What will happen if $$$r\geq2l$$$?

Suppose that $$$r\geq2l$$$, then at least we need to cover $$$[l,2l]$$$. Obviously, we must have $$$a>l$$$, since the length of the segment is already longer than $$$l$$$. Now if $$$l<a\leq2l$$$, there are at most $$$l$$$ modules of $$$a$$$ which are no less than $$$\frac{a}{2}$$$, which cannot cover the segment whose length is $$$l+1$$$. But if $$$a>2l$$$, then $$$l$$$ cannot be a good value.

On the contrary, if $$$r<2l$$$, we can always choose $$$a=2l$$$ which will be a valid answer.

So this problem can be simplified to judging whether $$$r<2l$$$ holds.

Time complexity is $$$O(1)$$$.

def read_int():
    return int(input())


def read_ints():
    return map(int, input().split(' '))


t = read_int()
for case_num in range(t):
    l, r = read_ints()
    print('YES' if l * 2 > r else 'NO')

How many $$$00$$$ or $$$11$$$ pairs can be reduced within one reverse?

We can reduce the number of $$$00$$$ or $$$11$$$ pairs by at most $$$2$$$ within one reverse. So we can simply count the total number of such pairs, and get the answer.

Time complexity is $$$O(N)$$$.

def read_int():
    return int(input())


def read_ints():
    return map(int, input().split(' '))


t = read_int()
for case_num in range(t):
    n = read_int()
    s = input()
    tot = 0
    for i in range(n - 1):
        if s[i] == s[i + 1]:
            tot += 1
    print((tot + 1) // 2)

1. We will never use numbers larger than $$$2n$$$.
2. The original sequence of the array does not matter.

We can sort the array first, then do simple dynamic programming, with $$$dp[last]$$$ meaning the total unpleasant values if the last oven is put out at $$$last$$$ minute.

Total time complexity is $$$O(N^3)$$$, which is enough to pass the time limit.

def read_int():
    return int(input())


def read_ints():
    return map(int, input().split(' '))


inf = int(1e9)
t = read_int()
for case_num in range(t):
    n = read_int()
    a = list(read_ints())
    a.sort()
    dp = [0 for _ in range(n * 2 + 1)]
    for t in a:
        ndp = [inf for _ in range(n * 2 + 1)]
        lo = inf
        for i in range(n * 2):
            lo = min(lo, dp[i])
            ndp[i + 1] = min(ndp[i + 1], lo + abs(i + 1 - t))
        dp = ndp
    print(min(dp))

Just go greedy.

We can arrange the nodes greedily. Unless the current number is smaller than the last one, we will keep connecting the numbers to the current parent node. Otherwise, we will move to the next parent node (which might be at the next level).

Time complexity is $$$O(N)$$$.

#include <cstdio>
#include <iostream>
#include <vector>

using namespace std;

template <typename T> void read(T &x) {
  x = 0;
  char c = getchar();
  T sig = 1;
  for (; !isdigit(c); c = getchar())
    if (c == '-')
      sig = -1;
  for (; isdigit(c); c = getchar())
    x = (x << 3) + (x << 1) + c - '0';
  x *= sig;
}

class Solution {
public:
  void solve() {
    int n;
    read(n);
    int height = 0;
    vector<int> level = {1};
    int col = 0, first, last = 0;
    read(first);
    for (int i = 0; i < n - 1; ++i) {
      int u;
      read(u);
      if (u < last) {
        col++;
        if (col == level[height]) {
          height++;
          col = 0;
        }
      }
      if (level.size() <= height + 1)
        level.emplace_back(0);
      level[height + 1]++;
      last = u;
    }
    printf("%d\n", (int)level.size() - 1);
  }
};

int main() {
  ios::sync_with_stdio(false);
  cin.tie(0);
  int t;
  read(t);
  while (t--) {
    Solution solution = Solution();
    solution.solve();
  }
}

1. We can add two sentinel elements at either end of $$$a$$$ and $$$b$$$. Then the original problem will be split into several subtasks.
2. For each subtask, we actually need to find the longest increasing subsequence.

The first step is to split the original problem into subtasks. Each subtask has four parameters, $$$l$$$ and $$$r$$$ for the start and the end of the interval, and $$$lo$$$ and $$$hi$$$ for the minimum valid value of the left end, and the maximum valid value of the right end.

Then for each subtask, we have three situations:

• $$$l>r$$$, which means the interval is empty. The answer is $$$0$$$.
• $$$l=r$$$, which means the length of the interval is $$$1$$$. The answer depends on whether $$$a[l]$$$ fits in $$$[lo,hi]$$$.
• $$$l<r$$$. In this case, we need to find the longest increasing subsequence. However, there are two differences compared to the classical LIS problem. One is that, if the current number does not fit in its valid range (which can be calculated using $$$lo$$$, $$$hi$$$, and $$$i$$$), it cannot be used in the LIS. The other difference is that we cannot use $$$a[i]$$$ directly. Considering the following case, $$$[2,5,3,1]$$$. Can we make an LIS with $$$[2,3]$$$? In this problem, we cannot, because there will be no room for the middle elements. How can we take the distance into consideration? We can use $$$a[i]-i$$$ instead of $$$a[i]$$$, and now we will find the longest non-decreasing subsequence instead of the longest increasing subsequence.

The algorithm for LIS is a classical DP enhanced with binary search, which takes $$$O(L\log L)$$$ time for an interval of length $$$L$$$.

So we have the total time complexity of $$$O(N\log N)$$$.

#include <algorithm>
#include <cstdio>
#include <iostream>
#include <vector>
#define INF 0x3f3f3f3f

using namespace std;

template <typename T> void read(T &x) {
  x = 0;
  char c = getchar();
  T sig = 1;
  for (; !isdigit(c); c = getchar())
    if (c == '-')
      sig = -1;
  for (; isdigit(c); c = getchar())
    x = (x << 3) + (x << 1) + c - '0';
  x *= sig;
}

class Solution {
public:
  void solve() {
    int n, k;
    read(n), read(k);
    vector<int> a(n + 2), b(k + 2);
    a[0] = -INF, a[n + 1] = INF;
    b[0] = 0, b[k + 1] = n + 1;
    for (int i = 1; i <= n; ++i)
      read(a[i]);
    for (int i = 1; i <= k; ++i) {
      read(b[i]);
      if (a[b[i]] - a[b[i - 1]] < b[i] - b[i - 1]) {
        printf("-1");
        return;
      }
    }
    int ans = 0;
    auto handle = [&](int l, int r, int lo, int hi) {
      if (l > r)
        return 0;
      if (l == r)
        return (a[l] >= lo && a[l] <= hi) ? 0 : 1;
      vector<int> LIS;
      for (int i = l; i <= r; ++i) {
        int clo = lo + i - l, chi = hi - r + i;
        if (a[i] < clo || a[i] > chi)
          continue;
        int pos = upper_bound(LIS.begin(), LIS.end(), a[i] - i) - LIS.begin();
        if (pos >= LIS.size())
          LIS.emplace_back(a[i] - i);
        else
          LIS[pos] = a[i] - i;
      }
      return r - l + 1 - (int)LIS.size();
    };
    for (int i = 1; i <= k + 1; ++i)
      ans += handle(b[i - 1] + 1, b[i] - 1, a[b[i - 1]] + 1, a[b[i]] - 1);
    printf("%d", ans);
  }
};

int main() {
  ios::sync_with_stdio(false);
  cin.tie(0);
  Solution solution = Solution();
  solution.solve();
}

...

#include <algorithm>
#include <cstdio>
#include <iostream>
#include <vector>
#define MOD 998244353

using namespace std;
typedef long long ll;

template <typename T> void read(T &x) {
  x = 0;
  char c = getchar();
  T sig = 1;
  for (; !isdigit(c); c = getchar())
    if (c == '-')
      sig = -1;
  for (; isdigit(c); c = getchar())
    x = (x << 3) + (x << 1) + c - '0';
  x *= sig;
}

class Solution {
public:
  void solve() {
    int n;
    read(n);
    vector<int> a(n);
    for (int i = 0; i < n; ++i)
      read(a[i]);
    sort(a.begin(), a.end());
    if (a[n - 2] * 2 > a[n - 1]) {
      printf("0");
      return;
    }

    vector<int> pre(n, -1);
    int l = n - 2;
    for (int r = n - 1; r >= 0; --r) {
      while (l >= 0 && a[l] * 2 > a[r])
        l--;
      if (l >= 0)
        pre[r] = l;
    }

    ll ans = 0;
    vector<ll> dp(n, 1), S(n + 1);
    for (int i = 1; i <= n; ++i)
      S[i] = i;
    for (int i = 1; i < n; ++i) {
      vector<ll> ndp(n, 0);
      for (int j = 0; j < n; ++j) {
        // Case 1: j to j
        int left = pre[j] == -1 ? 0 : pre[j] + 1;
        left -= i - 1;
        if (left > 0)
          ndp[j] = (ndp[j] + dp[j] * left % MOD);

        // Case 2: smaller to j
        if (pre[j] != -1)
          ndp[j] = (ndp[j] + S[pre[j] + 1]) % MOD;
      }
      for (int j = 1; j <= n; ++j)
        S[j] = (S[j - 1] + ndp[j - 1]) % MOD;
      dp = move(ndp);
    }
    printf("%lld", S[n]);
  }
};

int main() {
  ios::sync_with_stdio(false);
  cin.tie(0);
  Solution solution = Solution();
  solution.solve();
}

...

#include <iostream>
#include <queue>
#include <set>
#include <unordered_map>
#include <vector>

using namespace std;
struct Node {
  int idx = -1, fail = 0, children[26]{};
};

int main() {
  int n, q;
  cin >> n >> q;

  vector<string> names(n + 1);
  unordered_map<string, int> dict;
  unordered_map<int, int> id_dict;
  vector<set<pair<int, int>, greater<>>> heaps;
  vector<int> suspicion(n + 1);
  suspicion[0] = -1;
  vector<Node> nodes = {Node{}};
  int idx = 0;
  for (int i = 1; i <= n; ++i) {
    cin >> names[i];
    if (dict.count(names[i])) {
      heaps[dict[names[i]]].emplace(0, i);
      id_dict[i] = dict[names[i]];
      continue;
    }
    dict[names[i]] = idx;
    id_dict[i] = idx;
    heaps.push_back({{0, i}});
    int p = 0;
    for (char c : names[i]) {
      if (!nodes[p].children[c - 'a']) {
        nodes[p].children[c - 'a'] = nodes.size();
        nodes.emplace_back(Node{});
      }
      p = nodes[p].children[c - 'a'];
    }
    nodes[p].idx = idx++;
  }

  queue<int> que, visited;
  vector<bool> vis(nodes.size());
  for (int u : nodes[0].children)
    if (u)
      que.push(u);
  while (!que.empty()) {
    int u = que.front();
    que.pop();
    for (int i = 0; i < 26; ++i) {
      auto &v = nodes[u].children[i];
      if (v) {
        nodes[v].fail = nodes[nodes[u].fail].children[i];
        que.push(v);
      } else
        v = nodes[nodes[u].fail].children[i];
    }
  }

  string output;

  auto query = [&](string &s) {
    int ans = -1;
    int p = 0;
    int idx = 0;
    while (idx < s.size()) {
      char c = s[idx];
      if (nodes[p].children[c - 'a']) {
        p = nodes[p].children[c - 'a'];
        if (!vis[p]) {
          vis[p] = true;
          que.push(p);
        }
        idx++;
      } else {
        p = nodes[p].fail;
        if (!p)
          idx++;
      }
    }
    while (!que.empty()) {
      int u = que.front();
      que.pop();
      visited.push(u);
      if (nodes[u].idx != -1)
        ans = max(ans, heaps[nodes[u].idx].begin()->first);
      if (nodes[u].fail && !vis[nodes[u].fail]) {
        vis[nodes[u].fail] = true;
        que.push(nodes[u].fail);
      }
    }
    while (!visited.empty()) {
      vis[visited.front()] = false;
      visited.pop();
    }
    output += to_string(ans) + "\n";
  };

  while (q--) {
    int t;
    cin >> t;
    if (t == 1) {
      int i, x;
      cin >> i >> x;
      heaps[id_dict[i]].erase({suspicion[i], i});
      suspicion[i] = x;
      heaps[id_dict[i]].emplace(suspicion[i], i);
    } else {
      string s;
      cin >> s;
      query(s);
    }
  }

  cout << output;
}