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1. Explain how to implement two stacks in one array A[1..n] in such a way that neither stack overflows unless the total number of elements in both stacks together is n. The PUSH and POP operations should run in O(1) time. Justify your answer and give some examples.

1. Explain how to implement two stacks in one array A[1..n] in such a way that neither stack overflows unless the total number of elements in both stacks together is n. The PUSH and POP operations should run in O(1) time. Justify your answer and give some examples.

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
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Hello please answer the following question. Was really stuck on this one :(  Thank you so much!

**Implementing Two Stacks in a Single Array** To implement two stacks using a single array `A[1..n]`, you can use a two-pointer approach to ensure that neither stack overflows unless their combined total of elements is `n`. This setup also ensures that the push and pop operations run in O(1) time. **Approach:** 1. **Initialization:** - Use a single array `A[1..n]`. - Initialize two pointers: - `top1` starting at the beginning of the array (index 0). - `top2` starting at the end of the array (index n-1). 2. **Push Operation:** - **Stack 1 (`Push1`):** Add an element to the position `top1` and then increment `top1`. - **Stack 2 (`Push2`):** Add an element to the position `top2` and then decrement `top2`. 3. **Pop Operation:** - **Stack 1 (`Pop1`):** Decrement `top1` and then return the element at the new `top1` position. - **Stack 2 (`Pop2`):** Increment `top2` and then return the element at the new `top2` position. 4. **Overflow Condition:** - If `top1` exceeds `top2`, it means the array is full, and no more elements can be pushed into either stack. **Examples:** - **Example 1:** 1. Push an element onto Stack 1 by placing it at `A[top1]` and increment `top1`. 2. Push an element onto Stack 2 by placing it at `A[top2]` and decrement `top2`. 3. If `top1` becomes greater than `top2`, both stacks are full. - **Example 2:** - Start: `top1 = 0`, `top2 = n-1`. - Sequence: Push elements `1, 2, 3` onto Stack 1 and `10, 9, 8` onto Stack 2. - Final: `top1` could be `3`, `top2` could be `n-4` (assuming indices 3 and n-4 do not cross). This dual stack implementation using an array
Transcribed Image Text:**Implementing Two Stacks in a Single Array** To implement two stacks using a single array `A[1..n]`, you can use a two-pointer approach to ensure that neither stack overflows unless their combined total of elements is `n`. This setup also ensures that the push and pop operations run in O(1) time. **Approach:** 1. **Initialization:** - Use a single array `A[1..n]`. - Initialize two pointers: - `top1` starting at the beginning of the array (index 0). - `top2` starting at the end of the array (index n-1). 2. **Push Operation:** - **Stack 1 (`Push1`):** Add an element to the position `top1` and then increment `top1`. - **Stack 2 (`Push2`):** Add an element to the position `top2` and then decrement `top2`. 3. **Pop Operation:** - **Stack 1 (`Pop1`):** Decrement `top1` and then return the element at the new `top1` position. - **Stack 2 (`Pop2`):** Increment `top2` and then return the element at the new `top2` position. 4. **Overflow Condition:** - If `top1` exceeds `top2`, it means the array is full, and no more elements can be pushed into either stack. **Examples:** - **Example 1:** 1. Push an element onto Stack 1 by placing it at `A[top1]` and increment `top1`. 2. Push an element onto Stack 2 by placing it at `A[top2]` and decrement `top2`. 3. If `top1` becomes greater than `top2`, both stacks are full. - **Example 2:** - Start: `top1 = 0`, `top2 = n-1`. - Sequence: Push elements `1, 2, 3` onto Stack 1 and `10, 9, 8` onto Stack 2. - Final: `top1` could be `3`, `top2` could be `n-4` (assuming indices 3 and n-4 do not cross). This dual stack implementation using an array
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