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+---
+id: two-stack-queue
+title: Two-Stack Queue Implementation
+sidebar_label: Two-Stack Queue
+sidebar_position: 10
+description: A guide to implementing a queue using two stacks with a lazy transfer approach for efficient operations.
+tags: [data structure, queue, stack]
+---
+
+# Two-Stack Queue Implementation
+
+This project implements a queue data structure using two stacks with a lazy transfer approach. The implementation provides O(1) amortized time complexity for both enqueue and dequeue operations.
+
+## Algorithm Overview
+
+The implementation uses two stacks:
+- `inStack`: Used for enqueueing elements
+- `outStack`: Used for dequeueing elements
+
+### Key Operations
+
+1. **Enqueue**: Push elements directly onto `inStack` - O(1)
+2. **Dequeue**: 
+    - If `outStack` is empty, transfer all elements from `inStack` to `outStack`
+    - Pop and return the top element from `outStack`
+    - Amortized O(1)
+
+## Implementation
+
+```c
+#include <stdio.h>
+#include <stdlib.h>
+#include <stdbool.h>
+
+#define MAX_SIZE 100
+
+typedef struct {
+     int items[MAX_SIZE];
+     int top;
+} Stack;
+
+typedef struct {
+     Stack* inStack;
+     Stack* outStack;
+} Queue;
+
+// Stack operations
+void initStack(Stack* s) {
+     s->top = -1;
+}
+
+bool isEmpty(Stack* s) {
+     return s->top == -1;
+}
+
+bool isFull(Stack* s) {
+     return s->top == MAX_SIZE - 1;
+}
+
+void push(Stack* s, int value) {
+     if (!isFull(s)) {
+          s->items[++s->top] = value;
+     }
+}
+
+int pop(Stack* s) {
+     if (!isEmpty(s)) {
+          return s->items[s->top--];
+     }
+     return -1;  // Error value
+}
+
+Queue* createQueue() {
+     Queue* q = (Queue*)malloc(sizeof(Queue));
+     q->inStack = (Stack*)malloc(sizeof(Stack));
+     q->outStack = (Stack*)malloc(sizeof(Stack));
+     initStack(q->inStack);
+     initStack(q->outStack);
+     return q;
+}
+
+void enqueue(Queue* q, int value) {
+     push(q->inStack, value);
+}
+
+int dequeue(Queue* q) {
+     if (isEmpty(q->outStack)) {
+          // Transfer elements from inStack to outStack
+          while (!isEmpty(q->inStack)) {
+                push(q->outStack, pop(q->inStack));
+          }
+     }
+     return pop(q->outStack);
+}
+
+bool isQueueEmpty(Queue* q) {
+     return isEmpty(q->inStack) && isEmpty(q->outStack);
+}
+
+void destroyQueue(Queue* q) {
+     free(q->inStack);
+     free(q->outStack);
+     free(q);
+}
+```
+
+## Usage Example
+
+```c
+int main() {
+     Queue* q = createQueue();
+     
+     enqueue(q, 1);
+     enqueue(q, 2);
+     enqueue(q, 3);
+     
+     printf("%d\n", dequeue(q));  // Output: 1
+     printf("%d\n", dequeue(q));  // Output: 2
+     
+     enqueue(q, 4);
+     
+     printf("%d\n", dequeue(q));  // Output: 3
+     printf("%d\n", dequeue(q));  // Output: 4
+     
+     destroyQueue(q);
+     return 0;
+}
+```
+### Complexity
+
+- **Time Complexity**:
+    - **Enqueue**: $O(1)$
+    - **Dequeue**: Amortized $O(1)$
+- **Space Complexity**: $O(n)$, where $n$ is the number of elements in the queue.
+
+### Example
+
+Consider a queue with the following operations:
+
+1. **Enqueue 1**
+2. **Enqueue 2**
+3. **Dequeue**
+4. **Enqueue 3**
+5. **Dequeue**
+
+**Operations Breakdown**:
+
+- **Enqueue 1**:
+    - `inStack`: [1]
+    - `outStack`: []
+
+- **Enqueue 2**:
+    - `inStack`: [1, 2]
+    - `outStack`: []
+
+- **Dequeue**:
+    - Transfer elements from `inStack` to `outStack`: 
+        - `inStack`: []
+        - `outStack`: [2, 1]
+    - Pop from `outStack`: 1
+    - Result:
+        - `inStack`: []
+        - `outStack`: [2]
+
+- **Enqueue 3**:
+    - `inStack`: [3]
+    - `outStack`: [2]
+
+- **Dequeue**:
+    - Pop from `outStack`: 2
+    - Result:
+        - `inStack`: [3]
+        - `outStack`: []
+
+### Conclusion
+
+Implementing a queue using two stacks is an efficient way to achieve the FIFO behavior with LIFO structures. This approach ensures that both the **enqueue** and **dequeue** operations have an amortized constant time complexity, making it suitable for applications where performance and resource management are critical.
+
+## Flowchart
+
+```mermaid
+flowchart TD
+    A[Start] --> B{Operation Type}
+    
+    B -->|Enqueue| C[Push to inStack]
+    C --> D[Done]
+    
+    B -->|Dequeue| E{outStack Empty?}
+    E -->|Yes| F[Transfer all elements from inStack to outStack]
+    E -->|No| G[Pop from outStack]
+    F --> G
+    G --> D
+    
+    subgraph Transfer Process
+    H[Pop from inStack] --> I[Push to outStack]
+    I --> J{inStack Empty?}
+    J -->|No| H
+    J -->|Yes| K[Transfer Complete]
+    end
+
+    D --> K
+    K --> L[End]
+```
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