notebook/notes/hashing/index.md

---
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).
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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).
<!--ID: 1716307180959-->
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).
<!--ID: 1716307180961-->
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).
<!--ID: 1716307180962-->
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).
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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).
<!--ID: 1716307180965-->
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).
<!--ID: 1716307180967-->
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).
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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).
<!--ID: 1716307180970-->
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).
<!--ID: 1716307180971-->
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).
<!--ID: 1716307180973-->
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).
<!--ID: 1716307180974-->
END%%

%%ANKI
Basic
What does a hash table collision refer to?
Back: Two different keys hashing to the same slot.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716307180976-->
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).
<!--ID: 1716307180977-->
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).
<!--ID: 1716307180979-->
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).
<!--ID: 1716307180980-->
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).
<!--ID: 1718759188190-->
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).
<!--ID: 1718759188194-->
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).
<!--ID: 1718759188199-->
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).
<!--ID: 1718759188204-->
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).
<!--ID: 1718759188208-->
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).
<!--ID: 1718759188214-->
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).
<!--ID: 1718759188218-->
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).
<!--ID: 1718759188222-->
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).
<!--ID: 1718197741507-->
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).
<!--ID: 1718197741527-->
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).
<!--ID: 1718197741537-->
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).
<!--ID: 1718197741545-->
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).
<!--ID: 1718197741555-->
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).
<!--ID: 1719174576842-->
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).
<!--ID: 1719174576848-->
END%%

## Bibliography

* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).