}
}
inline void modify(int p, int x, int w) {
register int r = p / blockSize, c = p % blockSize;
if (a[r][c].first == x && a[r][c].second == w) return;
register int old = a[r][c].first, oldW = a[r][c].second;
a[r][c].first = x, a[r][c].second = w;
register int pos = std::lower_bound(sorted[r], sorted[r] + size[r],
std::make_pair(old, oldW)) -
sorted[r];
sorted[r][pos].first = x, sorted[r][pos].second = w;
if (std::make_pair(x, w) > std::make_pair(old, oldW)) {
register int begin = pos;
while (pos < size[r] - 1 && sorted[r][pos + 1] < sorted[r][pos])
std::swap(sorted[r][pos], sorted[r][pos + 1]), pos++;
if (begin == 0) sum[r][0] = sorted[r][begin].second, begin++;
for (; begin < size[r]; begin++)
sum[r][begin] = sum[r][begin - 1] + sorted[r][begin].second;
} else if (std::make_pair(x, w) < std::make_pair(old, oldW)) {
while (pos > 0 && sorted[r][pos - 1] > sorted[r][pos])
std::swap(sorted[r][pos], sorted[r][pos - 1]), pos--;
register int begin = pos;
if (begin == 0) sum[r][0] = sorted[r][begin].second, begin++;
for (; begin < size[r]; begin++)
sum[r][begin] = sum[r][begin - 1] + sorted[r][begin].second;
} else {
register int begin = pos;
if (begin == 0) sum[r][0] = sorted[r][begin].second, begin++;
for (; begin < size[r]; begin++)
sum[r][begin] = sum[r][begin - 1] + sorted[r][begin].second;
}
}
inline int query(int l, int r, int x) {
register int r1 = l / blockSize, r2 = r / blockSize;
register int c1 = l % blockSize, c2 = r % blockSize;
register int res = 0;
if (r1 == r2) {
while (c1 <= c2) {
if (a[r1][c1].first < x) res += a[r1][c1].second;
c1++;
}
} else {
register int tmp = size[r1];
while (c1 < tmp) {
if (a[r1][c1].first < x) res += a[r1][c1].second;
c1++;
}
r1++;
while (c2 >= 0) {
if (a[r2][c2].first < x) res += a[r2][c2].second;
c2--;
}
for (register int pos; r1 < r2;) {
pos = std::lower_bound(sorted[r1], sorted[r1] + size[r1],
std::make_pair(x, 0)) -
sorted[r1];
if (pos != 0) res += sum[r1][pos - 1];
r1++;
}
}
return res;
}
} block;
typedef std::set<int> Set;
Set set[MAXC + 1];
inline void init() {
for (register int i = 0; i < n; i++) {
if (set[a[i]].empty())
set[a[i]].insert(-1), set[a[i]].insert(set[a[i]].end(), n);
pre[i] = *--set[a[i]].insert(--set[a[i]].end(), i);
}
block.init(n, pre);
}
inline void modify(int p, int x) {
if (a[p] == x) return;
register int old = a[p];
a[p] = x, set[old].erase(p);
register int tp = *set[old].upper_bound(p);
if (tp != n)
pre[tp] = *--set[old].lower_bound(p), block.modify(tp, pre[tp], w[old]);
if (set[x].empty()) set[x].insert(-1), set[x].insert(set[x].end(), n);
set[x].insert(p), tp = *set[x].upper_bound(p);
if (tp != n) pre[tp] = p, block.modify(tp, p, w[x]);
pre[p] = *--set[x].lower_bound(p), block.modify(p, pre[p], w[x]);
}
inline void solve() {
io >> n >> m;
for (register int i = 0; i < n; i++) io >> a[i];
for (register int i = 1; i <= n; i++) io >> w[i];
init();
for (register int i = 1, l, r, cmd; i <= m; i++) {
io >> cmd;
switch (cmd) {
case 1:
io >> l >> r, modify(l - 1, r);
break;
case 2:
io >> l >> r, io << block.query(l - 1, r - 1, l - 1) << '\n';
break;
}
}
}
}
int main() {
// freopen("color.in", "r", stdin);
// freopen("color.out", "w", stdout);
solve();
return 0;
}

带修莫队

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/**
* Copyright (c) 2017, xehoth
* All rights reserved.
* 「SuperOJ 2022」颜色 30-10-2017
* 带修改莫队
* @author xehoth
*/
#include <bits/stdc++.h>
namespace IO {
inline char read() {
static const int IN_LEN = 1000000;
static char buf[IN_LEN], *s, *t;
s == t ? t = (s = buf) + fread(buf, 1, IN_LEN, stdin) : 0;
return s == t ? -1 : *s++;
}
template <typename T>
inline bool read(T &x) {
static char c;
static bool iosig;
for (c = read(), iosig = false; !isdigit(c); c = read()) {
if (c == -1) return false;
c == '-' ? iosig = true : 0;
}
for (x = 0; isdigit(c); c = read()) x = x * 10 + (c ^ '0');
iosig ? x = -x : 0;
return true;
}
inline void read(char &c) {
while (c = read(), isspace(c) && c != -1)
;
}
inline int read(char *buf) {
register int s = 0;
register char c;
while (c = read(), isspace(c) && c != -1)
;
if (c == -1) {
*buf = 0;
return -1;
}
do
buf[s++] = c;
while (c = read(), !isspace(c) && c != -1);
buf[s] = 0;
return s;
}
const int OUT_LEN = 1000000;
char obuf[OUT_LEN], *oh = obuf;
inline void print(char c) {
oh == obuf + OUT_LEN ? (fwrite(obuf, 1, OUT_LEN, stdout), oh = obuf) : 0;
*oh++ = c;
}
template <typename T>
inline void print(T x) {
static int buf[30], cnt;
if (x == 0) {
print('0');
} else {
x < 0 ? (print('-'), x = -x) : 0;
for (cnt = 0; x; x /= 10) buf[++cnt] = x % 10 | 48;
while (cnt) print((char)buf[cnt--]);
}
}
inline void print(const char *s) {
for (; *s; s++) print(*s);
}
inline void flush() { fwrite(obuf, 1, oh - obuf, stdout); }
struct InputOutputStream {
template <typename T>
inline InputOutputStream &operator>>(T &x) {
read(x);
return *this;
}
template <typename T>
inline InputOutputStream &operator<<(const T &x) {
print(x);
return *this;
}
~InputOutputStream() { flush(); }
} io;
}
namespace {
using IO::io;
const int MAXN = 100000;
const int MAXC = 100000;
int id[MAXN + 1];
struct Query {
int l, r, id, time;
long long *ans;
Query(int l = 0, int r = 0, int id = 0, int time = 0, long long *ans = NULL)
: l(l), r(r), id(id), time(time), ans(ans) {}
inline bool operator<(const Query &q) const {
return ::id[l] != ::id[q.l]
? ::id[l] < ::id[q.l]
: (::id[r] != ::id[q.r] ? (::id[r] < ::id[q.r])
: time < q.time);
}
};
struct Modify {
int p, c, last;
Modify(int p = 0, int c = 0, int last = 0) : p(p), c(c), last(last) {}
};
int n, m, blockSize;
int color[MAXN + 1], w[MAXN + 1];
Query query[MAXN + 1];
Modify modify[MAXN + 1];
int qcnt, mcnt;
int cnt[MAXC + 1];
long long nowAns, ans[MAXN + 1];
inline void shrink(int i) {
cnt[color[i]]--;
if (cnt[color[i]] == 0) nowAns -= w[color[i]];
}
inline void expand(int i) {
cnt[color[i]]++;
if (cnt[color[i]] == 1) nowAns += w[color[i]];
}
inline void inTime(int i, int l, int r) {
register int pos = modify[i].p, c = modify[i].c, &last = modify[i].last;
if (l <= pos && r >= pos) shrink(pos);
last = color[pos], color[pos] = c;
if (l <= pos && r >= pos) expand(pos);
}
inline void outTime(int i, int l, int r) {
register int pos = modify[i].p, &last = modify[i].last;
if (l <= pos && r >= pos) shrink(pos);
color[pos] = last;
if (l <= pos && r >= pos) expand(pos);
}
inline void solveCase() {
register int l = 1, r = 0, now = 0;
for (register int i = 1; i <= qcnt; i++) {
while (r < query[i].r) expand(++r);
while (r > query[i].r) shrink(r--);
while (l < query[i].l) shrink(l++);
while (l > query[i].l) expand(--l);
while (now < query[i].time) inTime(++now, l, r);
while (now > query[i].time) outTime(now--, l, r);
*query[i].ans = nowAns;
}
}
inline void solve() {
io >> n >> m;
for (register int i = 1; i <= n; i++) io >> color[i];
for (register int i = 1; i <= n; i++) io >> w[i];
for (register int i = 1, l, r, cmd; i <= m; i++) {
io >> cmd;
switch (cmd) {
case 1:
io >> l >> r;
modify[++mcnt] = Modify(l, r);
break;
case 2:
io >> l >> r, qcnt++;
query[qcnt] = Query(l, r, i, mcnt, &ans[qcnt]);
break;
}
}
if (abs(mcnt - qcnt) <= 10000) {
blockSize = pow(n, 2.0 / 3.0) * 1.8;
} else if (qcnt > mcnt) {
if (round((double)qcnt / mcnt) == 2)
blockSize = pow(n, 2.0 / 3.0) * 1.5;
else if (round((double)qcnt / mcnt) == 3)
blockSize = pow(n, 2.0 / 3.0) * 1.2;
else if (qcnt > 100 * mcnt)
blockSize = pow(n, 2.0 / 3.0) * 0.2;
else if (qcnt > 10 * mcnt)
blockSize = pow(n, 2.0 / 3.0) * 0.4;
else
blockSize = pow(n, 2.0 / 3.0);
} else {
if (round((double)mcnt / qcnt) == 2)
blockSize = pow(n, 2.0 / 3.0) * 2.7;
else
blockSize = pow(n, 2.0 / 3.0) * 2;
}
for (register int i = 1; i <= n; i++) id[i] = (i - 1) / blockSize + 1;
std::sort(query + 1, query + qcnt + 1);
solveCase();
for (register int i = 1; i <= qcnt; i++) io << ans[i] << '\n';
}
}
int main() {
// freopen("color.in", "r", stdin);
// freopen("color.out", "w", stdout);
solve();
return 0;
}
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