714 lines
26 KiB
Markdown
714 lines
26 KiB
Markdown
---
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title: Hashing
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TARGET DECK: Obsidian::STEM
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FILE TAGS: hashing
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tags:
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- hashing
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---
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## Overview
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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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%%ANKI
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Basic
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With respect to hashing, what does the "universe" of keys refer to?
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Back: Every potential key that may be provided to the hash function.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1716046153757-->
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END%%
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%%ANKI
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Basic
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What name is given to each position in a hash table?
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Back: A slot.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1716307180959-->
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END%%
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%%ANKI
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Basic
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Given a hash table with hash function $h$, the element at slot $k$ has what key?
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Back: A key $k'$ such that $h(k') = k$.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1716307180961-->
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END%%
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%%ANKI
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Basic
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Given a hash table with hash function $h$, an element with key $k$ is placed in what slot?
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Back: $h(k)$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1716307180962-->
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END%%
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%%ANKI
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Basic
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Given a hash table `T[0:m-1]`, what is the domain of a hash function?
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Back: The universe of keys.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1716307180964-->
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END%%
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%%ANKI
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Basic
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Given a hash table `T[0:m-1]`, what is the codomain of a hash function?
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Back: $\{0, \ldots, m - 1\}$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1716307180965-->
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END%%
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%%ANKI
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Basic
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What does a hash value refer to?
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Back: The result produced by a hash function.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1716307180967-->
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END%%
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%%ANKI
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Basic
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What distinguishes a slot from a hash value?
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Back: The former is a memory address. The latter is the result of a hash function.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1716307180968-->
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END%%
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%%ANKI
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Basic
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What relationship exists between slots and hash values?
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Back: A slot is often referred to by a hash value.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1716307180970-->
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END%%
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%%ANKI
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Cloze
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Given hash function $h$, key $k$ {hashes} to slot $h(k)$.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1716307180971-->
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END%%
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%%ANKI
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Basic
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What combinatorial concept is used to prove the presence of hash table collisions?
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Back: The pigeonhole principle.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1716307180973-->
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END%%
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%%ANKI
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Basic
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When *must* there exist hash table collisions?
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Back: When the number of hashed keys is greater than the number of slots.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1716307180974-->
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END%%
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%%ANKI
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Basic
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What does a hash table collision refer to?
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Back: Two different keys hashing to the same slot.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1716307180976-->
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END%%
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%%ANKI
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Basic
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With respect to hash tables, what imagery is invoked by the term "hash"?
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Back: Random mixing and chopping.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1716307180977-->
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END%%
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%%ANKI
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Basic
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Are hash tables or direct-address tables more general?
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Back: Hash tables.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1716307180979-->
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END%%
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%%ANKI
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Basic
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How is a direct-address table reinterpreted as a hash table?
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Back: It's a hash table with hash function $h(k) = k$.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1716307180980-->
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END%%
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## Load Factor
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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.
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%%ANKI
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Basic
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The load factor of a hash table is a ratio of what two numbers?
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Back: The number of entries in the table to the number of slots stored in the table.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1718759188190-->
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END%%
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%%ANKI
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Cloze
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The load factor of a hash table {increases} as the number of slots {decrease}.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1718759188194-->
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END%%
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%%ANKI
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Cloze
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The load factor of a hash table {decreases} as the number of total entries {decrease}.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1718759188199-->
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END%%
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%%ANKI
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Cloze
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The load factor of a hash table {increases} as the number of total entries {increase}.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1718759188204-->
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END%%
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%%ANKI
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Cloze
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The load factor of a hash table {decreases} as the number of slots {increase}.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1718759188208-->
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END%%
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%%ANKI
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Basic
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Let $n / m$ denote the load factor of a hash table. What does $n$ represent?
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Back: The total number of entries in the table.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1718759188214-->
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END%%
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%%ANKI
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Basic
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Let $n / m$ denote the load factor of a hash table. What does $m$ represent?
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Back: The number of slots in the table.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1718759188218-->
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END%%
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%%ANKI
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Basic
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*Why* is the load factor $\alpha$ of a hash table defined the way it is?
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Back: It represents the average number of entries stored at a slot.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1718759188222-->
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END%%
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## Static Hashing
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**Static hashing** refers to providing a single fixed hash function intended to work well on *any* data. Generally speaking, this should not be favored over random hashing.
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%%ANKI
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Basic
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What does static hashing refer to?
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Back: Providing a single hash function intended to work well on *any* data.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720821498619-->
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END%%
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%%ANKI
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Cloze
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{Static} hashing provides a {single hash function} intended to work well on any data.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720821498622-->
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END%%
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%%ANKI
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Basic
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What does it mean for static hashing to be independent?
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Back: Where a key hashes to is independent of where other keys hash to.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720821498628-->
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END%%
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%%ANKI
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Basic
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What about independent static hashing is a bit of a misnomer?
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Back: N/A.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720821498631-->
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END%%
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%%ANKI
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Basic
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What does it mean for static hashing to be uniform?
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Back: Each key has an equal likelihood of hashing to any slot.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720821498634-->
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END%%
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%%ANKI
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Basic
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What about uniform static hashing is a bit of a misnomer?
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Back: Where keys hash to depend on the input keys' probability distribution.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720821498637-->
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END%%
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%%ANKI
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Basic
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In static hashing, *why* is uniformity generally impossible?
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Back: Because we use a fixed hash function for *all* data.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720821498640-->
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END%%
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%%ANKI
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Basic
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Assuming $m$ slots, why is static hashing function $h(k) = \lfloor km \rfloor$ not generally "good"?
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Back: The probability distribution from which keys were drawn may not be uniform.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720821498644-->
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END%%
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%%ANKI
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Basic
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What property must an ideal static hashing function exhibit?
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Back: It must derive hash values independently of any patterns that may exist in the keys.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720821498648-->
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END%%
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%%ANKI
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Basic
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What randomization is available to static hashing?
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Back: The distribution of input keys.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720889385376-->
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END%%
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### Division Method
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The **division method** for creating hash functions maps a key $k$ into one of $m$ slots by taking the remainder of $k$ divided by $m$. That is, $h(k) = k \bmod{m}$.
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%%ANKI
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Basic
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The division method is used to produce what?
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Back: A hash function.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720889385404-->
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END%%
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%%ANKI
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Basic
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What hyperparameter(s) does the division method require?
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Back: The number of slots in the hash table.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720889385409-->
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END%%
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%%ANKI
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Basic
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Given $m$ slots, the division method produces what hash function?
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Back: $h(k) = k \bmod{m}$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720889385414-->
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END%%
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%%ANKI
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Basic
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Let $h$ be a division method hash function. What does $h(10)$ evaluate to?
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Back: $10 \bmod{m}$, where $m$ is the number of slots in the hash table.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720889385419-->
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END%%
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%%ANKI
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Basic
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Consider hash function $h(k) = k \bmod{m}$. What does $m$ likely represent?
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Back: The number of slots in the hash table.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720889385424-->
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END%%
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%%ANKI
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Basic
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Let $m$ be some number of slots. What $m$'s does the division method typically work best on?
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Back: A prime not too close to an exact power of $2$.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720889385429-->
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END%%
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%%ANKI
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Basic
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Why does the division method prefer a prime number of slots?
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Back: To operate as independently as possible of the input keys.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1721218408542-->
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END%%
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%%ANKI
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Basic
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Consider hash function $h(k) = k \bmod{m}$. What method was likely used to produce this?
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Back: The division method.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720891800562-->
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END%%
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%%ANKI
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Basic
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Is the division method an example of static or random hashing?
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Back: Static.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1720891800592-->
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END%%
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### Multiplication Method
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The **multiplication method** for creating hash functions first multiples a key $k$ by a constant $0 < A < 1$ and extracts the fractional part of $kA$. Then it multiplies this value by $m$ and takes the floor of the result. That is, $h(k) = \lfloor m(kA \bmod{1}) \rfloor$.
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%%ANKI
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Basic
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The multiplication method is used to produce what?
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Back: A hash function.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720891800597-->
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END%%
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%%ANKI
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Basic
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What hyperparameter(s) does the multiplication method require?
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Back: Slot count $m$ and some constant $0 < A < 1$.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720891800602-->
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END%%
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%%ANKI
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Basic
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Given $m$ slots and constant $A$, the multiplication method produces what hash function?
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Back: $h(k) = \lfloor m (kA \bmod{1}) \rfloor$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720891800607-->
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END%%
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%%ANKI
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Basic
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What range does the constant $A$ found in the multiplication method take on?
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Back: $0 < A < 1$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720891800612-->
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END%%
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%%ANKI
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Basic
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Consider hash function $h(k) = \lfloor m (kA \bmod{1}) \rfloor$. What does $m$ likely represent?
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Back: The number of slots in the hash table.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720891800617-->
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END%%
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%%ANKI
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Basic
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Consider hash function $h(k) = \lfloor m (kA \bmod{1}) \rfloor$. What does $A$ likely represent?
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Back: Some constant $0 < A < 1$.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720891800622-->
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END%%
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%%ANKI
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Basic
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Consider constant $A$ used in the multiplication method. *Why* shouldn't $A = 0$?
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Back: Then the produced hash function is constant.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720891800628-->
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END%%
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%%ANKI
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Basic
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Consider constant $A$ used in the multiplication method. *Why* shouldn't $A = 1$?
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Back: Then the produced hash function is constant.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720891800634-->
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END%%
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%%ANKI
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Basic
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Consider hash function $h(k) = \lfloor m (kA \bmod{1}) \rfloor$. What method was likely used to produce this?
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Back: The multiplication method.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::static
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<!--ID: 1720891800655-->
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END%%
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%%ANKI
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Basic
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Is the multiplication method an example of static or random hashing?
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Back: Static.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1720891800661-->
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END%%
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%%ANKI
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Basic
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For $x \in \mathbb{R}^+$, what does $x \bmod{1}$ represent?
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Back: The fractional part of $x$.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1720891800639-->
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END%%
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%%ANKI
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Basic
|
|
For $x \in \mathbb{R}^+$, what expression does $x \bmod{1}$ evaluate to?
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Back: $x - \lfloor x \rfloor$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1720891800644-->
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END%%
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%%ANKI
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Basic
|
|
For $x \in \mathbb{Z}^+$, what expression does $x \bmod{1}$ evaluate to?
|
|
Back: $0$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1720891800649-->
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END%%
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## Random Hashing
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**Random hashing** refers to choosing a hash function randomly in a way that is independent of the keys being stored.
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%%ANKI
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Basic
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|
What does random hashing refer to?
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|
Back: Choosing a hash function randomly and independently of the keys being stored.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::random
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<!--ID: 1721482558926-->
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END%%
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%%ANKI
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Basic
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|
What does random hashing avoid that static hashing doesn't?
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Back: Randomization guarantees no single input always evokes worst-case behavior.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::random
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<!--ID: 1721482558932-->
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END%%
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### Universal Hashing
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Let $\mathscr{H}$ be a finite family of hash functions that map a given universe $U$ of keys into range $\{0, 1, \ldots, m - 1\}$. Such a family is said to be **universal** if $$\forall x, y \in U, x \neq y \Rightarrow |\{h \in \mathscr{H} \mid h(x) = h(y)\}| \leq \frac{|\mathscr{H}|}{m}.$$
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%%ANKI
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Basic
|
|
Which of universal hashing or random hashing is more general?
|
|
Back: Random hashing.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: hashing::random hashing::universal
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<!--ID: 1721482558937-->
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END%%
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%%ANKI
|
|
Basic
|
|
With respect to universal hashing, what mathematical object is property "universal" attributed to?
|
|
Back: A finite set of hash functions.
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|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
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Tags: hashing::random hashing::universal
|
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<!--ID: 1721482558943-->
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END%%
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%%ANKI
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|
Basic
|
|
What does "family" refer to in the context of universal hashing?
|
|
Back: A finite set of hash functions.
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|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
Tags: hashing::random hashing::universal
|
|
<!--ID: 1721482558948-->
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END%%
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|
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%%ANKI
|
|
Basic
|
|
Consider a hash table with $m = 1$ slot. Which hash function families are universal?
|
|
Back: Any finite family of hash functions.
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|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
Tags: hashing::random hashing::universal
|
|
<!--ID: 1721482558957-->
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|
END%%
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|
|
%%ANKI
|
|
Basic
|
|
A "universal family" refers to a finite set of what?
|
|
Back: Hash functions.
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|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
Tags: hashing::random hashing::universal
|
|
<!--ID: 1721482558964-->
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|
END%%
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|
%%ANKI
|
|
Basic
|
|
Let $\mathscr{H}$ be a universal family and $h \in \mathscr{H}$. What is the domain of $h$?
|
|
Back: The universe of keys.
|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
Tags: hashing::random hashing::universal
|
|
<!--ID: 1721482558970-->
|
|
END%%
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|
%%ANKI
|
|
Basic
|
|
Let $\mathscr{H}$ be a universal family and $h \in \mathscr{H}$. What is the codomain of $h$?
|
|
Back: $\{0, 1, \ldots, m - 1\}$ (or similar), where $m$ refers to the number of hash table slots.
|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
Tags: hashing::random hashing::universal
|
|
<!--ID: 1721482558977-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider universal family $\mathscr{H}$ and universe $U$. What number does the following evaluate to? $$|\{h \in \mathscr{H} \mid h(x) = h(y)\}| \text{ for distinct } x, y \in U$$
|
|
Back: A value between $0$ and $|\mathscr{H}|$ inclusive.
|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
Tags: hashing::random hashing::universal
|
|
<!--ID: 1721482558983-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $\mathscr{H} = \{h \mid U \rightarrow \{0, 1, \ldots, m - 1\}\}$ be universal. What first-order logic statement holds?
|
|
Back: $$\forall x, y \in U, x \neq y \Rightarrow |\{h \in \mathscr{H} \mid h(x) = h(y)\}| \leq \frac{|\mathscr{H}|}{m}$$
|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
Tags: hashing::random hashing::universal
|
|
<!--ID: 1721482558988-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $\mathscr{H} = \{h \mid U \rightarrow \{0, 1, \ldots, m - 1\}\}$ be universal. What does $m > |\mathscr{H}|$ imply?
|
|
Back: For any distinct $x, y \in U$, $h(x) \neq h(y)$ for all $h \in \mathscr{H}$.
|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
Tags: hashing::random hashing::universal
|
|
<!--ID: 1721482558992-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Independent uniform hashing is equivalent to picking a function from what universal family?
|
|
Back: $^U\{0, 1, \ldots, m\}$, i.e. the set of functions from $U$ to $\{0, 1, \ldots, m\}$.
|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
Tags: hashing::random hashing::universal
|
|
<!--ID: 1721482559002-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider universe $U$ and $\mathscr{H} = \{I_U\}$. Is $\mathscr{H}$ universal?
|
|
Back: Yes.
|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
Tags: hashing::random hashing::universal
|
|
<!--ID: 1721482559008-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider universe $U$ and $\mathscr{H} = \{I_U\}$. *Why* is $\mathscr{H}$ universal?
|
|
Back: Because for any distinct $x, y \in U$, $I_U(x) \neq I_U(y)$.
|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
Tags: hashing::random hashing::universal
|
|
<!--ID: 1721482559014-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider universe $U$ and $\mathscr{H} = \{h\}$ where $h(x) = 0$. Is $\mathscr{H}$ universal?
|
|
Back: Indeterminate.
|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
Tags: hashing::random hashing::universal
|
|
<!--ID: 1721482559021-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider universe $U$ and $\mathscr{H} = \{h\}$ where $h(x) = 0$. *When* is $\mathscr{H}$ universal?
|
|
Back: When there exists only one slot in the relevant hash table.
|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
Tags: hashing::random hashing::universal
|
|
<!--ID: 1721482559031-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Consider universe $U$ and $\mathscr{H} = \{h\}$ where $h(x) = 0$. *When* is $\mathscr{H}$ not universal?
|
|
Back: When there exists more than one slot in the relevant hash table.
|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
Tags: hashing::random hashing::universal
|
|
<!--ID: 1721482559043-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $\mathscr{H} = \{h \mid U \rightarrow \{0, 1, \ldots, m - 1\}\}$ be universal. What number decreases as $m$ increases?
|
|
Back: The number of permitted conflicts for each $h \in \mathscr{H}$.
|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
Tags: hashing::random hashing::universal
|
|
<!--ID: 1721482559053-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Let $\mathscr{H} = \{h \mid U \rightarrow \{0, 1, \ldots, m - 1\}\}$ be universal. What number increases as $|\mathscr{H}|$ increases?
|
|
Back: The number of permitted conflicts for each $h \in \mathscr{H}$.
|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
Tags: hashing::random hashing::universal
|
|
<!--ID: 1721482559059-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Is $\varnothing$ a universal family?
|
|
Back: Yes.
|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
Tags: hashing::random hashing::universal
|
|
<!--ID: 1721482559064-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How might we redefine "universal" to prevent $\varnothing \subseteq \{h \mid h \colon U \rightarrow \{0, 1, \ldots, m - 1\}$ being considered universal?
|
|
Back: $$\forall x, y \in U, x \neq y \Rightarrow \frac{|\varnothing|}{|\varnothing|} \leq \frac{1}{m}$$
|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
Tags: hashing::random hashing::universal
|
|
<!--ID: 1721482559069-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is it that universal hashing makes impossible?
|
|
Back: Forcing the worst-case running time of hash table operations.
|
|
Tags: hashing::random hashing::universal
|
|
<!--ID: 1722080163399-->
|
|
END%%
|
|
|
|
## Bibliography
|
|
|
|
* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
* “Universal Hashing,” in _Wikipedia_, April 18, 2024, [https://en.wikipedia.org/w/index.php?title=Universal_hashing](https://en.wikipedia.org/w/index.php?title=Universal_hashing&oldid=1219538176). |