Complete Merge Sort Note

lessons/mergesort.md

@@ -5,3 +5,193 @@ order: "8E"
 section: "Recursion"
 description: "learn Recursion from scratch"
 ---
+
+# 🧩 Merge Sort – Complete Notes
+
+> **Merge Sort** is a sorting algorithm based on the **Divide and Conquer** paradigm.  
+It divides the input array into smaller subarrays, recursively sorts them, and then merges the sorted subarrays to produce the final sorted array.
+
+---
+
+## 📌 Why Merge Sort?
+- Guaranteed performance
+- Stable sorting algorithm
+- Efficient for large datasets
+- Ideal for external sorting (files, linked lists)
+
+---
+
+## ⚙️ Core Idea (Divide and Conquer)
+
+1. **Divide** the array into two halves  
+2. **Recursively sort** each half  
+3. **Merge** the sorted halves  
+
+
+---
+
+## 🔍 Step-by-Step Example
+
+### Given Array
+[38, 27, 43, 3, 9, 82, 10]
+
+### Step 1: Divide Phase
+
+[38, 27, 43, 3] [9, 82, 10]
+
+[38, 27] [43, 3] [9, 82] [10]
+
+[38] [27] [43] [3] [9] [82] [10]
+
+---
+
+### Step 2: Merge Phase (Bottom-Up)
+
+[38] + [27] → [27, 38]
+[43] + [3] → [3, 43]
+[9] + [82] → [9, 82]
+
+
+[27, 38] + [3, 43] → [3, 27, 38, 43]
+[9, 82] + [10] → [9, 10, 82]
+
+Final Sorted Array:
+[3, 9, 10, 27, 38, 43, 82]
+
+### 🧠 Algorithm (Pseudocode)
+
+
+    MergeSort(arr):
+        if length(arr) ≤ 1:
+            return arr
+
+        mid = length(arr) / 2
+        left  = MergeSort(arr[0..mid])
+        right = MergeSort(arr[mid+1..end])
+
+        return Merge(left, right)
+
+
+### 🔗 Merge Procedure
+
+
+    Merge(left, right):
+        result = empty array
+
+        while left and right are not empty:
+            if left[0] ≤ right[0]:
+                add left[0] to result
+            else:
+                add right[0] to result
+
+        append remaining elements of left
+        append remaining elements of right
+
+        return result
+
+
+### 💻 Java Implementation
+
+
+    class MergeSort {
+
+        void merge(int[] arr, int left, int mid, int right) {
+            int n1 = mid - left + 1;
+            int n2 = right - mid;
+
+            int[] L = new int[n1];
+            int[] R = new int[n2];
+
+            for (int i = 0; i < n1; i++)
+                L[i] = arr[left + i];
+
+            for (int j = 0; j < n2; j++)
+                R[j] = arr[mid + 1 + j];
+
+            int i = 0, j = 0, k = left;
+
+            while (i < n1 && j < n2) {
+                if (L[i] <= R[j])
+                    arr[k++] = L[i++];
+                else
+                    arr[k++] = R[j++];
+            }
+
+            while (i < n1)
+                arr[k++] = L[i++];
+
+            while (j < n2)
+                arr[k++] = R[j++];
+        }
+
+        void sort(int[] arr, int left, int right) {
+            if (left < right) {
+                int mid = left + (right - left) / 2;
+
+                sort(arr, left, mid);
+                sort(arr, mid + 1, right);
+                merge(arr, left, mid, right);
+            }
+        }
+    }
+
+
+## ⏱️ Time and Space Complexity
+
+### ⏱️ Time Complexity
+
+| Case     | Complexity |
+|---------|------------|
+| Best    | O(n log n) |
+| Average | O(n log n) |
+| Worst   | O(n log n) |
+
+### 📦 Space Complexity
+
+| Property      | Value |
+|--------------|-------|
+| Extra Space  | O(n) |
+| In-place     | ❌ No |
+| Stable       | ✅ Yes |
+
+---
+
+## ✅ Advantages
+
+- Guaranteed performance
+- Stable sorting
+- Suitable for large datasets
+- Efficient for linked lists and external sorting
+
+---
+
+## ❌ Disadvantages
+
+- Requires additional memory
+- Slower than Quick Sort for small inputs
+- Not an in-place algorithm
+
+---
+
+## 🎯 Interview Quick Points
+
+- Based on **Divide and Conquer**
+- Always runs in **O(n log n)**
+- **Stable**, but **not in-place**
+- Preferred in **external sorting**
+
+---
+
+## 📌 When to Use Merge Sort?
+
+✔ Large datasets  
+✔ Stability is required  
+✔ Linked lists  
+✔ File-based data sorting  
+
+---
+
+## 📝 Summary
+
+> Merge Sort is a **stable, efficient, and predictable** sorting algorithm.  
+Its guaranteed time complexity makes it suitable for large-scale and real-world applications despite higher memory usage.