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I am programming in C, so I do not want to implement a hash-based datastructure such as a hashset or hashmap/dictionary. However, I need to solve the following task in linear time.

Given two arrays $a$ and $b$ which contain the same set of distinct integers, determine for every element of $a$ the index of the same element in $b$.

For example, if $a=[9,4,3,7]$ and $b=[3,4,7,9]$, then the output should be $[3,1,0,2]$.

Note that this becomes a very easy task when you have a hashset, because you can simply store for every element in $b$ the index, and then query the hashmap for every element of $a$.

So my question is whether there is a linear algorithm for this task that does not use any hashsets.

If the only operation allowed between any two (possibly the same) elements in the two arrays is to determine which one is the smaller one, then it will take $Theta(nlog n)$ time in worst case for any algorithm. This can be seen in the case when array $a$ is sorted while array $b$ is not. Then knowing the index $I(k)$ of the same element in $b$ for the $k$-th element of $a$ for all $k$, we can sort $b$ in linear time by putting $b_I(k)$ in $k$-th position.

The following is a formal formulation of the conclusion above in the comparison computation model.

Let $mathcal O$ be an oracle that can tell a fixed strict linear ordering on $E$, a set of $n$ elements. That is, on input $e,fin E$, $mathcal O$ outputs -1 if $eprec f$, 0 if $e$ is $f$ and 1 otherwise. Let $a$ and $b$ are two bijections from $0, 1,cdots, n-1$ to $E$. To output $I(0), I(1), cdots, I(n-1)$ in that order such that $a(k)=b(I(k))$ for all $0le kle n-1$, it will take $Theta(nlog n)$ queries against $mathcal O$ in the worst case.

whether there is a linear algorithm for this task that does not use any hashsets.

A computation model that is defined by no usage of hashset is not a well-defined computation mode. How can you check there is no usage of hashset? There are literally hundreds of ways to implement a data structure that is a hashset or looks like a hashset or looks like a hashset partially. In general, a well-defined computation model must be defined by what can be done formally.