notebook/notes/hashing/addressing.md

title	TARGET DECK	FILE TAGS	tags
Addressing	Obsidian::STEM	hashing::addressing	addressing hashing

Overview

Direct

Given a universe of keys U = \{0, 1, \ldots, m - 1\}, a direct-address table has m slots. Each slot corresponds to a key in universe U.

%%ANKI Basic Given universe U, how many slots must a direct-address table have? Back: |U| 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 direct-address 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 direct-address table, the element at slot k has what key? Back: k. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Given a direct-address table, an element with key k is placed in what slot? Back: The $k$th slot. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Write pseudocode to test membership of x in direct-address table T[0:m-1]. Back:

bool membership(T, x) {
  return T[x.key] != NIL;
}

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

END%%

%%ANKI Basic What is the worst-cast runtime complexity of direct-address table searches? Back: O(1) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Write pseudocode to insert x into direct-address table T[0:m-1]. Back:

void insert(T, x) {
  T[x.key] = x;
}

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

END%%

%%ANKI Basic What is the worst-case runtime complexity of direct-address table insertions? Back: O(1) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Write pseudocode to delete x from direct-address table T[0:m-1]. Back:

void delete(T, x) {
  T[x.key] = NIL;
}

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

END%%

%%ANKI Basic What is the worst-cast runtime complexity of direct-address table deletions? Back: O(1) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In what situation does direct addressing waste space? Back: When the number of keys used is less than the size of the universe. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In what situation is direct addressing impossible? Back: When the size of the universe is too large to hold in memory. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What distinguishes direct addressing from closed and open addressing? Back: Direct addressing isn't concerned with conflicting keys. Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, https://programming.guide/hash-tables-open-vs-closed-addressing.html.

END%%

%%ANKI Basic Direct addressing sits between what other addressing types? Back: Open and closed addressing. Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, https://programming.guide/hash-tables-open-vs-closed-addressing.html.

END%%

%%ANKI Basic What is the theoretical maximum load factor in direct addressing? Back: 1 Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, https://programming.guide/hash-tables-open-vs-closed-addressing.html.

END%%

Closed

In closed addressing, a key is always stored in the bucket it's hashed to. Collisions are dealt with using separate data structures on a per-bucket basis.

%%ANKI Basic What does "closed" refer to in term "closed addressing"? Back: A key is always stored in the slot it hashes to. Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, https://programming.guide/hash-tables-open-vs-closed-addressing.html.

END%%

%%ANKI Basic What does "open" refer to in term "open hashing"? Back: A key may resides in a data structure separate from the hash table. Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, https://programming.guide/hash-tables-open-vs-closed-addressing.html.

END%%

%%ANKI Cloze {Closed} addressing is also known as {open} hashing. Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, https://programming.guide/hash-tables-open-vs-closed-addressing.html.

END%%

%%ANKI Cloze The following is an example of {closed} addressing. ! Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, https://programming.guide/hash-tables-open-vs-closed-addressing.html.

END%%

%%ANKI Cloze The following is an example of {open} hashing. ! Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, https://programming.guide/hash-tables-open-vs-closed-addressing.html.

END%%

%%ANKI Basic What is the theoretical maximum load factor in closed addressing? Back: N/A Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, https://programming.guide/hash-tables-open-vs-closed-addressing.html.

END%%

%%ANKI Basic Why is the theoretical maximum load factor of closed addressing unbounded? Back: A closed addressing hash table can always have more entries inserted into it. Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, https://programming.guide/hash-tables-open-vs-closed-addressing.html.

END%%

%%ANKI Basic When is the load factor of a closed addressing hash table 0? Back: When no entries are stored in the table. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic When is the load factor of a closed addressing hash table 1? Back: When there exist the same number of total entries as slots. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic When is the load factor of a closed addressing hash table > 1? Back: When there exist more total entries than number of slots. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider a closed addressing hash table of m slots. What is its hash function's domain? Back: U, the universe of keys. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider a closed addressing hash table of m slots. What is its hash function's codomain? Back: \{0, 1, \ldots, m - 1\}, i.e. the m slots. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider open hashing in a table of m slots. What is the hash function's domain? Back: U, the universe of keys. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider open hashing in a table of m slots. What is its hash function's codomain? Back: \{0, 1, \ldots, m - 1\}, i.e. the m slots. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

Ideal Hashing

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 the ideal closed addressing 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: Hashing of a key always produces the same hash value. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is "uniform" in independent uniform hashing? Back: 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" in independent uniform 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%%

Chaining

The most common form of closed addressing is chaining. In this scheme, each slot j is a (nullable) pointer to the head of a linked list containing all the elements with hash value j.

%%ANKI Basic What is the most common implementation of closed addressing? Back: Chaining. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What data structure is typically used in a hash table with chaining? Back: Linked lists. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider a hash table with chaining. What is in an empty slot? Back: A NIL pointer. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider a hash table with chaining. What is in a nonempty slot? Back: A pointer to the head of a linked list. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider a hash table with chaining. How many linked list instances exist? Back: One for each slot in the hash table. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze A hash table with chaining is an example of {closed} addressing. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze A hash table with chaining is an example of {open} hashing. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the worst-case behavior of hashing with chaining? Back: All keys hash to the same slot. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the load factor of a hash table in which all n keys hash to one of m slots? Back: n / m (the load factor is a property of the table, not the distribution of keys). Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a hash table with chaining and independent uniform hashing, what is the average unsuccessful search runtime? Back: Given load factor \alpha, \Theta(1 + \alpha). Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a hash table with chaining and independent uniform hash function h, which elements are examined in an unsuccessful search for element x? Back: All the elements in slot h(x.key). Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a hash table with chaining and independent uniform hashing, what is the average successful search runtime? Back: Given load factor \alpha, \Theta(1 + \alpha). Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a hash table with chaining and independent uniform hash function h, which elements are examined in a successful search for element x? Back: x and the elements preceding x in slot h(x.key). Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a hash table with chaining and independent uniform hashing, what is the average seach runtime? Back: Given load factor \alpha, \Theta(1 + \alpha). Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In a hash table with chaining and independent uniform hashing, when is the average runtime of search O(1)? Back: When the number of entries is at most proportional to 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 Suppose n is at most proportional to m. How is this denoted in complexity notation? Back: n = O(m) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider a hash table of m slots with n = O(m) elements. How is the load factor described in complexity notation? Back: \alpha = O(1) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

Open

In open addressing, keys always reside in the hash table. Collisions are dealt with by searching for other empty buckets within the hash table.

%%ANKI Basic What does "closed" refer to in term "closed hashing"? Back: A key must reside in the hash table. Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, https://programming.guide/hash-tables-open-vs-closed-addressing.html.

END%%

%%ANKI Basic What does "open" refer to in term "open addressing"? Back: A key is not necessarily stored in the slot it hashes to. Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, https://programming.guide/hash-tables-open-vs-closed-addressing.html.

END%%

%%ANKI Cloze {Open} addressing is also known as {closed} hashing. Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, https://programming.guide/hash-tables-open-vs-closed-addressing.html.

END%%

%%ANKI Cloze The following is an example of {closed} hashing. ! Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, https://programming.guide/hash-tables-open-vs-closed-addressing.html.

END%%

%%ANKI Cloze The following is an example of {open} addressing. ! Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, https://programming.guide/hash-tables-open-vs-closed-addressing.html.

END%%

%%ANKI Basic What is the theoretical maximum load factor in open addressing? Back: 1 Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, https://programming.guide/hash-tables-open-vs-closed-addressing.html.

END%%

%%ANKI Basic Why is the theoretical maximum load factor of open addressing bounded? Back: An open addressing hash table can only store as many entries as slots. Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, https://programming.guide/hash-tables-open-vs-closed-addressing.html.

END%%

%%ANKI Basic When is the load factor of an open addressing hash table 0? Back: When no entries are stored in the table. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic When is the load factor of a open addressing hash table 1? Back: When there exist the same number of total entries as slots. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic When is the load factor of an open addressing hash table > 1? Back: N/A Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

Sequential examination of slots during dictionary operations is called probing. Given hash function h, the probe sequence refers to the sequence \langle h(k, 0), h(k, 1), \ldots, h(k, m - 1) \rangle visited when probing. Every probe sequence is expected to be a permutation of \langle 0, 1, \ldots, m - 1 \rangle.

%%ANKI Basic Consider an open addressing hash table of m slots. What is its hash function's domain? Back: Given universe of keys U, U \times \{0, 1, \ldots, m\}. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider an open addressing hash table of m slots. What is its hash function's codomain? Back: \{0, 1, \ldots, m - 1\}, i.e. the m slots. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider closed hashing in a table of m slots. What is the hash function's domain? Back: Given universe of keys U, U \times \{0, 1, \ldots, m\}. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider closed hashing in a table of m slots. What is its hash function's codomain? Back: \{0, 1, \ldots, m - 1\}, i.e. the m slots. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Cloze {Probing} refers to the {sequential examining of slots} performed in open addressing. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In open addressing, probing produces what kind of sequence? Back: A probe sequence. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider open addressed hash table with m slots. What condition must every probe sequence satisfy? Back: Each sequence must be a permutation of \langle 0, 1, \ldots, m - 1 \rangle. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In open addressing, when does probing usually stop? Back: When encountering an empty slot. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic In open addressing, when must probing stop? Back: When every slot was iterated over. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is probing related to open or closed addressing? Back: Open. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Is probing related to open or closed hashing? Back: Closed. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Why are probe sequences expected to be permutations of hash table slots? Back: So every hash table slot is considered as the table fills up. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

Ideal Hashing

An independent uniform permutation hash function is the ideal theoretical abstraction in open addressing. The probe sequence of each key is equally likely to be any of the m! permutations of \langle 0, 1, \ldots, m - 1 \rangle.

%%ANKI Basic What is considered the ideal open addressing hash function? Back: An independent uniform permutation 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 permutation hash function h, what about h is "independent"? Back: Each key's probe sequence is 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 permutation hash function h, what about h is "uniform"? Back: Every probe sequence is equally likely to be any permutation of \langle 0, 1, \ldots, m - 1 \rangle. 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 permutation hash functions? Back: The probe sequence for any key is fixed. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is "uniform" in independent uniform permutation hashing? Back: An input key's probe sequence is equally likely to be any permutation of slots. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is "independent" in independent uniform permutation hashing? Back: An element's probe sequence is independent of those of other elements. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

Bibliography

• “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, https://programming.guide/hash-tables-open-vs-closed-addressing.html.
• Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).