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4. Question 4: Say that elements have two keys. key? is not unique, namely, there are may be many elements with the same key?. But each element also has key? so that the second keys are pairwise disjoint. The keys are numbers. Show how to implement the following operations: Find(S, key?(x), key?(x)) that finds the unique element with these two keys. Also, implement Insert(S, $k_1$, $k_2$) that adds into S an elements with first key $k_1$ and second key $k_2$. Finally the data structure should be able to do Delete(S, $k_1$) with $k_1$ some valid first key. This means that you want to remove all whose first key is $k_1$. Remark: Its possible to do all the above so that every operation will take much less than $\Omega(n)$.

          4. Question 4: Say that elements have two keys. key? is not unique, namely, there are
may be many elements with the same key?. But each element also has key? so that
the second keys are pairwise disjoint. The keys are numbers.
Show how to implement the following operations: Find(S, key?(x), key?(x)) that finds
the unique element with these two keys. Also, implement Insert(S, $k_1$, $k_2$) that adds
into S an elements with first key $k_1$ and second key $k_2$. Finally the data structure
should be able to do Delete(S, $k_1$) with $k_1$ some valid first key. This means that you
want to remove all whose first key is $k_1$.
Remark: Its possible to do all the above so that every operation will take
much less than $\Omega(n)$.

4. Question 4: Say that elements have two keys. key? is not unique, namely, there are
may be many elements with the same key?. But each element also has key? so that
the second keys are pairwise disjoint. The keys are numbers.
Show how to implement the following operations: Find(S, key?(x), key?(x)) that finds
the unique element with these two keys. Also, implement Insert(S, k1, k2) that adds
into S an elements with first key k1 and second key k2. Finally the data structure
should be able to do Delete(S, k1) with k1 some valid first key. This means that you
want to remove all whose first key is k1.
Remark: Its possible to do all the above so that every operation will take
much less than Ω(n).

Added by Christina P.

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