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What is the time complexity for inserting a key or integer in Van Emde Boas data structure?

(a) O (log M!)

(b) O (M!)

(c) O (M^2)

(d) O (log (log M))

My doubt stems from Trees in section Trees of Data Structures & Algorithms I

I had been asked this question in a job interview.

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Correct answer is (a) O (log M!)

The best explanation: In order to insert a key or integer in the Van Emde Boas data structure, the operation can be performed on an associative array. Hence, the time complexity for inserting a key or integer in Van Emde Boas data structure is O (log (log M)).

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