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You, Alice and Bob are working on recursive search algorithms and have been studying a variant of binary search called trinary search. Alice has created the following pseudocode for this algorithm:

You, Alice and Bob are working on recursive search algorithms and have been studying a variant of binary search called trinary search. Alice has created the following pseudocode for this algorithm:

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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**Understanding Trinary Search Algorithms and Their Analysis** You, Alice and Bob are working on recursive search algorithms and have been studying a variant of a binary search called trinary search. Alice has created the following pseudocode for this algorithm: ```pseudo TSearch(A[a...b], t) If a > b return -1 Let p1 = a + Floor((b - a)/3) If A[p1] = t return p1 If A[p1] > t return TSearch(A[a...p1-1],t) Let p2 = a + Ceiling(2(b - a)/3) If A[p2] = t return p2 If A[p2] > t return TSearch(A[p1+1...p2-1],t) Return TSearch(A[p2+1...b],t) EndTSearch ``` **Task and Analysis** a) **State a recurrence relation that expresses the number of operations carried out by this recursive algorithm when called on an input array of size \( n \).** b) **Bob has heard that trinary search is no more efficient than binary search when considering asymptotic growth. Help prove him correct by using induction to show that your recurrence relation is in \( \Theta(\log_2 n) \) as well.** **Steps to Prove Using Induction:** i. **Split the tight bound into an upper (big-O) and lower (big-Ω).** ii. **For each bound select a function from \( \Theta(\log_2 n) \) to use in your proof, like \( a \log_2 n \) or \( a \log_2 n - b \). Remember there are typically multiple ways to prove the theorem using different choices of functions.** iii. **Use induction to prove your bound. Include all parts of the proof including base case, inductive hypothesis, and inductive case. Be as precise as possible with your language and your math. Remember it’s possible to get stuck at this point if you have selected the wrong function in the last step.** **Key Elements of Trinary Search Pseudocode:** 1. **Base Case:** - If the sub-array to search is invalid (`a > b`), return -1. 2. **Partitioning:** - Calculate two partition indices (`p1` and `p
Transcribed Image Text:**Understanding Trinary Search Algorithms and Their Analysis** You, Alice and Bob are working on recursive search algorithms and have been studying a variant of a binary search called trinary search. Alice has created the following pseudocode for this algorithm: ```pseudo TSearch(A[a...b], t) If a > b return -1 Let p1 = a + Floor((b - a)/3) If A[p1] = t return p1 If A[p1] > t return TSearch(A[a...p1-1],t) Let p2 = a + Ceiling(2(b - a)/3) If A[p2] = t return p2 If A[p2] > t return TSearch(A[p1+1...p2-1],t) Return TSearch(A[p2+1...b],t) EndTSearch ``` **Task and Analysis** a) **State a recurrence relation that expresses the number of operations carried out by this recursive algorithm when called on an input array of size \( n \).** b) **Bob has heard that trinary search is no more efficient than binary search when considering asymptotic growth. Help prove him correct by using induction to show that your recurrence relation is in \( \Theta(\log_2 n) \) as well.** **Steps to Prove Using Induction:** i. **Split the tight bound into an upper (big-O) and lower (big-Ω).** ii. **For each bound select a function from \( \Theta(\log_2 n) \) to use in your proof, like \( a \log_2 n \) or \( a \log_2 n - b \). Remember there are typically multiple ways to prove the theorem using different choices of functions.** iii. **Use induction to prove your bound. Include all parts of the proof including base case, inductive hypothesis, and inductive case. Be as precise as possible with your language and your math. Remember it’s possible to get stuck at this point if you have selected the wrong function in the last step.** **Key Elements of Trinary Search Pseudocode:** 1. **Base Case:** - If the sub-array to search is invalid (`a > b`), return -1. 2. **Partitioning:** - Calculate two partition indices (`p1` and `p
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