--- title: Hashing TARGET DECK: Obsidian::STEM FILE TAGS: hashing tags: - hashing --- ## Overview 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" $h(k) = k$). %%ANKI Basic With respect to hashing, what does the "universe" of keys refer to? Back: Every potential key that may be provided to the hash function. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Basic What name is given to each position in a hash table? Back: A slot. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Basic Given a hash table with hash function $h$, the element at slot $k$ has what key? Back: A key $k'$ such that $h(k') = k$. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Basic Given a hash table with hash function $h$, an element with key $k$ is placed in what slot? Back: $h(k)$ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Basic Given a hash table `T[0:m-1]`, what is the domain of a hash function? Back: The universe of keys. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Basic Given a hash table `T[0:m-1]`, what is the codomain of a hash function? Back: $\{0, \ldots, m - 1\}$ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Basic What does a hash value refer to? Back: The result produced by a hash function. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Basic What distinguishes a slot from a hash value? Back: The former is a memory address. The latter is the result of a hash function. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Basic What relationship exists between slots and hash values? Back: A slot is often referred to by a hash value. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Cloze Given hash function $h$, key $k$ {hashes} to slot $h(k)$. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Basic What combinatorial concept is used to prove the presence of hash table collisions? Back: The pigeonhole principle. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Basic When *must* there exist hash table collisions? Back: When the number of hashed keys is greater than the number of slots. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Basic What does a hash table collision refer to? Back: Two keys hashing to the same slot. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Basic With respect to hash tables, what imagery is invoked by the term "hash"? Back: Random mixing and chopping. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Basic Are hash tables or direct-address tables more general? Back: Hash tables. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Basic How is a direct-address table reinterpreted as a hash table? Back: It's a hash table with hash function $h(k) = k$. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% ## Load Factor Consider hash table $T$ with $m$ slots that stores $n$ entries. Then the **load factor** $\alpha$ for $T$ is defined to be $n / m$, i.e. the average number of entries that map to the same slot. %%ANKI Basic The load factor of a hash table is a ratio of what two numbers? Back: The number of entries in the table to the number of slots stored in the table. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Cloze The load factor of a hash table {increases} as the number of slots {decrease}. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Cloze The load factor of a hash table {decreases} as the number of total entries {decrease}. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Cloze The load factor of a hash table {increases} as the number of total entries {increase}. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Cloze The load factor of a hash table {decreases} as the number of slots {increase}. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Basic Let $n / m$ denote the load factor of a hash table. What does $n$ represent? Back: The total number of entries in the table. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Basic Let $n / m$ denote the load factor of a hash table. What does $m$ represent? Back: The number of slots in the table. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Basic *Why* is the load factor $\alpha$ of a hash table defined the way it is? Back: It represents the average number of entries stored at a slot. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% 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 $h(k)$. Independent uniform hashing is **universal**, meaning the chance of any two distinct keys colliding is at most $1 / m$. %%ANKI 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). END%% %%ANKI 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). END%% %%ANKI 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). END%% %%ANKI 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). END%% %%ANKI 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). END%% %%ANKI Basic What is uniform hashing? Back: Any given element is equally likely to hash into any slot. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% %%ANKI Basic What is independent hashing? Back: The slot an element hashes to is independent of where other elements hash to. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%% ## Bibliography * Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).