ALT Q1. Two Sum HashMap stores value → index. Find complement (target - num) while traversing. Q2. 4Sum Sort array and fix first two numbers. Use two pointers for remaining pair, skip duplicates. Q3. Longest Consecutive Sequence Store all numbers in HashSet. Start from sequence beginnings and count forward.

ALT Q1. majority element use Boyer-Moore Voting to find up to two potential elements occurring more than n/3 times. Verify both candidates by counting their actual frequencies in a second traversal. Q2. unique paths From each cell, recursively explore moving right and down until reaching the destination. Use memoization (DP) to store previously computed results and avoid repeated calculations. Q3. reverse pairs Apply Merge Sort to divide the array into sorted halves recursively. During merging, use two pointers to count pairs satisfying arr[i] > 2 × arr[j] efficiently.

ALT Q1. Search a 2D Matrix: Treat the matrix as a sorted 1D array and perform Binary Search. Convert the middle index into row and column coordinates to access elements efficiently. T.C: O(log(m×n)), S.C: O(1). Q2. Pow(x, n): Use Binary Exponentiation by repeatedly squaring the base and processing exponent bits. Multiply the answer when the current bit is set and handle negative powers using reciprocals. T.C: O(log n), S.C: O(1). Q3. Majority Element: Apply Boyer-Moore Voting. Maintain a candidate and count; matching elements increase count, mismatches decrease it. The remaining candidate is guaranteed to be the majority element. T.C: O(n), S.C: O(1).

ALT 1. Find Duplicate Number → Floyd’s Cycle Detection (slow/fast pointers), treating the array as a linked list. O(n) time, O(1) space. 2. Missing & Repeated Value → HashSet to detect duplicate, then find the missing number. O(n²) elements, O(n²) space. 3. Inversion Count → Merge Sort; count inversions during merge when a right element goes for remaining left elements. O(n log n).

ALT 1. Merge Sorted Array Start from the end of both sorted arrays. Place the larger element at the last available position in nums[i]. 2. Merge Intervals Sort intervals by their start time. Merge intervals whenever the next interval overlaps with the current one. 3. Rotate Image Transpose the matrix (convert rows to columns). Reverse each row to achieve a 90° clockwise rotation.