Consider an implementation of a hash table with linear probing where the hash function runs in constant time and the cost of comparison is also constant. What is the best-case time complexity of the __getitem__ method? Consider n is the number of items in the hash table.

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BlackTom AI · Accepted Answer

To approach this question, I’ll evaluate the time complexity of a hash table lookup under linear probing when certain operations are constant.
Option a: O(log n). This would imply that the get operation grows logarithmically with the number of items. In a well-designed hash table with constant-time hash function and constant-time comparisons, lookups do not require traversing data in a way that scales with log n. Linear probing can degrade in worst cases, but under the stated assumptions (constant-time hashing and comparisons, and best-case behavior), O(log n) is not justified.
Option b: O(n). This would mean a full scan in the worst case, which would occur if hashing were poor or all slots collided and we had to probe many positions. However, the problem specifies a hash function that runs in constant time and constant-cost comparisons, and asks for the best-case time. The best case would avoid scanning many slots, so O(n) would not be the best-case.
Option c: O(n^2). This would imply a quadratic-time behavior, such as repeatedly scanning or nested processes for each item. In the context of a single __getitem__ call with a proper hash table and linear probing, the best case would not exhibit quadratic growth. Hence this is not correct for the best case.
Option d: O(1). This is the hallmark of a well-implemented hash table lookup in the ideal scenario: compute the hash in constant time, perform a constant number of probes (often just one), and compare in constant time. In the best case of linear probing, the item is found immediately at its hashed position with no or minimal probing, yielding constant time. The setup assumptions (constant-time hash and comparison) support this.
In summary, among the given choices, the best-case time complexity for __getitem__ in this scenario is O(1), which corresponds to option d. The explanations above contrast the possible complexities and explain why they are not appropriate for the best case under the stated conditions.

FIT1008-FIT1054-FIT2085 Fundamentals of algorithms - S2 2025

Consider an implementation of a hash table with linear probing where the hash function runs in constant time and the cost of comparison is also constant. What is the best-case time complexity of the getitem method? Consider n is the number of items in the hash table.

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FIT1008-FIT1054-FIT2085 Fundamentals of algorithms - S2 2025

Consider an implementation of a hash table with linear probing where the hash function runs in constant time and the cost of comparison is also constant. What is the best-case time complexity of the __getitem__ method? Consider n is the number of items in the hash table.

View Explanation

Log in for full answers

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Consider an implementation of a hash table with linear probing where the hash function runs in constant time and the cost of comparison is also constant. What is the best-case time complexity of the getitem method? Consider n is the number of items in the hash table.

HT_Rev_6 A key 48 is to be inserted into a hash table using the hash function h(k)=k mod 10, open addressing, and linear probing for collision handling. Given the current state of the hash table below, where will key 48 be stored?