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Use induction to prove that the number of elements in any SList of depth p is 2P.

Use induction to prove that the number of elements in any SList of depth p is 2P.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Title: Proof by Induction for Elements in an SList of Depth p** **Objective:** Use mathematical induction to demonstrate that the number of elements in any SList (Skip List) of depth \( p \) is \( 2^p \). **Introduction:** In computer science, a Skip List is a data structure that allows for efficient search, insertion, and deletion operations. The Skip List can be seen as a layered, linked list structure with multiple levels. In this exercise, we aim to prove that the number of elements in a Skip List of depth \( p \) is \( 2^p \). **Proof by Induction:** 1. **Base Case:** - Consider the simplest Skip List of depth \( p = 0 \). - At this level, there is just one element. - Therefore, the number of elements \( = 2^0 = 1 \). - Thus, the base case holds. 2. **Inductive Step:** - Assume that for a Skip List of depth \( k \), the number of elements is \( 2^k \). - Now, consider a Skip List of depth \( k+1 \). - At this new level, each element on level \( k \) is linked to two additional elements in level \( k+1 \). - Thus, the number of elements at level \( k+1 \) is twice the number at level \( k \), hence \( 2 \times 2^k = 2^{k+1} \). - Thus, by the principle of mathematical induction, the number of elements in any Skip List of depth \( p \) is \( 2^p \). **Conclusion:** Through our inductive proof, we conclusively show that a Skip List of depth \( p \) contains \( 2^p \) elements, corroborating the efficiency and scalability of Skip Lists in data organization.
Transcribed Image Text:**Title: Proof by Induction for Elements in an SList of Depth p** **Objective:** Use mathematical induction to demonstrate that the number of elements in any SList (Skip List) of depth \( p \) is \( 2^p \). **Introduction:** In computer science, a Skip List is a data structure that allows for efficient search, insertion, and deletion operations. The Skip List can be seen as a layered, linked list structure with multiple levels. In this exercise, we aim to prove that the number of elements in a Skip List of depth \( p \) is \( 2^p \). **Proof by Induction:** 1. **Base Case:** - Consider the simplest Skip List of depth \( p = 0 \). - At this level, there is just one element. - Therefore, the number of elements \( = 2^0 = 1 \). - Thus, the base case holds. 2. **Inductive Step:** - Assume that for a Skip List of depth \( k \), the number of elements is \( 2^k \). - Now, consider a Skip List of depth \( k+1 \). - At this new level, each element on level \( k \) is linked to two additional elements in level \( k+1 \). - Thus, the number of elements at level \( k+1 \) is twice the number at level \( k \), hence \( 2 \times 2^k = 2^{k+1} \). - Thus, by the principle of mathematical induction, the number of elements in any Skip List of depth \( p \) is \( 2^p \). **Conclusion:** Through our inductive proof, we conclusively show that a Skip List of depth \( p \) contains \( 2^p \) elements, corroborating the efficiency and scalability of Skip Lists in data organization.
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