notebook/notes/hashing/index.md

title	TARGET DECK	FILE TAGS	tags
Hashing	Obsidian::STEM	hashing	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 different 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).

%%ANKI Basic What is considered an ideal (though theoretical) 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 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: Hasing in which an input key 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%%

%%ANKI Basic In practice, hash functions are designed to handle keys of what two types? Back: A fixed-width nonnegative integer or a vector of them. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How does Cormen et al. define a "good" hash function? Back: It satisfies (approximately) the assumption of independent uniform hashing. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

Static Hashing

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.

%%ANKI Basic What does static hashing refer to? Back: Providing a single hash function intended to work well on any data. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Cloze {Static} hashing provides a {single hash function} intended to work well on any data. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic What does it mean for static hashing to be independent? Back: Where a key hashes to is independent of where other keys hash to. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic What about independent static hashing is a bit of a misnomer? Back: N/A. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic What does it mean for static hashing to be uniform? Back: Each key has an equal likelihood of hashing to any slot. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic What about uniform static hashing is a bit of a misnomer? Back: Where keys hash to depend on the input keys' probability distribution. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic In static hashing, why is uniformity generally impossible? Back: Because we use a fixed hash function for all data. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic Assuming m slots, why is static hashing function h(k) = \lfloor km \rfloor not generally "good"? Back: The probability distribution from which keys were drawn may not be uniform. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic What property must an ideal static hashing function exhibit? Back: It must derive hash values independently of any patterns that may exist in the keys. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic What randomization is available to static hashing? Back: The distribution of input keys. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

Division Method

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}.

%%ANKI Basic The division method is used to produce what? Back: A hash function. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic What hyperparameter(s) does the division method require? Back: The number of slots in the hash table. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic Given m slots, the division method produces what hash function? Back: h(k) = k \bmod{m} Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic Let h be a division method hash function. What does h(10) evaluate to? Back: To 10 \bmod{m}, where m is the number of slots in the hash table. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic Consider hash function h(k) = k \bmod{m}. What does m likely represent? Back: The number of slots in the hash table. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic Let m be some number of slots. What m's does the division method typically work best on? Back: A prime not too close to an exact power of 2. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic Why does the division method prefer a prime number of slots? Back: To operate as independently as possible of the input keys. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider hash function h(k) = k \bmod{m}. What method was likely used to produce this? Back: The division method. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic Is the division method an example of static or random hashing? Back: Static. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

Multiplication Method

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.

%%ANKI Basic The multiplication method is used to produce what? Back: A hash function. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic What hyperparameter(s) does the multiplication method require? Back: Slot count m and some constant 0 < A < 1. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic Given m slots and constant A, the multiplication method produces what hash function? Back: h(k) = \lfloor m (kA \bmod{1}) \rfloor Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic What range does the constant A found in the multiplication method take on? Back: 0 < A < 1 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic Consider hash function h(k) = \lfloor m (kA \bmod{1}) \rfloor. What does m likely represent? Back: The number of slots in the hash table. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic Consider hash function h(k) = \lfloor m (kA \bmod{1}) \rfloor. What does A likely represent? Back: Some constant 0 < A < 1. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic Consider constant A used in the multiplication method. Why shouldn't A = 0? Back: Then the produced hash function is constant. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic Consider constant A used in the multiplication method. Why shouldn't A = 1? Back: Then the produced hash function is constant. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic Consider hash function h(k) = \lfloor m (kA \bmod{1}) \rfloor. What method was likely used to produce this? Back: The multiplication method. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::static

END%%

%%ANKI Basic Is the multiplication method an example of static or random hashing? Back: Static. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic For x \in \mathbb{R}^+, what does x \bmod{1} represent? Back: The fractional part of x. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic For x \in \mathbb{R}^+, what expression does x \bmod{1} evaluate to? Back: x - \lfloor x \rfloor Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic For x \in \mathbb{Z}^+, what expression does x \bmod{1} evaluate to? Back: 0 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

Random Hashing

Random hashing refers to choosing a hash function randomly in a way that is independent of the keys being stored.

%%ANKI Basic What does random hashing refer to? Back: Choosing a hash function randomly and independently of the keys being stored. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::random

END%%

%%ANKI Basic What does random hashing avoid that static hashing doesn't? Back: Randomization guarantees no single input always evokes worst-case behavior. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::random

END%%

Universal Hashing

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}.

%%ANKI Basic Which of universal hashing or random hashing more general? Back: Random hashing. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::random hashing::universal

END%%

%%ANKI Basic With respect to universal hashing, what mathematical object is property "universal" attributed to? Back: A finite set of hash functions. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::random hashing::universal

END%%

%%ANKI Basic What does "family" refer to in the context of universal hashing? Back: A finite set of hash functions. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::random hashing::universal

END%%

%%ANKI Basic Consider a hash table with m = 1 slot. Which hash function families are universal? Back: Finite families of hash functions mapping to e.g. \{0\}. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::random hashing::universal

END%%

%%ANKI Basic A "universal family" refers to a finite set of what? Back: Hash functions. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::random hashing::universal

END%%

%%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

END%%

%%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

END%%

%%ANKI Basic Consider universal family \mathscr{H} and universe U. What 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

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

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

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

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

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

END%%

%%ANKI Basic Consider universe U and \mathscr{H} = \{h\} where h(x) = 0. Is \mathscr{H} universal? Back: Not necessarily. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: hashing::random hashing::universal

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

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

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

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

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

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

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.