Add explanation and solution for LeetCode problem 2119 (Minimum Number of Operations to Make Array Continuous)

explanations/2119/en.md

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+## 2119. Minimum Number of Operations to Make Array Continuous [Hard]
+
+https://leetcode.com/problems/minimum-number-of-operations-to-make-array-continuous
+
+## Description
+You are given an integer array `nums`. In one operation, you can replace **any** element in `nums` with **any** integer.
+
+`nums` is considered **continuous** if both of the following conditions are fulfilled:
+
+- All elements in `nums` are **unique**.
+- The difference between the **maximum** element and the **minimum** element in `nums` equals `nums.length - 1`.
+
+For example, `nums = [4, 2, 5, 3]` is **continuous**, but `nums = [1, 2, 3, 5, 6]` is **not continuous**.
+
+Return *the **minimum** number of operations to make* `nums` **continuous**.
+
+**Examples**
+
+```tex
+Example 1:
+Input: nums = [4,2,5,3]
+Output: 0
+Explanation: nums is already continuous.
+
+Example 2:
+Input: nums = [1,2,3,5,6]
+Output: 1
+Explanation: One possible solution is to change the last element to 4.
+The resulting array is [1,2,3,5,4], which is continuous.
+
+Example 3:
+Input: nums = [1,10,100,1000]
+Output: 3
+Explanation: One possible solution is to:
+- Change the second element to 2.
+- Change the third element to 3.
+- Change the fourth element to 4.
+The resulting array is [1,2,3,4], which is continuous.
+```
+
+**Constraints**
+```tex
+- 1 <= nums.length <= 10^5
+- 1 <= nums[i] <= 10^9
+```
+
+## Explanation
+
+### Strategy
+Let's restate the problem: You're given an array, and you need to find the minimum number of operations to make it continuous. A continuous array has unique elements where the difference between max and min equals the length minus 1.
+
+This is a **sliding window problem** that requires understanding what makes an array continuous and finding the optimal window of elements to keep.
+
+**What is given?** An array of integers that can be very large (up to 100,000 elements).
+
+**What is being asked?** Find the minimum operations to make the array continuous.
+
+**Constraints:** The array can be up to 100,000 elements long, with values up to 10⁹.
+
+**Edge cases:** 
+- Single element array
+- Already continuous array
+- Array with all identical values
+- Very large gaps between elements
+
+**High-level approach:**
+The solution involves sorting the array and using a sliding window approach to find the optimal subset of elements that can form a continuous array with minimal operations.
+
+**Decomposition:**
+1. **Sort the array**: This helps us identify potential continuous subsequences
+2. **Use sliding window**: Find the longest window that can be made continuous
+3. **Calculate operations**: Determine how many elements need to be changed
+4. **Find minimum**: Try different window sizes to find the optimal solution
+
+**Brute force vs. optimized strategy:**
+- **Brute force**: Try all possible subsets. This is extremely inefficient.
+- **Optimized**: Use sliding window with binary search. This takes O(n log n) time.
+
+### Steps
+Let's walk through the solution step by step using the second example: `nums = [1,2,3,5,6]`
+
+**Step 1: Sort the array**
+- Original: [1,2,3,5,6]
+- Sorted: [1,2,3,5,6] (already sorted)
+
+**Step 2: Understand what makes an array continuous**
+- For length 5, we need: max - min = 5 - 1 = 4
+- So we need elements that span exactly 4 values
+- Example: [1,2,3,4,5] has max=5, min=1, difference=4 ✓
+
+**Step 3: Use sliding window approach**
+- Start with window size = array length
+- Try to find a window that can be made continuous
+- For each window, calculate how many operations are needed
+
+**Step 4: Calculate operations for different windows**
+- **Window [1,2,3,5,6]**: 
+  - Current span: 6 - 1 = 5
+  - Need span: 5 - 1 = 4
+  - Gap: 5 - 4 = 1
+  - Operations needed: 1 (change 6 to 4)
+
+- **Window [1,2,3,5]**:
+  - Current span: 5 - 1 = 4
+  - Need span: 4 - 1 = 3
+  - Gap: 4 - 3 = 1
+  - Operations needed: 1 (change 5 to 3)
+
+- **Window [2,3,5,6]**:
+  - Current span: 6 - 2 = 4
+  - Need span: 4 - 1 = 3
+  - Gap: 4 - 3 = 1
+  - Operations needed: 1 (change 6 to 4)
+
+**Step 5: Find optimal solution**
+- Minimum operations: 1
+- Optimal window: [1,2,3,5] → change 5 to 4 → [1,2,3,4]
+
+**Why this works:**
+The sliding window approach works because:
+1. **Optimal substructure**: The best solution for a larger array must include the best solution for a smaller subarray
+2. **Monotonicity**: If a window of size k can be made continuous, then a window of size k-1 can also be made continuous
+3. **Efficiency**: We only need to check O(n) different window sizes instead of all possible subsets
+
+> **Note:** The key insight is that we can use a sliding window to find the longest subsequence that can be made continuous, and then calculate the minimum operations needed. This avoids the need to try all possible combinations.
+
+**Time Complexity:** O(n log n) - sorting takes O(n log n), sliding window takes O(n)  
+**Space Complexity:** O(1) - we only need a few variables for the sliding window

solutions/2119/01.py

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+def minOperations(nums):
+    """
+    Find the minimum number of operations to make the array continuous.
+
+    Args:
+        nums: List[int] - Array of integers
+
+    Returns:
+        int - Minimum number of operations needed
+    """
+    # Handle edge case
+    if len(nums) <= 1:
+        return 0
+
+    # Sort the array to work with ordered elements
+    nums.sort()
+
+    # Remove duplicates to work with unique elements
+    unique_nums = []
+    for i, num in enumerate(nums):
+        if i == 0 or num != nums[i-1]:
+            unique_nums.append(num)
+
+    n = len(unique_nums)
+    min_operations = float('inf')
+
+    # Try different window sizes
+    for i in range(n):
+        # For each starting position, find the longest window that can be made continuous
+        left = unique_nums[i]
+
+        # Binary search for the rightmost element that can be part of a continuous sequence
+        # starting from left
+        right = left + len(nums) - 1
+
+        # Find how many elements in unique_nums fall within [left, right]
+        # This gives us the size of the window that can be made continuous
+        count = 0
+        for j in range(i, n):
+            if unique_nums[j] <= right:
+                count += 1
+            else:
+                break
+
+        # Calculate operations needed
+        # We need to change (len(nums) - count) elements
+        operations = len(nums) - count
+        min_operations = min(min_operations, operations)
+
+    return min_operations