Record - Nov. 28st, 2020 - Exam. REC

Prob. 1

Desc. & Link.

暴力为 \(\Theta(NK)\)。

正解（也许）：

把每一个全为正整数的子段找出来。

然后判断一下中间连接的情况即可。

但是这样决策情况太多了。

我们需要考虑贪心。

把所有整数段的个数记为 \(totP\)，每个子段的区间记为 \([posL_{i},posR_{i}]\)，区间和记为 \(sumP_{i}\)

把其他的负数段个数记为 \(totN\)，区间和记为 \(sumN_{i}\)。

当 \(totP\le k\) 答案显然。

我们需要考虑的是 \(totP>k\) 的情况。

我们把整数段、负数段缩成点。

然后问题还是最多选 \(k\) 段的最大子段和。

不过我们的序列有个性质：相邻数的正负性不同。（gu）

好了放弃以上想法。

模拟 \(k\) 轮找全局最大子段和，找到一次把子段乘上 \(-1\)。

#include <cstdio>

typedef long long LL;

const int MAXN = 1e5 + 5;

int rint () {
	int x = 0, f = 1; char c = getchar ();
	for ( ; c < '0' || c > '9'; c = getchar () )	f = c == '-' ? -1 : f;
	for ( ; c >= '0' && c <= '9'; c = getchar () )	x = ( x << 3 ) + ( x << 1 ) + ( c & 15 );
	return x * f;
}

template<typename _T>
void wint ( _T x ) {
	if ( x < 0 )	putchar ( '-' ), x = ~ x + 1;
	if ( x > 9 )	wint ( x / 10 );
	putchar ( x % 10 ^ '0' );
}

template<typename _T> _T MAX ( const _T x, const _T y ) { return x < y ? y : x; }
template<typename _T> void swapp ( _T &x, _T &y ) { _T w = x; x = y; y = w; }

struct nodeS {
	LL val, dat, p, s;
	int l, r, pl, pr, sl, sr;
	nodeS ( LL V = 0, LL D = 0, LL P = 0, LL S = 0,
			int L = 0, int R = 0, int Pl = 0, int Pr = 0, int Sl = 0, int Sr = 0 ) {
				val = V, dat = D, p = P, s = S, l = L, r = R, pl = Pl, pr = Pr, sl = Sl, sr = Sr; }
} nodes[MAXN * 4][2];

int n, k, a[MAXN];
bool tag[MAXN * 4];

nodeS Merge ( const nodeS lch, const nodeS rch ) {
	nodeS ret;
	ret.val = lch.val + rch.val;
	ret.p = MAX ( lch.p, lch.val + rch.p );
	if ( ret.p == lch.p )	ret.pl = lch.pl, ret.pr = lch.pr;
	else	ret.pl = lch.pl, ret.pr = rch.pr;
	ret.s = MAX ( rch.s, rch.val + lch.s );
	if ( ret.s == rch.s )	ret.sl = rch.sl, ret.sr = rch.sr;
	else	ret.sl = lch.sl, ret.sr = rch.sr;
	ret.dat = MAX ( lch.s + rch.p, MAX ( lch.dat, rch.dat ) );
	if ( ret.dat == lch.dat )	ret.l = lch.l, ret.r = lch.r;
	else if ( ret.dat == rch.dat )	ret.l = rch.l, ret.r = rch.r;
	else	ret.l = lch.sl, ret.r = rch.pr;
	return ret;
}

void Upt ( const int x ) {
	nodes[x][0] = Merge ( nodes[x << 1][0], nodes[x << 1 | 1][0] );
	nodes[x][1] = Merge ( nodes[x << 1][1], nodes[x << 1 | 1][1] );
}

void Spr ( const int x ) {
	if ( ! tag[x] )	return;
	swapp ( nodes[x << 1][0], nodes[x << 1][1] );
	swapp ( nodes[x << 1 | 1][0], nodes[x << 1 | 1][1] );
	tag[x << 1] ^= 1, tag[x << 1 | 1] ^= 1, tag[x] = 0;
}

void Build ( const int x, const int l, const int r ) {
	if ( l == r ) {
		nodes[x][0] = nodeS ( a[l], a[l], a[l], a[l], l, l, l, l, l, l );
		nodes[x][1] = nodeS ( -a[l], -a[l], -a[l], -a[l], l, l, l, l, l, l );
		return;
	}
	int mid = ( l + r ) >> 1;
	Build ( x << 1, l, mid );
	Build ( x << 1 | 1, mid + 1, r );
	Upt ( x );
}

void Modify ( const int x, const int l, const int r, const int segL, const int segR ) {
	if ( l > segR || r < segL )	return;
	if ( l >= segL && r <= segR ) {
		swapp ( nodes[x][0], nodes[x][1] );
		tag[x] ^= 1;
		return;
	}
	int mid = ( l + r ) >> 1;
	Spr ( x );
	Modify ( x << 1, l, mid, segL, segR );
	Modify ( x << 1 | 1, mid + 1, r, segL, segR );
	Upt ( x );
}

int main () {
	n = rint (), k = rint ();
	for ( int i = 1; i <= n; ++ i )	a[i] = rint ();
	Build ( 1, 1, n );
	LL ans = 0;
	while ( k -- > 0 ) {
		nodeS ret = nodes[1][0];
		if ( ret.dat < 0 )	break;
		Modify ( 1, 1, n, ret.l, ret.r );
		ans += ret.dat;
	}
	wint ( ans ), putchar ( '\n' );
	return 0;
}

Prob. 2

Desc. & Link.

设 \(f_{i,0/1}\) 表示把 \(a_{i}\) 往头/尾放可以得到的最多的上升子序列。

\[f_{i,0}=\begin{cases}\max\{f_{j,0},f_{j,1}\}+1,a_{i}<a_{j} \\\max\{f_{j,0},f_{j,1}\},a_{i}\ge a_{j}\end{cases} \\f_{i,1}=\begin{cases}\max\{f_{j,0},f_{j,1}\},a_{i}<a_{j} \\\max\{f_{j,0},f_{j,1}\}+1,a_{i}\ge a_{j}\end{cases}
\]

不行。

考虑普通的 LIS 怎么做。

\[f_{i}=\max\{f_{j}\}+1,a_{i}>a_{j}
\]

\[a_{i}<a_{j},i<j
\]

选择往前放的元素，放得越晚越靠前。

选择往后放的元素，放得越晚越靠后。

那么需要做的是把相对较大的元素往后，相对较小的元素往前。

连边，把李三花后的 \(a_{i}\) 连向 \(\text{trans}(i,0),\text{trans}(i,1),\text{trans}(x,0/1)=n-x+1/x+n\)。

#include <cstdio>
#include <algorithm>

using namespace std;

const int MAXN = 2e5 + 5;

int rint () {
	int x = 0, f = 1; char c = getchar ();
	for ( ; c < '0' || c > '9'; c = getchar () )	f = c == '-' ? -1 : f;
	for ( ; c >= '0' && c <= '9'; c = getchar () )	x = ( x << 3 ) + ( x << 1 ) + ( c & 15 );
	return x * f;
}

template<typename _T>
void wint ( _T x ) {
	if ( x < 0 )	putchar ( '-' ), x = ~ x + 1;
	if ( x > 9 )	wint ( x / 10 );
	putchar ( x % 10 ^ '0' );
}

template<typename _T> _T MAX ( const _T x, const _T y ) { return x < y ? y : x; }

struct Value {
	int val, pos;
	Value ( int V = 0, int P = 0 ) { val = V, pos = P; }
	bool operator < ( const Value &another ) { return val < another.val; }
} vals[MAXN];

struct GraphSet {
	int to, nx;
	GraphSet ( int T = 0, int N = 0 ) { to = T, nx = N; }
} as[MAXN * 2];

int n, cnt, len, degin[MAXN], a[MAXN], b[MAXN], buc[MAXN], sywf[MAXN];

void makeEdge ( const int u, const int v ) { as[++ cnt] = GraphSet ( v, degin[u] ), degin[u] = cnt; }
int Trans ( const int x, const int y ) { return ! y ? n - x + 1 : x + n; }

void ADD ( int p, const int x ) { for ( ; p <= ( n << 1 ); p += p & -p )	sywf[p] = MAX ( sywf[p], x ); }
int ASK ( int p ) { int res = 0; for ( ; p; p -= p & -p )	res = MAX ( res, sywf[p] ); return res; }
int CMP ( const int x, const int y ) { return x > y; }

int main () {
//	freopen ( "dequexlis.in", "r", stdin );
//	freopen ( "dequexlis.out", "w", stdout );
	n = rint ();
	for ( int i = 1; i <= n; ++ i )	a[i] = b[i] = rint ();
	sort ( b + 1, b + 1 + n );
	len = unique ( b + 1, b + 1 + n ) - b - 1;
	for ( int i = 1; i <= n; ++ i )	a[i] = lower_bound ( b + 1, b + 1 + len, a[i] ) - b;
	for ( int i = 1; i <= n; ++ i )	vals[i] = Value ( a[i], i );
	for ( int i = 1; i <= n; ++ i ) {
		makeEdge ( a[i], Trans ( i, 0 ) );
		makeEdge ( a[i], Trans ( i, 1 ) );
	}
	int BUC = 0;
	for ( int x_x = 1; x_x <= n; ++ x_x ) {
		int u = x_x;
		BUC = 0;
		for ( int i = degin[u]; i; i = as[i].nx ) {
			int v = as[i].to;
			buc[++ BUC] = v;
		}
		sort ( buc + 1, buc + 1 + BUC, CMP );
		for ( int i = 1; i <= BUC; ++ i )	ADD ( buc[i], 1 + ASK ( buc[i] - 1 ) );
	}
	wint ( ASK ( n << 1 ) ), putchar ( '\n' );
	return 0;
}

/* Jesus bless all */

Prob. 3

放弃人生打了个 50 的记搜。

#include <cstdio>
#include <map>
#define mod ( 998244853 )

using namespace std;

template<typename _T> _T MAX ( const _T x, const _T y ) { return x < y ? y : x; }

int n, m;

namespace Course {
const int MAXN = 255;
int f[MAXN][MAXN][MAXN];
bool vis[MAXN][MAXN][MAXN];

int dfs ( const int mx, const int a, const int b ) {
	if ( vis[a][b][mx] )	return f[a][b][mx];
	vis[a][b][mx] = 1;
	if ( a == n && b == m )	return f[a][b][mx] = mx;
	if ( a < n )	f[a][b][mx] = ( f[a][b][mx] + dfs ( MAX ( mx, a - b + 1 ), a + 1, b ) ) % mod;
	if ( b < m )	f[a][b][mx] = ( f[a][b][mx] + dfs ( mx, a, b + 1 ) ) % mod;
	return f[a][b][mx];
}
}

int main () {
	// freopen ( "maxpsum.in", "r", stdin );
	// freopen ( "maxpsum.out", "w", stdout );
	scanf ( "%d%d", &n, &m );
	if ( n <= 250 && m <= 250 )	printf ( "%d\n", Course :: dfs ( 0, 0, 0 ) );
	// else	printf ( "%d\n", Might :: dfs ( 0, 0, 0 ) );
	return 0;
}

Prob. 4

Desc. & Link.