Element distinctness problem
In computational complexity theory, the element distinctness problem or element uniqueness problem is the problem of determining whether all the elements of a list are distinct.
It is a well studied problem in many different models of computation. The problem may be solved by sorting the list and then checking if there are any consecutive equal elements; it may also be solved in linear expected time by a randomized algorithm that inserts each item into a hash table and compares only those elements that are placed in the same hash table cell.
Several lower bounds in computational complexity are proved by reducing the element distinctness problem to the problem in question, i.e., by demonstrating that the solution of the element uniqueness problem may be quickly found after solving the problem in question.
It is known that, for lists of numbers, the problem's time complexity is Θ(n log n), i.e., both the upper and lower bounds on its time complexity are of order of the linearithmic function in the algebraic decision tree model of computation, a model of computation in which the elements may not be used to index the computer's memory (as in the hash table solution) but may only be accessed by computing and comparing simple algebraic functions of their values. In other words, an asymptotically optimal algorithm of linearithmic time complexity is known for this model. The algebraic computation tree model basically means that the allowable algorithms are only the ones that can perform polynomial operations of bounded degree on the input data and comparisons of the results of these computations.
The same lower bound was later proved for the randomized algebraic decision tree model.
It is also known that quantum algorithms can solve this problem faster in queries. The optimal algorithm is by Andris Ambainis.Yaoyun Shi first proved a tight lower bound when the size of the range is sufficiently large. Ambainis and Kutin independently (and via different proofs) extended his work to obtain the lower bound for all functions.
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