715 lines
28 KiB
Markdown
715 lines
28 KiB
Markdown
---
|
|
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%%
|
|
|
|
%%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).
|
|
<!--ID: 1722080163402-->
|
|
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).
|
|
<!--ID: 1722080163405-->
|
|
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).
|
|
<!--ID: 1722081955435-->
|
|
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).
|
|
<!--ID: 1722081955439-->
|
|
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).
|
|
<!--ID: 1718197741507-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Given independent uniform hash function $h$, what about $h$ is "independent"?
|
|
Back: Each key $k$ has output $h(k)$ determined independently from other keys.
|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
<!--ID: 1718197741527-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Given independent uniform hash function $h$, what about $h$ is "uniform"?
|
|
Back: Every output of $h$ is equally likely to be any of the values in its range.
|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
<!--ID: 1718197741537-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
With respect to hashing, a random oracle refers to what kind of hash function?
|
|
Back: An independent uniform hash function.
|
|
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
|
|
<!--ID: 1718197741545-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Where does "memory" come into play with independent uniform hash functions?
|
|
Back: 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).
|
|
<!--ID: 1718197741555-->
|
|
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).
|
|
<!--ID: 1719174576842-->
|
|
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).
|
|
<!--ID: 1719174576848-->
|
|
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).
|
|
<!--ID: 1720821498614-->
|
|
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).
|
|
<!--ID: 1720821498625-->
|
|
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%%
|
|
|
|
%%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).
|
|
<!--ID: 1722080163409-->
|
|
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%%
|
|
|
|
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).
|
|
<!--ID: 1722080163416-->
|
|
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).
|
|
<!--ID: 1722080163421-->
|
|
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).
|
|
<!--ID: 1722081955442-->
|
|
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).
|
|
<!--ID: 1722081955446-->
|
|
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).
|
|
<!--ID: 1722080563925-->
|
|
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).
|
|
<!--ID: 1722080563934-->
|
|
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).
|
|
<!--ID: 1722080563937-->
|
|
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).
|
|
<!--ID: 1722080563941-->
|
|
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).
|
|
<!--ID: 1722081955449-->
|
|
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).
|
|
<!--ID: 1722080563945-->
|
|
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).
|
|
<!--ID: 1722080563930-->
|
|
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).
|
|
<!--ID: 1722081955453-->
|
|
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).
|
|
<!--ID: 1722081955457-->
|
|
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).
|
|
<!--ID: 1722081955461-->
|
|
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).
|
|
<!--ID: 1722081955464-->
|
|
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).
|
|
<!--ID: 1722081955468-->
|
|
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).
|
|
<!--ID: 1722081955472-->
|
|
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).
|
|
<!--ID: 1722081955476-->
|
|
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). |