**Trisection Search Algorithm Explanation** Consider the algorithm referred to as trisection search. Trisection search determines whether a specified integer \( x \) is part of a sorted list \( L \) of integers. Assume that the length of \( L \) is \( 3^p \) for some integer \( p \). ### Algorithm: TrisectionSearch(\( x, L \)) 1. If \( L \) has a length of 1, return `True` if the single element equals \( x \), and return `False` otherwise. 2. Divide \( L \) into three equal sections, named \( L_1, L_2, \) and \( L_3 \). Denote the last elements of these lists by \( \ell_1, \ell_2, \ell_3 \), respectively. - If \( x \leq \ell_1 \), then return TrisectionSearch(\( x, L_1 \)). - Else, if \( \ell_1 < x \leq \ell_2 \), then return TrisectionSearch(\( x, L_2 \)). - Else, if \( \ell_2 < x \), then return TrisectionSearch(\( x, L_3 \)). ### Part A Provide a big-\( O \) estimate for the total integer comparisons made by the trisection search on a list of length \( n = 3^p \). Follow these steps: 1. Establish a recurrence relation justifying the comparison count (referencing a helpful example). 2. Use a theorem to achieve a big-\( O \) estimate. ### Part B Evaluate each statement with "true" or "false" and give a brief justification: 1. Trisection search requires the same number of integer comparisons as binary search. 2. Trisection search necessitates no more than 10% additional comparisons than binary search. 3. If \( n \) rises by a factor of 10, both binary and trisection search comparison counts will grow similarly (though not exactly equivalently).
**Trisection Search Algorithm Explanation** Consider the algorithm referred to as trisection search. Trisection search determines whether a specified integer \( x \) is part of a sorted list \( L \) of integers. Assume that the length of \( L \) is \( 3^p \) for some integer \( p \). ### Algorithm: TrisectionSearch(\( x, L \)) 1. If \( L \) has a length of 1, return `True` if the single element equals \( x \), and return `False` otherwise. 2. Divide \( L \) into three equal sections, named \( L_1, L_2, \) and \( L_3 \). Denote the last elements of these lists by \( \ell_1, \ell_2, \ell_3 \), respectively. - If \( x \leq \ell_1 \), then return TrisectionSearch(\( x, L_1 \)). - Else, if \( \ell_1 < x \leq \ell_2 \), then return TrisectionSearch(\( x, L_2 \)). - Else, if \( \ell_2 < x \), then return TrisectionSearch(\( x, L_3 \)). ### Part A Provide a big-\( O \) estimate for the total integer comparisons made by the trisection search on a list of length \( n = 3^p \). Follow these steps: 1. Establish a recurrence relation justifying the comparison count (referencing a helpful example). 2. Use a theorem to achieve a big-\( O \) estimate. ### Part B Evaluate each statement with "true" or "false" and give a brief justification: 1. Trisection search requires the same number of integer comparisons as binary search. 2. Trisection search necessitates no more than 10% additional comparisons than binary search. 3. If \( n \) rises by a factor of 10, both binary and trisection search comparison counts will grow similarly (though not exactly equivalently).
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
Problem 1PE
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Transcribed Image Text:**Trisection Search Algorithm Explanation**
Consider the algorithm referred to as trisection search. Trisection search determines whether a specified integer \( x \) is part of a sorted list \( L \) of integers. Assume that the length of \( L \) is \( 3^p \) for some integer \( p \).
### Algorithm: TrisectionSearch(\( x, L \))
1. If \( L \) has a length of 1, return `True` if the single element equals \( x \), and return `False` otherwise.
2. Divide \( L \) into three equal sections, named \( L_1, L_2, \) and \( L_3 \). Denote the last elements of these lists by \( \ell_1, \ell_2, \ell_3 \), respectively.
- If \( x \leq \ell_1 \), then return TrisectionSearch(\( x, L_1 \)).
- Else, if \( \ell_1 < x \leq \ell_2 \), then return TrisectionSearch(\( x, L_2 \)).
- Else, if \( \ell_2 < x \), then return TrisectionSearch(\( x, L_3 \)).
### Part A
Provide a big-\( O \) estimate for the total integer comparisons made by the trisection search on a list of length \( n = 3^p \). Follow these steps:
1. Establish a recurrence relation justifying the comparison count (referencing a helpful example).
2. Use a theorem to achieve a big-\( O \) estimate.
### Part B
Evaluate each statement with "true" or "false" and give a brief justification:
1. Trisection search requires the same number of integer comparisons as binary search.
2. Trisection search necessitates no more than 10% additional comparisons than binary search.
3. If \( n \) rises by a factor of 10, both binary and trisection search comparison counts will grow similarly (though not exactly equivalently).
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