notebook/notes/hashing/addressing.md

---
title: Addressing
TARGET DECK: Obsidian::STEM
FILE TAGS: hashing::addressing
tags:
  - 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).
<!--ID: 1716046153762-->
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).
<!--ID: 1716046153766-->
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).
<!--ID: 1716046153770-->
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).
<!--ID: 1716046153775-->
END%%

%%ANKI
Basic
Write pseudocode to test membership of $x$ in direct-address table `T[0:m-1]`.
Back:
```c
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).
<!--ID: 1716046153781-->
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).
<!--ID: 1716307180982-->
END%%

%%ANKI
Basic
Write pseudocode to insert $x$ into direct-address table `T[0:m-1]`.
Back:
```c
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).
<!--ID: 1716046153785-->
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).
<!--ID: 1716307180983-->
END%%

%%ANKI
Basic
Write pseudocode to delete $x$ from direct-address table `T[0:m-1]`.
Back:
```c
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).
<!--ID: 1716046153789-->
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).
<!--ID: 1716307180984-->
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).
<!--ID: 1716307180986-->
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).
<!--ID: 1716307180987-->
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](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718199205862-->
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](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718199205872-->
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](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718759188227-->
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](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718198717474-->
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](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718198717484-->
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](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718198717495-->
END%%

%%ANKI
Cloze
The following is an example of {closed} addressing.
![[closed-addressing.png]]
Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718198717506-->
END%%

%%ANKI
Cloze
The following is an example of {open} hashing.
![[closed-addressing.png]]
Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718198755496-->
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](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718759188231-->
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](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718759188234-->
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).
<!--ID: 1718759188238-->
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).
<!--ID: 1718759188241-->
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).
<!--ID: 1718759188245-->
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).
<!--ID: 1718759188249-->
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).
<!--ID: 1718759188252-->
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).
<!--ID: 1718759188256-->
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).
<!--ID: 1718759188261-->
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).
<!--ID: 1718759188269-->
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).
<!--ID: 1718759188275-->
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).
<!--ID: 1718759188281-->
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).
<!--ID: 1719174576856-->
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).
<!--ID: 1719174576860-->
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).
<!--ID: 1719174576864-->
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).
<!--ID: 1719176493045-->
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).
<!--ID: 1719176493050-->
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).
<!--ID: 1719176797748-->
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).
<!--ID: 1719176797752-->
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).
<!--ID: 1719176797756-->
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).
<!--ID: 1719176797760-->
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](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718198717434-->
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](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718198717447-->
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](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718198717455-->
END%%

%%ANKI
Cloze
The following is an example of {closed} hashing.
![[open-addressing.png]]
Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718198717464-->
END%%

%%ANKI
Cloze
The following is an example of {open} addressing.
![[open-addressing.png]]
Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718198755486-->
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](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718759188171-->
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](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718759188176-->
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).
<!--ID: 1718759188179-->
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).
<!--ID: 1718759188182-->
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).
<!--ID: 1718759188186-->
END%%

## Bibliography

* “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](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).