codeharborhub · ajay-dhangar · Jun 1, 2024 · Jun 1, 2024 · Jun 1, 2024 · Jun 1, 2024
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+---
+id: Median of Two Sorted Arrays
+title: Median of Two Sorted Arrays(Leetcode)
+sidebar_label: 0004 - Median of Two Sorted Arrays
+tags: 
+    - Array
+    - Binary Search
+    - Divide and Conquer
+description: Given two sorted arrays nums1 and nums2 of size m and n respectively, return the median of the two sorted arrays. The overall run time complexity should be O(log (m+n)).
+---
+
+
+## Problem Description
+
+| Problem Statement                                                                          | Solution Link                                                                                                                                      | LeetCode Profile                                   |
+| :----------------------------------------------------------------------------------------- | :------------------------------------------------------------------------------------------------------------------------------------------------- | :------------------------------------------------- |
+| [Median of Two Sorted Arrays on LeetCode](https://leetcode.com/problems/median-of-two-sorted-arrays/description/) | [Median of Two Sorted Arrays Solution on LeetCode](https://leetcode.com/problems/median-of-two-sorted-arrays/editorial/) | [gabaniyash846](https://leetcode.com/u/gabaniyash846/) |
+
+### Problem Statement
+
+Given two sorted arrays `nums1` and `nums2` of size `m` and `n` respectively, return the median of the two sorted arrays.
+
+The overall run time complexity should be `O(log (m+n))`.
+
+**Example 1:**
+```
+Input: nums1 = [1,3], nums2 = [2]
+Output: 2.00000
+Explanation: merged array = [1,2,3] and median is 2.
+```
+
+**Example 2:**
+```
+Input: nums1 = [1,2], nums2 = [3,4]
+Output: 2.50000
+Explanation: merged array = [1,2,3,4] and median is (2 + 3) / 2 = 2.5.
+```
+
+### Constraints
+- `nums1.length == m`
+- `nums2.length == n`
+- `0 <= m <= 1000`
+- `0 <= n <= 1000`
+- `1 <= m + n <= 2000`
+- `-10^6 <= nums1[i], nums2[i] <= 10^6`
+
+### Solution
+
+#### Approach 1: Merge Sort (O(m + n))
+
+**Algorithm:**
+1. Merge `nums1` and `nums2` into a single sorted array.
+2. Find the median of the merged array:
+   - If the total number of elements is odd, return the middle element.
+   - If the total number of elements is even, return the average of the two middle elements.
+
+**Python Implementation:**
+```python
+def findMedianSortedArrays(nums1, nums2):
+    merged = []
+    i = j = 0
+    while i < len(nums1) and j < len(nums2):
+        if nums1[i] < nums2[j]:
+            merged.append(nums1[i])
+            i += 1
+        else:
+            merged.append(nums2[j])
+            j += 1
+    merged.extend(nums1[i:])
+    merged.extend(nums2[j:])
+
+    n = len(merged)
+    if n % 2 == 1:
+        return merged[n // 2]
+    else:
+        return (merged[n // 2 - 1] + merged[n // 2]) / 2.0
+```
+
+#### Approach 2: Binary Search (O(log(min(m, n))))
+
+**Intuition:**
+To achieve O(log(min(m, n))) complexity, use binary search on the smaller array. The goal is to find a partition where the left half of both arrays combined is less than or equal to the right half.
+
+**Algorithm:**
+1. Ensure `nums1` is the smaller array.
+2. Define `imin` and `imax` for binary search on `nums1`.
+3. Calculate `partitionX` and `partitionY` such that:
+   - `partitionX + partitionY = (m + n + 1) // 2`
+4. Check conditions to find the correct partition:
+   - If `maxLeftX <= minRightY` and `maxLeftY <= minRightX`, you've found the correct partition.
+   - If `maxLeftX > minRightY`, move `imax` to the left.
+   - If `maxLeftY > minRightX`, move `imin` to the right.
+5. Calculate the median based on the combined lengths of the arrays.
+
+**Python Implementation:**
+```python
+def findMedianSortedArrays(nums1, nums2):
+    if len(nums1) > len(nums2):
+        nums1, nums2 = nums2, nums1
+
+    m, n = len(nums1), len(nums2)
+    imin, imax, half_len = 0, m, (m + n + 1) // 2
+
+    while imin <= imax:
+        partitionX = (imin + imax) // 2
+        partitionY = half_len - partitionX
+
+        maxLeftX = float('-inf') if partitionX == 0 else nums1[partitionX - 1]
+        minRightX = float('inf') if partitionX == m else nums1[partitionX]
+
+        maxLeftY = float('-inf') if partitionY == 0 else nums2[partitionY - 1]
+        minRightY = float('inf') if partitionY == n else nums2[partitionY]
+
+        if maxLeftX <= minRightY and maxLeftY <= minRightX:
+            if (m + n) % 2 == 1:
+                return max(maxLeftX, maxLeftY)
+            else:
+                return (max(maxLeftX, maxLeftY) + min(minRightX, minRightY)) / 2.0
+        elif maxLeftX > minRightY:
+            imax = partitionX - 1
+        else:
+            imin = partitionX + 1
+```
+
+### Complexity Analysis
+
+- **Time Complexity:**
+  - Approach 1: O(m + n) because it involves merging both arrays into one sorted array.
+  - Approach 2: O(log(min(m, n))) because it performs binary search on the smaller array.
+
+- **Space Complexity:**
+  - Approach 1: O(m + n) due to the additional space needed for the merged array.
+  - Approach 2: O(1) because it uses only a constant amount of additional space.
+
+### Summary
+
+Approach 1 is straightforward but not optimal in terms of time complexity for large input sizes. Approach 2 leverages binary search to efficiently find the median with logarithmic time complexity, making it suitable for large arrays.
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+---
+id: Longest Palindromic Substring
+title: Longest Palindromic Substring(LeetCode)
+sidebar_label:  0005-Longest Palindrome Substring
+tags: 
+    - String
+    -  Dynamic Programming
+description: Given a string s, return the longest palindromic substring in s.
+---
+
+## Problem Description
+
+| Problem Statement                                                                 | Solution Link                                                                                                                                          | LeetCode Profile                                   |
+| :-------------------------------------------------------------------------------- | :----------------------------------------------------------------------------------------------------------------------------------------------------- | :------------------------------------------------- |
+| [Longest Palindromic Substring on LeetCode](https://leetcode.com/problems/longest-palindromic-substring/description/) | [Longest Palindromic Substring Solution on LeetCode](https://leetcode.com/problems/longest-palindromic-substring/editorial/) | [gabaniyash846](https://leetcode.com/u/gabaniyash846/) |
+
+## Problem Statement
+
+Given a string `s`, return the longest palindromic substring in `s`.
+
+### Examples
+
+**Example 1:**
+```plaintext
+Input: s = "babad"
+Output: "bab"
+Explanation: "aba" is also a valid answer.
+```
+
+**Example 2:**
+```plaintext
+Input: s = "cbbd"
+Output: "bb"
+```
+
+### Constraints
+- `1 <= s.length <= 1000`
+- `s` consist of only digits and English letters.
+
+## Solution
+
+### Approach 1: Check All Substrings
+
+#### Intuition
+We can start with a brute-force approach. We will simply check if each substring is a palindrome, and take the longest one that is.
+
+#### Algorithm
+1. Create a helper method `check(i, j)` to determine if a substring is a palindrome.
+2. Use two pointers to iterate from the start and end of the substring towards the center.
+3. If characters at both pointers are equal, move the pointers inward.
+4. Use nested loops to check all possible substrings and use the helper method to determine if they are palindromes.
+5. Keep track of the longest palindrome found.
+
+#### Implementation
+```python
+def longestPalindrome(s: str) -> str:
+    def check(l, r):
+        while l >= 0 and r < len(s) and s[l] == s[r]:
+            l -= 1
+            r += 1
+        return s[l+1:r]
+
+    longest = ""
+    for i in range(len(s)):
+        odd_palindrome = check(i, i)
+        even_palindrome = check(i, i + 1)
+        longest = max(longest, odd_palindrome, even_palindrome, key=len)
+
+    return longest
+```
+
+### Complexity Analysis
+- **Time complexity**: `O(n^3)` - We check each substring, and checking each substring takes `O(n)` time.
+- **Space complexity**: `O(1)` - We use a constant amount of extra space.
+
+### Approach 2: Dynamic Programming
+
+#### Intuition
+If we know a substring is a palindrome, we can extend it if the characters on both ends match.
+
+#### Algorithm
+1. Initialize a 2D DP table with `False`.
+2. Mark all single-character substrings as palindromes.
+3. Check substrings of length 2 and mark them if characters match.
+4. Use the table to extend palindromes to longer substrings.
+5. Keep track of the longest palindrome found.
+
+#### Implementation
+```python
+def longestPalindrome(s: str) -> str:
+    n = len(s)
+    dp = [[False] * n for _ in range(n)]
+    longest = ""
+
+    for i in range(n):
+        dp[i][i] = True
+        longest = s[i]
+
+    for i in range(n-1):
+        if s[i] == s[i+1]:
+            dp[i][i+1] = True
+            longest = s[i:i+2]
+
+    for length in range(3, n+1):
+        for i in range(n-length+1):
+            j = i + length - 1
+            if s[i] == s[j] and dp[i+1][j-1]:
+                dp[i][j] = True
+                longest = s[i:j+1]
+
+    return longest
+```
+
+### Complexity Analysis
+- **Time complexity**: `O(n^2)` - We fill an `n * n` table.
+- **Space complexity**: `O(n^2)` - We use an `n * n` table to store the DP results.
+
+### Approach 3: Expand From Centers
+
+#### Intuition
+A palindrome mirrors around its center. We can expand around potential centers to find palindromes.
+
+#### Algorithm
+1. Consider each character and pair of characters as potential centers.
+2. Expand around each center to find the longest palindrome.
+3. Keep track of the longest palindrome found.
+
+#### Implementation
+```python
+def longestPalindrome(s: str) -> str:
+    def expandAroundCenter(s, left, right):
+        while left >= 0 and right < len(s) and s[left] == s[right]:
+            left -= 1
+            right += 1
+        return s[left + 1:right]
+
+    if not s:
+        return ""
+
+    longest = s[0]
+
+    for i in range(len(s)):
+        odd_palindrome = expandAroundCenter(s, i, i)
+        even_palindrome = expandAroundCenter(s, i, i + 1)
+        longest = max(longest, odd_palindrome, even_palindrome, key=len)
+
+    return longest
+```
+
+### Complexity Analysis
+- **Time complexity**: `O(n^2)` - We expand around each center, which takes `O(n)` time.
+- **Space complexity**: `O(1)` - We use a constant amount of extra space.
+
+### Approach 4: Manacher's Algorithm
+
+#### Intuition
+Manacher's algorithm can find the longest palindromic substring in linear time.
+
+#### Algorithm
+1. Transform the string to handle even-length palindromes.
+2. Use a helper array to store the length of the palindrome at each position.
+3. Use the properties of palindromes and the helper array to efficiently find the longest palindrome.
+
+#### Implementation
+```python
+def longestPalindrome(s: str) -> str:
+    if not s:
+        return ""
+
+    # Transform s into new string with inserted boundaries
+    T = '#'.join(f"^{s}$")
+    n = len(T)
+    P = [0] * n
+    C = R = 0
+
+    for i in range(1, n - 1):
+        P[i] = (R > i) and min(R - i, P[2 * C - i])
+        while T[i + P[i] + 1] == T[i - P[i] - 1]:
+            P[i] += 1
+        if i + P[i] > R:
+            C, R = i, i + P[i]
+
+    max_len, center_index = max((n, i) for i, n in enumerate(P))
+    return s[(center_index - max_len) // 2: (center_index + max_len) // 2]
+```
+
+### Complexity Analysis
+- **Time complexity**: `O(n)` - Manacher's algorithm runs in linear time.
+- **Space complexity**: `O(n)` - We use an array of length `n` to store the lengths of palindromes.
+
+### Conclusion
+We discussed four approaches to solve the "Longest Palindromic Substring" problem, each with varying complexities and trade-offs. The dynamic programming and center expansion approaches provide a good balance of simplicity and efficiency for practical use cases, while Manacher's algorithm offers the optimal theoretical performance.
+```
+