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Asked: 2026-06-10T15:15:37+00:00

(I’m banging my head here. Let X={x1,x2,…,xn} is an integer set. Let A1,A2,…Am be

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(I’m banging my head here. Let X={x1,x2,…,xn} is an integer set. Let A1,A2,…Am be the m subsets of X. For any i and j, Ai and Aj are not necessarily disjoint. Now the goal is to find the maximal value on each Ai (i=1,…,m) efficiently, with the number of operations as fewer as possible.

For example, given X={2,4,6,3,1}, and its subsets A1={2,3,1}, A2={2,6,3,1}, A3={4,2,3,1}. We need to find Max{A1}, Max{A2}, Max{A3}, respectively.

The brute-force way for finding Max{A1}, Max{A2}, Max{A3} is to scan all the elements in each Ai, and (m*d) operations are required, with m the number of subsets of X, and d the average length of the subsets {Ai} of X.

Now, I have some observations:

(1) For any set Y⊆X, max{Y}≤max{X},

For instance, since Max{X}=6 and 6 is in A2, then Max{A2}=6 can be found directly.

(2) For any two sets A and B, if A∩B is non-empty, Max{A} and Max{B} can be identified as follows:

First, we find the common parts between A and B, deonted as c=max{A∩B}.

Then, we find Max{A}=Max{Max{A-(A∩B)}, c} and Max{B}=Max{Max{B-(A∩B)}, c}.

I am not sure whether there are some other interesting obervations for find these max values.
Any ideas are warmly welcome!

My question is what if for the general case when X={x1,x2,…,xn} and there are m subsets of X, denoted as A1,A2,…Am, is there some more efficient techniques to find such max values Max{Ai} (i=1,…,m) ?

Your help will be highly appreciated!

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1 Answer

  1. Editorial Team

    There is no method asymptotically better than brute force, assuming a typical representation of the given sets. Simply scanning through the sets to find the largest member of each requires linear time and linear time is optimal since every member of the set must be read in order to determine the maximum value.

    Now if the input representation is not simply a listing of the elements in each set, than other bounds and algorithms may apply. For example, if we know the input sets are sorted and the length of the set is given as part of the input, we can obviously find the maximum elements in time linear only on the number of subsets but not on their length.

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