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[Caml-list] how to calculate a "xor"
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Date: -- (:)
From: Jean-Baptiste Rouquier <jean-baptiste.rouquier@e...>
Subject: Re: [Caml-list] how to calculate a "xor"
Damien wrote:

> Hi algorithmers,
> Given two sets A and B, I want to calculate A\B _and_ B\A.
> The sets are represented by lists.

Consider the case A = [a_1; a_3; a_5; ...] and B = [a_2; a_4; a_6; ...] 
where a_i < a__{i+1}.
I think it's a worst case for any algorithm.

> without using an order to sort the lists,

that is, if the only allowed comparison is "="

> is there something better than (...) (O(n2)) ?

In my example, any algorithm has to test if a_{2i} = a_{2j+1}. So the 
worst case is O(n2).

> with an order, is there something better than (O(n*ln n)) ?

Consider the case where an algorithm M hasn't sorted B. So there are 2 
consecutive elements, say a_4 and a_6, that hasn't been compared 
(directly or with transitivity). a_5 can't have been compared to both 
a_4 and a_6, let's consider a_5 hasn't been compared to a_4. Then M 
can't decide whether a_4 = a_5 or not, even using transitivity.

This proove that any algorithm has to sort both A and B in my example, 
so the worst case is
O(n ln n).

See you soon,

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