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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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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: TSearch (A[a...b], t) If a b return -1 Let pl = 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 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 (log₂ n) as well. i. Split the tight bound into and upper (big-O) and lower (big-1). ii. For each bound select a function from (log₂ n) to use in your proof, like alog₂ n or alog₂ 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.
Transcribed Image Text: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: TSearch (A[a...b], t) If a b return -1 Let pl = 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 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 (log₂ n) as well. i. Split the tight bound into and upper (big-O) and lower (big-1). ii. For each bound select a function from (log₂ n) to use in your proof, like alog₂ n or alog₂ 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.
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