To solve this problem efficiently, we simulate the sequence of operations while maintaining two values: the current number of ice cream packs and the count of distressed children. We start with $$$x$$$ packs and iterate through $$$n$$$ operations. If a carrier ($$$+d$$$) arrives, we increase our stock by $$$d$$$. If a child ($$$-d$$$) requests d packs, we check if we have enough ice cream. If yes, we serve the request and decrease our stock; otherwise, we increment the count of distressed kids. By processing all operations sequentially, we determine the final ice cream count and the number of distressed kids. This approach runs in $$$O(n)$$$ time, making it efficient given the constraints.

def solve():
    n, cur = map(int, input().split())
    distress = 0
    
    for _ in range(n):
        op, d = input().split()
        d = int(d)
        
        if op == '+':
            cur += d
        elif cur >= d:
            cur -= d
        else:
            distress += 1
    
    print(cur, distress)

To solve this problem, we need to construct a sequence $$$b$$$ that satisfies the given conditions. The key idea is to ensure that $$$b$$$ is strictly increasing and does not contain any elements from $$$a$$$.

Steps to Solve: Initialize a variable \t{Last_elem} to $$$0$$$, which will keep track of the last value added to the sequence $$$b$$$. Iterate through each element in the sequence $$$a$$$. For each element $$$a_i$$$: Increment \t{Last_elem} by $$$1$$$. If \t{Last_elem} is equal to $$$a_i$$$, increment \t{Last_elem} by $$$1$$$ again to ensure $$$b_i \neq a_i$$$. After processing all elements, \t{Last_elem} will hold the minimum value of $$$b_n$$$.

t = int(input())
for _ in range(t):
    n = int(input())
    a = list(map(int, input().split()))
    last_elem = 0
    for num in a:
        last_elem += 1
        if last_elem == num:
            last_elem += 1
    print(last_elem)

We need the deque data structure to solve this problem. We iterate the given permutation starting from the first element and we append to the left of our deque either if it is empty or the element at the 0-th index of our deque is larger than the current element we are about to add to our deque. Otherwise we append to the end of our deque because appending a larger number than what we currently have at the 0-th index of our deque will definitly make our final answer lexicographically larger.

from collections import deque

t = int(input())
for _ in range(t):
    n = int(input())
    arr = list(map(int, input().split()))

    ans = deque()
    for num in arr:
        if ans and num < ans[0]:
            ans.appendleft(num)
        else:
            ans.append(num)
    
    print(*ans)

• Think of a queue (FIFO structure) to store the most recent k unique chats.
• Use a set to quickly check if a friend's chat is already on the screen.

When a new chat arrives and the screen is full, remove the oldest chat before adding the new one.

The problem requires maintaining the most recent k unique chat messages in order.
Key observations:
1. If a friend sends multiple messages, only the first occurrence matters.
2. If the chat screen is full (k elements), the oldest message is removed when a new unique message arrives.

Approach:

• Use a deque (chat_screen) to store recent unique chats while maintaining order.
• Use a set (seen) to track which friends are currently displayed.
• Process each message:
• If it's already in seen, ignore it.
• Otherwise:
  • If chat_screen is full, remove the oldest chat (last element).
  • Add the new chat to the front of chat_screen and update seen.
• Print the final chat list.

This method ensures an O(n) time complexity, which is efficient for large inputs.

from collections import deque

def chat_screen(n, k, messages):
    chat_screen = deque()
    seen = set()

    for id_i in messages:
        if id_i not in seen:
            if len(chat_screen) == k:
                removed = chat_screen.pop()
                seen.remove(removed)
            
            chat_screen.appendleft(id_i)
            seen.add(id_i)

    print(len(chat_screen))
    print(*chat_screen)

n, k = map(int, input().split())
messages = list(map(int, input().split()))
chat_screen(n, k, messages)

• Think of maintaining two lists to track subsequences ending in 0 and 1.
• A greedy approach helps in assigning characters to subsequences optimally.

• If a 1 appears, assign it to a subsequence that previously ended in 0.
• If a 0 appears, assign it to a subsequence that previously ended in 1.

The problem requires dividing the binary string into the minimum number of subsequences such that each follows an alternating 0-1 or 1-0 pattern.

1. Each character belongs to exactly one subsequence.
2. If a 1 appears, it should ideally continue a subsequence ending in 0, and vice versa.
3. We can efficiently track and assign subsequences using two lists:

• zero → Stores subsequences ending in 0.
• one → Stores subsequences ending in 1.

• Iterate through the binary string:

• If 1 is encountered:
  • Assign it to a subsequence from zero (if available).
  • Otherwise, create a new subsequence.
• If 0 is encountered:
  • Assign it to a subsequence from one (if available).
  • Otherwise, create a new subsequence.
• The maximum number used in our subsequences is the answer.

Complexity:

• Each character is processed once and assigned in O(n) time.
• Given the constraints, this approach is efficient.

import sys 

def solve():
    
    # Read input: length of binary string
    n = int(sys.stdin.readline().strip())
    
    # Convert input string into a list of integers (0s and 1s)
    nums = [int(i) for i in sys.stdin.readline().strip()]

    zero = []   # Tracks subsequences ending in 0
    one = []    # Tracks subsequences ending in 1
    ans = []    # Stores the assigned subsequence number for each character
    maxi = 0    # Keeps track of the total number of subsequences used

    # Process each character in the binary string
    for val in nums:
        if val == 1:  # If the current character is '1'
            if zero:
                # If there exists a subsequence ending in '0', use it
                temp = zero.pop()
            else:
                # Otherwise, create a new subsequence
                maxi += 1
                temp = maxi 
              
            # Mark that this subsequence now ends in '1'
            one.append(temp)
            ans.append(temp)

        else:  # If the current character is '0'
            if one:
                # If there exists a subsequence ending in '1', use it
                temp = one.pop()
            else:
                # Otherwise, create a new subsequence
                maxi += 1
                temp = maxi 

            # Mark that this subsequence now ends in '0'
            zero.append(temp)
            ans.append(temp)

    # Print the total number of subsequences used
    print(max(ans))
    
    # Print the assigned subsequence number for each character
    print(*ans)

q = int(sys.stdin.readline().strip())

for _ in range(q):
    solve()

If we have a list of subarrays where the element at index $$$i$$$ is the max, which subarrays should we check to be sufficient?

Checking subarrays which end or start at index $$$i$$$ is sufficient, so we can optimize our solution with this observation as the basis.

Let's look at the problem from the perspective of each $$$a_i$$$. We want to check whether the sum of the subarrays, where $$$a_i$$$ is the maximum element, exceeds $$$a_i$$$ or not.

Firstly, we must find out in which subarrays is $$$a_i$$$ the maximum. This involves finding the previous greater element index and the next greater element index of $$$i$$$, which can be done for all indices in $$$O(n)$$$ using monotonic stack. Take these indices as $$$x_i$$$, $$$y_i$$$.

Take $$$(j,k)$$$, which represents the sum of a subarray which starts at index $$$j$$$ and ends at index $$$k$$$, where $$$j∈[x_i +1, i]$$$, $$$k∈[i,y_i−1]$$$. If $$$(j,k)>a_i$$$, then $$$(j,i−1)+(i,i)+(i+1,k)>a_i$$$, giving us $$$(j,i−1)+(i+1,k)>0$$$. Hence, at least one of the subarrays, $$$(j,i−1)$$$ or $$$(i+1,k)$$$ has a sum greater than $$$0$$$, which implies that one of subarrays $$$(j,i)$$$, $$$(i,k)$$$ has sum greater than $$$a_i$$$, so only checking subarrays which start or end at index $$$i$$$ suffices.

To determine whether there exists a subarray where the sum exceeds its maximum value, we utilize a Monotonic Stack to efficiently find the previous greater element indices and Prefix Sums to enable quick subarray sum queries.

For any index $$$i$$$, the monotonic stack ensures that:

While processing $$$a_i$$$ , we pop all elements from the stack that are less than or equal to $$$a_i$$$. This guarantees that for all popped indices, $$$a_i$$$ is the first greater element to their right. For the current popped index $$$v$$$, we check the sum from $$$v$$$ to $$$(i-1)$$$ > $$$0$$$ If true, it implies that some subarray sum exceeds its maximum, violating the required condition. In this case, we immediately return $$$NO$$$ since at least one problematic subarray exists.

$$$Why$$$ $$$This$$$ $$$is$$$ $$$Sufficient?$$$

The monotonic stack guarantees that we only check valid subarrays where the current element is the maximum, ensuring:

No problematic subarray is missed.

Minimal subarray checks, avoiding an exhaustive search.

Since we apply the same check on both the original and reversed array, we ensure correctness across all possible subarrays.

$$$Final$$$ $$$Steps$$$

Process the array from left to right: Use a monotonic stack to track the next greater elements. As elements are popped, check if the suffix sum (from the popped index up to $$$i−1$$$) is greater than 0. If true, print $$$NO$$$ and terminate early. Reverse the array and repeat the process to handle Next greater elements efficiently.

Time Complexity: $$$O(n)$$$

import sys

def solve():
    n = int(sys.stdin.readline().strip())
    nums = list(map(int, sys.stdin.readline().strip().split()))

    def prevGreater(nums):
        stack = []
        pref = [0]
        
        for num in nums:
            pref.append(pref[-1] + num)
    
        for i in range(n):

            while stack and nums[stack[-1]] <= nums[i]:
                v = stack.pop()

                if  pref[i] - pref[v]> 0:
                    return False
                
            stack.append(i)

        return True
    
    if prevGreater(nums) and prevGreater(nums[::-1]):
        return "YES"
    
    return "NO"

t = int(sys.stdin.readline().strip())
for _ in range(t):
    print(solve())