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POHow to prove a lower bound logn for a data structure?
 

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  1. POHow to prove a lower bound logn for a data structure?
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    <p>I have a homework question as follows (note that I am not looking for exact answers, just looking for simple suggestions to move on). </p> <blockquote> <p>S is a data structure which supports Insert(x,S), Delete(x, S) and Find_Smallest_Item(S) in time &lt;= T(n). Prove a lower bound for T(n) such as Ω(logn).</p> </blockquote> <p>Here is my thoughts so far:</p> <p>I guess I need to find a reduction where I will reduce this problem into an easier one, and prove it cannot be lower than logn. I read many tutorials about lower bounds and most of them reduced a problem into sorting, then they used sorting as a black box and proved the algorithm cannot be lower than nlogn.</p> <p>But here, we are dealing with logn and I don't know such an algorithm to reduce to. Maybe it's gotta do something with the depth of a tree structure, logn. But I couldn't figure out where to start.</p> <p>Can you suggest me some hints? </p> <p><strong>Edit</strong>: Actually something came to my mind but I don't know if I am supposed to prove a lower bound with a trick like that. So, I assume I have insert, delete and find_smallest operations, each of them has a logn time complexity.</p> <p>For example, to construct a sorted list, I can use delete and find_smallest functions, e.g. I can run find_smallest for the first time, and after finding the smallest element in the list, then I will delete the element. I will run it once again and therefore I will find 2nd smallest element, and so on. </p> <p>Therefore, I can implement sorting by using deletion and find_smallest functions. So if I continue doing this n times, each of them will take logn (for deletion) + logn (for finding smallest), so in overall, sorting will take nlogn.</p> <p>I don't know how to tweak this for insertion, though.</p> <p><strong>Edit 2:</strong> In order to use insertion for the proof: after finding the ith smallest element in the list, what if I insert it to ith place? e.g. after finding 3rd smallest element with the procedure above, I can insert it to 3rd index of the data structure. Therefore, at the end I will obtain a sorted data structure.</p>
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