SPOJ GSS系列(数据结构维护技巧入门)
题目链接 GSS
$GSS1$
对于每个询问$l$, $r$,查询$a_{l}$, $a_{l+1}$, $a_{l+2}$, ..., $a_{r}$这个序列的最大字段和。
建立线段树,每个节点维护四个信息
$c$:当前区间的元素和
$lc$:当前区间左端点开始的最大子序列和
$rc$:当前区间右端点结束的最大子序列和
$ret$:当前区间的答案
于是我们建立线段树的时候预处理出每个节点的四个信息,查询的时候返回一个节点,这个节点的$ret$即为答案。
#include <bits/stdc++.h>
using namespace std;
#define rep(i, a, b) for (int i(a); i <= (b); ++i)
#define dec(i, a, b) for (int i(a); i >= (b); --i)
#define lson i << 1, L, mid
#define rson i << 1 | 1, mid + 1, R
#define ls i << 1
#define rs i << 1 | 1
typedef long long LL;
const int N = 1e5 + 10;
struct node{
int c, lc, rc, ret;
} t[N << 2];
int n, q, l, r;
void pushup(int i){
t[i].c = t[ls].c + t[rs].c;
t[i].lc = max(t[ls].lc, t[ls].c + t[rs].lc);
t[i].rc = max(t[rs].rc, t[rs].c + t[ls].rc);
t[i].ret = max(max(t[ls].ret, t[rs].ret), t[ls].rc + t[rs].lc);
}
void build(int i, int L, int R){
if (L == R){
scanf("%d", &t[i].ret);
t[i].c = t[i].lc = t[i].rc = t[i].ret;
return;
}
int mid = (L + R) >> 1;
build(lson);
build(rson);
pushup(i);
}
node query(int i, int L, int R, int l, int r){
if (l == L && R == r) return t[i];
int mid = (L + R) >> 1;
if (r <= mid) return query(lson, l, r);
if (l > mid) return query(rson, l, r);
node ta = query(lson, l, mid);
node tb = query(rson, mid + 1, r);
node ans;
ans.c = ta.c + tb.c;
ans.lc = max(ta.lc, ta.c + tb.lc);
ans.rc = max(tb.rc, tb.c + ta.rc);
ans.ret = max(max(ta.ret, tb.ret), ta.rc + tb.lc);
return ans;
}
int main(){
while (~scanf("%d", &n)){
build(1, 1, n);
scanf("%d", &q);
while (q--){
scanf("%d%d", &l, &r);
node ans = query(1, 1, n, l, r);
printf("%d\n", ans.ret);
}
}
return 0;
}
$GSS2$
$GSS1$的去重版
什么意思呢
询问的形式还是完全一样的,就是子序列的价值稍微变了一下。
$GSS1$中子序列的价值是该子序列的元素和,$GSS2$这里是该子序列出现过的元素的和
做法和$GSS1$完全不一样。
我们对询问离线。然后排序,按照右端点升序。
然后从左往右扫过去,每次加入$a_{i}$的时候, 先把$a_{i}$加进去。
接着我们对$last[a_{i}] + 1$到$i-1$这段区间都加上$a_{i}$,$last[x]$表示$x$上一次出现的位置,初值为$0$。
维护线段树的话我们需要四个信息。
$now$:当前这个时候区间的最大值
$mx$:历史区间的最大值
$h$:历史lazy标记(传给后代)的最大值
$add$:当前lazy标记(需传给后代)的值
然后查询一遍就可以了。
#include <bits/stdc++.h>
using namespace std;
#define rep(i, a, b) for (int i(a); i <= (b); ++i)
#define dec(i, a, b) for (int i(a); i >= (b); --i)
#define lson i << 1, L, mid
#define rson i << 1 | 1, mid + 1, R
#define ls i << 1
#define rs i << 1 | 1
#define MP make_pair
#define fi first
#define se second
typedef long long LL;
const int N = 1e5 + 10;
const int d = 1e5;
int n, m, x, y;
int a[N], c[N << 1];
vector <pair <int, int> > v[N];
LL mx[N << 2], now[N << 2], add[N << 2], h[N << 2];
LL ans[N];
void pushup(int i){
mx[i] = max(mx[ls], mx[rs]);
now[i] = max(now[ls], now[rs]);
}
void pushdown(int i){
h[ls] = max(h[ls], add[ls] + h[i]);
h[rs] = max(h[rs], add[rs] + h[i]);
add[ls] += add[i];
add[rs] += add[i];
mx[ls] = max(mx[ls], now[ls] + h[i]);
mx[rs] = max(mx[rs], now[rs] + h[i]);
now[ls] += add[i];
now[rs] += add[i];
add[i] = 0;
h[i] = -1e16;
}
void update(int i, int L, int R, int x, int val){
if (L == R && L == x){
mx[i] = now[i] = val;
return;
}
int mid = (L + R) >> 1;
pushdown(i);
if (x <= mid) update(lson, x, val);
else update(rson, x, val);
pushup(i);
}
void Add(int i, int L, int R, int l, int r, int val){
if (l <= L && R <= r){
now[i] += val;
mx[i] = max(mx[i], now[i]);
add[i] += val;
h[i] = max(h[i], add[i]);
return ;
}
int mid = (L + R) >> 1;
pushdown(i);
if (l <= mid) Add(lson, l, r, val);
if (r > mid) Add(rson, l, r, val);
pushup(i);
}
LL query(int i, int L, int R, int l, int r){
if (l <= L && R <= r) return mx[i];
int mid = (L + R) >> 1;
LL ret = -1e16;
pushdown(i);
if (l <= mid) ret = max(ret, query(lson, l, r));
if (r > mid) ret = max(ret, query(rson, l, r));
pushup(i);
return ret;
}
int main(){
scanf("%d", &n);
rep(i, 1, n) scanf("%d", a + i);
scanf("%d", &m);
rep(i, 1, ((n << 2) - 1)) mx[i] = h[i] = now[i] = -1e16;
rep(i, 1, m){
scanf("%d%d", &x, &y);
v[y].push_back(MP(x, i));
}
rep(i, 1, n){
update(1, 1, n, i, a[i]);
if (c[a[i] + d] + 1 < i) Add(1, 1, n, c[a[i] + d] + 1, i - 1, a[i]);
c[a[i] + d] = i;
for (auto u : v[i]) ans[u.se] = query(1, 1, n, u.fi, i);
}
rep(i, 1, m) printf("%lld\n", max(0ll, ans[i]));
return 0;
}
$GSS3$
$GSS1$加个单点更新,没了。
#include <bits/stdc++.h>
using namespace std;
#define rep(i, a, b) for (int i(a); i <= (b); ++i)
#define dec(i, a, b) for (int i(a); i >= (b); --i)
#define lson i << 1, L, mid
#define rson i << 1 | 1, mid + 1, R
#define ls i << 1
#define rs i << 1 | 1
typedef long long LL;
const int N = 1e5 + 10;
struct node{
int c, lc, rc, ret;
} t[N << 2];
int n, q, l, r;
int op, x, y;
void pushup(int i){
t[i].c = t[ls].c + t[rs].c;
t[i].lc = max(t[ls].lc, t[ls].c + t[rs].lc);
t[i].rc = max(t[rs].rc, t[rs].c + t[ls].rc);
t[i].ret = max(max(t[ls].ret, t[rs].ret), t[ls].rc + t[rs].lc);
}
void build(int i, int L, int R){
if (L == R){
scanf("%d", &t[i].ret);
t[i].c = t[i].lc = t[i].rc = t[i].ret;
return;
}
int mid = (L + R) >> 1;
build(lson);
build(rson);
pushup(i);
}
void update(int i, int L, int R, int x, int val){
if (L == x && L == R){
t[i].c = t[i].lc = t[i].rc = t[i].ret = val;
return;
}
int mid = (L + R) >> 1;
if (x <= mid) update(lson, x, val);
else update(rson, x, val);
pushup(i);
}
node query(int i, int L, int R, int l, int r){
if (l == L && R == r) return t[i];
int mid = (L + R) >> 1;
if (r <= mid) return query(lson, l, r);
if (l > mid) return query(rson, l, r);
node ta = query(lson, l, mid);
node tb = query(rson, mid + 1, r);
node ans;
ans.c = ta.c + tb.c;
ans.lc = max(ta.lc, ta.c + tb.lc);
ans.rc = max(tb.rc, tb.c + ta.rc);
ans.ret = max(max(ta.ret, tb.ret), ta.rc + tb.lc);
return ans;
}
int main(){
scanf("%d", &n);
build(1, 1, n);
scanf("%d", &q);
while (q--){
scanf("%d", &op);
if (op == 1){
scanf("%d%d", &l, &r);
node ans = query(1, 1, n, l, r);
printf("%d\n", ans.ret);
}
else{
scanf("%d%d", &x, &y);
update(1, 1, n, x, y);
}
}
return 0;
}
$GSS4$
这题做法很暴力
对那些值已经全部是1的区间结点打上永久标记即可。
#include <bits/stdc++.h>
using namespace std;
#define rep(i, a, b) for (int i(a); i <= (b); ++i)
#define dec(i, a, b) for (int i(a); i >= (b); --i)
#define ls i << 1
#define rs i << 1 | 1
#define lson ls, L, mid
#define rson rs, mid + 1, R
typedef long long LL;
const int N = 1e5 + 10;
LL s[N << 2];
int t[N << 2];
int n;
int q;
int ca = 0;
inline void pushup(int i){
s[i] = s[ls] + s[rs];
t[i] = t[ls] & t[rs];
}
void build(int i, int L, int R){
if (L == R){
scanf("%lld", s + i);
t[i] = s[i] <= 1;
return;
}
int mid = (L + R) >> 1;
build(lson);
build(rson);
pushup(i);
}
void update(int i, int L, int R, int l, int r){
if (t[i]) return;
if (L == R){
s[i] = sqrt(s[i]);
t[i] = s[i] <= 1;
return;
}
int mid = (L + R) >> 1;
if (l <= mid) update(lson, l, r);
if (r > mid) update(rson, l, r);
pushup(i);
}
LL query(int i, int L, int R, int l, int r){
if (l <= L && R <= r) return s[i];
int mid = (L + R) >> 1;
LL ret = 0;
if (l <= mid) ret += query(lson, l, r);
if (r > mid) ret += query(rson, l, r);
return ret;
}
int main(){
while (~scanf("%d", &n)){
printf("Case #%d:\n", ++ca);
build(1, 1, n);
scanf("%d", &q);
while (q--){
int op, x, y;
scanf("%d%d%d", &op, &x, &y);
if (x > y) swap(x, y);
if (op == 1) printf("%lld\n", query(1, 1, n, x, y));
else update(1, 1, n, x, y);
}
putchar(10);
}
return 0;
}
$GSS5$
分类讨论,注意细节就可以了。
#include <bits/stdc++.h> using namespace std; #define rep(i, a, b) for (int i(a); i <= (b); ++i) #define dec(i, a, b) for (int i(a); i >= (b); --i) #define lson i << 1, L, mid #define rson i << 1 | 1, mid + 1, R #define ls i << 1 #define rs i << 1 | 1 const int N = 1e5 + 10; struct node{ int c, lc, rc, ret; } t[N << 2]; int T; int n, q, l, r; int a[N], s[N]; int ans; void pushup(int i){ t[i].c = t[ls].c + t[rs].c; t[i].lc = max(t[ls].lc, t[ls].c + t[rs].lc); t[i].rc = max(t[rs].rc, t[rs].c + t[ls].rc); t[i].ret = max(max(t[ls].ret, t[rs].ret), t[ls].rc + t[rs].lc); } void build(int i, int L, int R){ if (L == R){ t[i].ret = a[L]; t[i].c = t[i].lc = t[i].rc = t[i].ret; return; } int mid = (L + R) >> 1; build(lson); build(rson); pushup(i); } node query(int i, int L, int R, int l, int r){ if (l == L && R == r) return t[i]; int mid = (L + R) >> 1; if (r <= mid) return query(lson, l, r); if (l > mid) return query(rson, l, r); node ta = query(lson, l, mid); node tb = query(rson, mid + 1, r); node ans; ans.c = ta.c + tb.c; ans.lc = max(ta.lc, ta.c + tb.lc); ans.rc = max(tb.rc, tb.c + ta.rc); ans.ret = max(max(ta.ret, tb.ret), ta.rc + tb.lc); return ans; } int main(){ scanf("%d", &T); while (T--){ scanf("%d", &n); rep(i, 1, n) scanf("%d", a + i); rep(i, 1, n) s[i] = s[i - 1] + a[i]; build(1, 1, n); scanf("%d", &q); while (q--){ int x1, y1, x2, y2; scanf("%d%d%d%d", &x1, &y1, &x2, &y2); ans = 0; if (x2 == y1){ ans = query(1, 1, n, x1, y1).rc + query(1, 1, n, x2, y2).lc - a[y1]; printf("%d\n", ans); } else if (y1 < x2){ ans = s[x2 - 1] - s[y1]; ans += query(1, 1, n, x1, y1).rc + query(1, 1, n, x2, y2).lc; printf("%d\n", ans); } else{ ans = query(1, 1, n, x2, y1).ret; ans = max(ans, s[y1 - 1] - s[x2] + query(1, 1, n, x1, x2).rc + query(1, 1, n, y1, y2).lc); ans = max(ans, query(1, 1, n, x1, x2).rc + query(1, 1, n, x2, y1).lc - a[x2]); ans = max(ans, query(1, 1, n, x2, y1).rc + query(1, 1, n, y1, y2).lc - a[y1]); printf("%d\n", ans); } } } return 0; }