A **hash table**`T[0:m-1]` uses a **hash function** to map a universe of keys into slots of the hash table. It can be seen as a generalization of direct addressing (which has "hash function" $h(k) = k$).
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Basic
With respect to hashing, what does the "universe" of keys refer to?
An **independent uniform hash function** is the ideal theoretical abstraction. For each possible input $k$ in universe $U$, an output $h(k)$ is produced randomly and independently chosen from range $\{0, 1, \ldots, m - 1\}$. Once a value $h(k)$ is chosen, each subsequent call to $h$ with the same input $k$ yields the same output $h(k)$.
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Basic
What is considered the ideal (though only theoretical) hash function?
Back: The independent uniform hash function.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Basic
Given independent uniform hash function $h$, what about $h$ is "independent"?
Back: Each key $k$ has output $h(k)$ determined independently from other keys.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Basic
Given independent uniform hash function $h$, what about $h$ is "uniform"?
Back: Every output of $h$ is equally likely to be any of the values in its range.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Basic
With respect to hashing, a random oracle refers to what kind of hash function?
Back: An independent uniform hash function.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Basic
Where does "memory" come into play with independent uniform hash functions?
Back: Once $h(k)$ is determined, subsequent calls to $h$ with $k$ always yield the same value.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).