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John tells you that his algorithm runs in 0(n³+ n), and Bill says that the same algorithm runs in ©(n²). Can they both be correct? If correct, show one example. If not, justify your answer.

John tells you that his algorithm runs in 0(n³+ n), and Bill says that the same algorithm runs in ©(n²). Can they both be correct? If correct, show one example. If not, justify your answer.

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
Problem 1PE
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**Question:** John tells you that his algorithm runs in \(O(n^3 + n)\), and Bill says that the same algorithm runs in \(\Theta(n^2)\). Can they both be correct? If correct, show one example. If not, justify your answer. **Explanation:** - **Big O Notation (\(O\))** describes an upper bound on the time complexity, indicating the maximum growth rate of the algorithm. - **Theta Notation (\(\Theta\))** describes a tight bound, indicating that the growth rate is both an upper and lower bound. **Analysis:** 1. **John's Claim:** \(O(n^3 + n)\) implies the algorithm's maximum growth rate is similar to a cubic function. The \(n^3\) term dominates the \(n\) term for large \(n\), so \(O(n^3 + n) = O(n^3)\). 2. **Bill's Claim:** \(\Theta(n^2)\) indicates a tight bound around a quadratic function, meaning the algorithm must grow like \(n^2\) for large inputs. **Conclusion:** - Both claims cannot be correct because: - If John is accurate with \(O(n^3)\), then \(\Theta(n^2)\) by Bill is too low since \(\Theta(n^2)\) suggests a quadratic growth rate rather than cubic. - In other words, for Bill's claim to hold, the dominant term should be \(n^2\), contradicting John's \(n^3\) term. Thus, both cannot be simultaneously correct for the same algorithm.
Transcribed Image Text:**Question:** John tells you that his algorithm runs in \(O(n^3 + n)\), and Bill says that the same algorithm runs in \(\Theta(n^2)\). Can they both be correct? If correct, show one example. If not, justify your answer. **Explanation:** - **Big O Notation (\(O\))** describes an upper bound on the time complexity, indicating the maximum growth rate of the algorithm. - **Theta Notation (\(\Theta\))** describes a tight bound, indicating that the growth rate is both an upper and lower bound. **Analysis:** 1. **John's Claim:** \(O(n^3 + n)\) implies the algorithm's maximum growth rate is similar to a cubic function. The \(n^3\) term dominates the \(n\) term for large \(n\), so \(O(n^3 + n) = O(n^3)\). 2. **Bill's Claim:** \(\Theta(n^2)\) indicates a tight bound around a quadratic function, meaning the algorithm must grow like \(n^2\) for large inputs. **Conclusion:** - Both claims cannot be correct because: - If John is accurate with \(O(n^3)\), then \(\Theta(n^2)\) by Bill is too low since \(\Theta(n^2)\) suggests a quadratic growth rate rather than cubic. - In other words, for Bill's claim to hold, the dominant term should be \(n^2\), contradicting John's \(n^3\) term. Thus, both cannot be simultaneously correct for the same algorithm.
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