Merge pull request #1103 from PradnyaGaitonde/PradnyaGaitonde-patch-1

Create 0034-Find-first-and-last-position-in-sorted-array.md

Author: ajay-dhangar

dsa-solutions/lc-solutions/0000-0099/0034-Find-first-and-last-position-in-sorted-array.md

@@ -0,0 +1,170 @@
+---
+id: find-first-and-last-position-of-element-in-sorted-array
+title: Find First and Last Position of Element in Sorted Array(LeetCode)
+sidebar_label: 0034-Find First and Last Position of Element in Sorted Array
+tags:
+  - Array
+  - Binary Search
+description: Given an array of integers nums sorted in non-decreasing order, find the starting and ending position of a given target value.
+sidebar_position: 34
+---
+
+## Problem Statement
+
+Given an array of integers `nums` sorted in non-decreasing order, find the starting and ending position of a given `target` value.
+
+If `target` is not found in the array, return [-1, -1].
+
+You must write an algorithm with O(log n) runtime complexity.
+
+### Examples
+
+**Example 1:**
+
+```plaintext
+Input: nums = [5,7,7,8,8,10], target = 8
+Output: [3,4]
+```
+
+**Example 2:**
+
+```plaintext
+Input: nums = [5,7,7,8,8,10], target = 6
+Output: [-1,-1]
+```
+
+**Example 3:**
+
+```plaintext
+Input: nums = [], target = 0
+Output: [-1,-1]
+```
+
+### Constraints
+
+- `0 <= nums.length <= 105`
+- `109 <= nums[i] <= 109`
+- `nums` is a non-decreasing array.
+- `109 <= target <= 109`
+
+## Solution
+
+When solving the problem of finding the starting and ending position of a target value in a sorted array, we can use 
+two main approaches: Linear Search (Brute Force) and Binary Search (Optimized). Below, both approaches are explained along with their time and space complexities.
+
+### Approach 1: Linear Search (Brute Force)
+
+#### Explanation:
+
+1. Traverse the array from the beginning to find the first occurrence of the target.
+2. Traverse the array from the end to find the last occurrence of the target.
+3. Return the indices of the first and last occurrences.
+
+#### Algorithm
+
+1. Initialize `startingPosition` and `endingPosition` to -1.
+2. Loop through the array from the beginning to find the first occurrence of the target and set `startingPosition`.
+3. Loop through the array from the end to find the last occurrence of the target and set `endingPosition`.
+4. Return `{startingPosition, endingPosition}`.
+
+#### Implementation
+
+```C++
+class Solution {
+public:
+    vector<int> searchRange(vector<int>& nums, int target) {
+        int startingPosition = -1, endingPosition = -1;
+        int n = nums.size();
+        for(int i = 0; i < n; i++) {
+            if(nums[i] == target) {
+                startingPosition = i;
+                break;
+            }
+        }
+        for(int i = n - 1; i >= 0; i--) {
+            if(nums[i] == target) {
+                endingPosition = i;
+                break;
+            }
+        }
+        return {startingPosition, endingPosition};
+    }
+};
+```
+
+### Complexity Analysis
+
+- **Time complexity**: O(N), where N is the size of the array. In the worst case, we might traverse all elements of the array.
+- **Space complexity**: O(1), as we are using a constant amount of extra space.
+
+### Approach 2: Binary Search (Optimized)
+
+#### Explanation:
+
+1. Use binary search to find the lower bound (first occurrence) of the target.
+2. Use binary search to find the upper bound (last occurrence) of the target by searching for the target + 1 and subtracting 1 from the result.
+3. Check if the target exists in the array and return the indices of the first and last occurrences.
+
+#### Algorithm
+
+1. Define a helper function `lower_bound` to find the first position where the target can be inserted.
+2. Use `lower_bound` to find the starting position of the target.
+3. Use `lower_bound` to find the position where `target + 1` can be inserted, then subtract 1 to get the ending position.
+4. Check if `startingPosition` is within bounds and equals the target.
+5. Return `{startingPosition, endingPosition}` if the target is found, otherwise return `{-1, -1}`.
+
+#### Implementation (First Version)
+
+```C++
+class Solution {
+private:
+    int lower_bound(vector<int>& nums, int low, int high, int target) {
+        while(low <= high) {
+            int mid = (low + high) >> 1;
+            if(nums[mid] < target) {
+                low = mid + 1;
+            } else {
+                high = mid - 1;
+            }
+        }
+        return low;
+    }
+public:
+    vector<int> searchRange(vector<int>& nums, int target) {
+        int low = 0, high = nums.size() - 1;
+        int startingPosition = lower_bound(nums, low, high, target);
+        int endingPosition = lower_bound(nums, low, high, target + 1) - 1;
+        if(startingPosition < nums.size() && nums[startingPosition] == target) {
+            return {startingPosition, endingPosition};
+        }
+        return {-1, -1};
+    }
+};
+```
+#### Implementation (Second Version)
+
+```C++
+class Solution {
+public:
+    vector<int> searchRange(vector<int>& nums, int target) {
+        int startingPosition = lower_bound(nums.begin(), nums.end(), target) - nums.begin();
+        int endingPosition = lower_bound(nums.begin(), nums.end(), target + 1) - nums.begin() - 1;
+        if(startingPosition < nums.size() && nums[startingPosition] == target) {
+            return {startingPosition, endingPosition};
+        }
+        return {-1, -1};
+    }
+};
+```
+
+### Complexity Analysis
+
+- **Time complexity**: O(log N), where N is the size of the array. We perform binary search, which has a logarithmic time complexity.
+- **Space complexity**: O(1), as we are using a constant amount of extra space.
+
+### Conclusion
+
+1. Linear Search is straightforward but less efficient with a time complexity of O(N).
+2. Binary Search is more efficient with a time complexity of O(log N), making it suitable for large datasets.
+Both approaches provide a clear way to find the starting and ending positions of a target value in a sorted array, with the Binary Search approach being the
+optimized solution.