Prerequisites (click to view)
graph LR
ALG(["fas:fatrophy Hash Table fas:fatrophy #160;"])
ASY_ANA(["fas:facheck Asymptotic Analysis #160;"])
click ASY_ANA "./mathasymptoticanalysis"
ARRAY(["fas:facheck Array #160;"])
click ARRAY "./listdatastructarray"
LL(["fas:facheck Linked List #160;"])
click LL "./listdatastructlinkedlist"
MF(["fas:facheck Math Functions #160;"])
click MF "./mathfunctions"
BW(["fas:facheck Bitwise Logic #160;"])
click BW "./mathbitwise"
ASY_ANA>ALG
MF>ALG
BW>ALG
ARRAY>ALG
LL>ALG
All the data structures covered to this point (stack, queue, and priority queue) have organized data to optimize retrieval order. This section expands our algorithmic toolbox to accommodate use cases that require locating data at will using a key value. For instance, consider an address book: a desideratum is to locate any contact by name. That is, locate data in memory using a key. A hash table, aka associative array^{1}, hash map or dictionary, is perfectly suited to this purpose. What’s even better is that a hash table is capable of storing, deleting, and locating data in constant time ($\Theta(1)$)^{2}. Many algorithms can be reduced to repeated lookups; therefore, the ability to perform a search in constant time is invaluable.
History
Hans Peter Luhn, an IBM engineer, is typically credited^{3} with the discovery of hashing^{4}. He released a seminal IBM internal memorandum outlining the concept in 1953. He described placing data into “buckets” and calculating a bucket index by “hashing” a key value. IBM immediately began bringing his idea to fruition. Five years later, IBM rocked the computing world by releasing machines capable of indexing large bodies of text. The achievement made front page news.
While hash table is an appropriate name given the data structure’s internal
workings, dictionary is a more suitable characterization based on the abstract
interface. Just like a physical dictionary stores definitions retrievable via a
word, hash tables store objects that are retrievable via an arbitrary key. There
are three typical operations^{5}: put(key, value)
, get(key)
, and
remove(key)
. The put
operation performs an upsert^{6}, get
returns
the value associated with the supplied key, and remove
deletes the key value
pair. This is illustrated below.
With a firm grasp on the abstract interface, it’s time to examine the internal workings of hash tables. There are three associated concepts: direct addressing, hashing^{7}, and collision management. See the sections below.
Direct Addressing
Hash tables store data in arrays. Recall that arrays are contagious segments of memory that support direct addressing. That is to say, assuming a known ordinal index, the memory address of any item is easily calculable in constant ($\Theta(1)$) time. Going back to the example in the previous section, all of the data is stored in an array as shown below.
Assuming the knowledge that the key Richie
is stored at index 1, it can be
retrieved in constant time. Recall from the interface definition that the index
is NOT known. The only information available to the algorithm is the key. How
is it possible to divine the index given only the key? That’s where hashing
comes into play.
Hashing
Hashing literally means chop and mix together. A hash function^{8} accepts an arbitrary sized piece of data, chops it up, and commingles it into a fixed length value. In the case of a hash table, the value is an array index. See the image below.
To demonstrate, consider the naive hash function implementation defined in the succeeding pseudo code. There are a couple things to note. The first is that the choice of accepting a byte array is deliberate. Almost any value is convertible to a byte array so this function is capable of generating an index from almost any type of key. The second is that is uses an $\oplus$ (exclusive or aka xor)^{9} to “hash” individual bytes. In a nutshell, $\oplus$ deterministically scrambles the bits.
\begin{algorithm}
\caption{Naive Hash Function}
\begin{algorithmic}
\REQUIRE key = byte array
\REQUIRE bound = array size
\OUTPUT integer $\geq$ 0 and $\lt$ bound
\FUNCTION{hash}{key}
\STATE hashed $\gets$ key[0]
\FOR{byte $\in$ key}
\STATE hashed $\gets$ hashed $\oplus$ byte
\ENDFOR
\RETURN hashed $\bmod$ bound
\ENDFUNCTION
\end{algorithmic}
\end{algorithm}
WARNING: Naive hash function is a purely pedagogical device aimed at simplicity. It is not suitable for realworld use.
To summarize, a hash table employs a hash function similar to this one to determine the array index in which to store\retrieve constituent data items. There is however a glaring problem with this implementation. A good hash function has the following proprieties:
 Efficiently computable.
 Simple uniform hashing. That is, every output value is equally likely given the input value.
The naive hash function satisfies property one but fails miserably at property two. The return value is derived from a single byte which has a max value of 255; therefore, output values will be clustered in that range. Stated differently, the distribution of output values generated from all possible input values isn’t highly varied. Some indices will occur frequently while others wont occur at all. See the image below.
Another property of simple uniform hashing is that similar inputs must generate
widely varied outputs. It’s reasonable to assume that a consumer will store
tightly correlated keys in a hash table. It’s important that small key
variations generate highly disparate indices. In other words, the keys foo
and
fop
should not produce similar hashes.
To elucidate the uneven distribution problem, consider a hash table backed by an
array of size 8 as depicted below. If the key Hamming
hashes to the same index
as Dijkstra
, they must both somehow be stored at the same index. This is known
as a collision. An uneven distribution of keys increases the likelihood of
collisions.
Creating a good hash function requires statistics expertise and extensive testing. Luckily, there are several simple and effective noncryptographic implementations freely available. For instance djb2^{10}, sbdm, FarmHash, MurmurHash3, and SpookyHash are some popular viable options. Using one of these hash algorithms will probabilistically reduce the likelihood of collisions. The following section explores the possibility of eliminating collisions.
Collisions
Collisions occur anytime a hash function generates the same index for two distinct keys. To create a generalpurpose hash table, there are two options:
 Eliminate collisions
 Store multiple data items in a single array location
This subsection explores the first option.
Assuming one knows every possible key in advance and one has access to a sufficiently sized array, it is theoretically possible to eliminate collisions. Unfortunately, such a scenario is exceptionally rare and collisions are otherwise unavoidable. The proof of said fact is demonstrable with a single succinct thought experiment: the pigeonhole principle^{11}. Given $x$ pigeonholes and $x + 1$ pigeons, it’s impossible to place all the pigeons in distinct pigeonholes. See the illustration below.
To demonstrate the relationship to hashing, imagine that an array index is a pigeonhole, a key is a pigeon, and a hash function accepts a pigeon and assigns it to a pigeonhole. Consider a hash table using keys constrained to 10 lower case letters. This equates to $26^{10}$ possible keys. Each key requires a distinct array location. Even if each array item is a mere 16 bytes in size, the total size of the array would be $26^{10} \times 16 \approx 2259 \space \text{terabyte}$. Obviously, this course of action isn’t feasible.
As mentioned in the previous subsection, it is possible to probabilistically reduce the likelihood of collisions with simple uniform hashing. This is true; however, an actor with intimate knowledge of the employed hashing algorithm can create a pathological data set. A pathological data set is a collection of key values purposefully engineered to prevent even distribution of data. Stated differently, a data set specifically designed to generate a small set of output values given a large set of input values. As the pigeonhole principle illustrates, there will always be multiple input values that generate the same output value. Pathological data is always a possibility^{12}.
To further exacerbate the collision dilemma, the probability of collisions in unintuitively high even with simple uniform hashing. The concept is best illustrated by the infamous birthday paradox^{13}. Suppose there are 367 randomly chosen people in a room. Because there are 365 days in a year, by the pigeonhole principal, there is a 100% probability that two people share the same birthday. This is intuitive; however, the reality that if there are 70 people in the room there is a 99.9% probability of a birthday collision is not so intuitive. In fact, with just 23 people in the room, there is a 50% probability of a collision. The salient point is that if there is a hash table backed by an array of 365 elements, inserting 23 items yields a 50% probability of a collision. Inserting 70 items yields a 99.9% probability of collision.
The bottom line is that it’s theoretically impossible to generate unique indices from keys when the total possible number of keys is larger than the number of indices. Furthermore, the statistical likelihood of collisions is high. The only option is to manage collisions when they inevitably occur.
Chaining
As established, collisions are unavoidable. The only option is to devise a way to map multiple items to a single array index. The most common remedy is chaining^{14}, which is storing a linked list at each array item. The image below depicts a hash table with integer keys. The array has eight available slots and eight stored items. There were four collisions at index two, so four items are stored in that bucket. Locating an item in the hash table involves generating a hash, locating the desired linked list, and searching it.
The challenge of chaining^{15} is that performance degrades as the number of collisions increase. The larger the linked lists becomes, the more searching is required.
Reducing Collisions
There are three implementation details that have impact on collision frequency.
 Array size
 Choice of hash function
 Method to compress hash codes into array indices
They are explored in detail in the subsections below.
Array Size
It goes without saying, the larger the array the lower the probability of collisions. A common metric for array sizing is load. Load is the ratio of the count of stored objects to the length of the array. In the chaining image above, the load is $1$ because there are $8$ stored objects and the array has a length of $8$. A rule of thumb is to keep the load below $0.7$. In practice, this means resizing the array and rehashing all the items when the load gets too high. Obviously, this is an expensive operation that should be avoided when possible. Choosing a suitable starting size that balances the cost of resizing with memory consumption is critical.
Choice of Hash Function
Much has already been said about hash functions in the hashing subsection. Recall that simple uniform hashing reduces the likelihood of collisions so this is the highest concern. There are several freely available hashing algorithms to choose from, all of which have similar characteristics. Choose whichever is most convenient and available for your platform. It’s not recommended to “roll your own” hashing algorithm.
As a final matter of note, hashing functions come in two flavors: cryptographic and noncryptographic. Cryptographic functions have stronger hashing characteristics but are less performant. It’s recommend to stick with noncryptographic hashing for hash tables because they are more efficiency computable.
Hash Compression
Hash functions typically return a number with a large range. For instance, FarmHash generates a 64bit unsigned integer^{16} that has a range of $0$ to $18446744073709551615$ inclusive. The hash value must be further compressed into an array bound. This is depicted graphically below.
Compressing a hash value to an array index has implications on the distribution of array indices. There are two common compression techniques: division and multiplication.
The most common reduction method is known as the division method. If $h$ is the hash value and $m$ is the size of the array, the compression formula is $h \bmod m$. This works well is many cases. However, one has to carefully consider the value of $m$. Any common factors between the input keys and $m$ will result in a nonuniform distribution of data. The best way to mitigate this risk to ensure that $m$ has as few factors as possible; therefore, prime numbers are ideal.
The multiplication method is another means of compressing a hash into an array index. It is more complex but has the advantage that the array size is not critical to the distribution of data. The formula for the multiplication method is $\left \lfloor m(hA \bmod 1) \right \rfloor$ where $A$ is a constant, $h$ is the hash value, and $m$ is the array size. The value of $A$ can be anything between 0 and 1; however, Donald Knuth suggests that a good general purpose value is $\frac{\sqrt{5}1}{2}$^{17}.
In conclusion, making intelligent decisions for the three most important design considerations will ensure a performant hash table. Unfortunately, there is a great deal of nuance involved that is easy to overlook. Hash tables should not be implemented hastily. Properly implemented, hash tables allow for insertion, deletion, and search in constant time.
Applications
 Countless general purpose programming tasks
 DeDuplication: Inserting items into a hash table automatically removes duplicates by nature of the way they work. This has many applications such as tracking unique visitors to a web site.
 Symbol Tables: Compilers commonly track symbols using hash tables.
Pseudo Code
\begin{algorithm}
\caption{Hash Table Abstract Interface}
\begin{algorithmic}
\REQUIRE key = key to use for future lookups
\REQUIRE value = value to add to the queue
\FUNCTION{put}{key, value}
\IF{key $\in$ HT}
\STATE HT[key] $\gets$ value
\ELSE
\STATE HT $\gets$ HT $\cup$ (key, value)
\ENDIF
\ENDFUNCTION
\REQUIRE key = key to search for
\OUTPUT value assoicated with key
\FUNCTION{get}{ht, key}
\RETURN{HT[key]}
\ENDFUNCTION
\REQUIRE key = key to remove
\FUNCTION{remove}{ht, key}
\STATE HT $\gets$ HT  (key, value)
\ENDFUNCTION
\end{algorithmic}
\end{algorithm}
Asymptotic Complexity
Assuming a minimal load, nonpathological data, and simple uniform hashing.
 Put: $O(1)$
 Get: $O(1)$
 Remove: $O(1)$
Source Code
Relevant Files:
Click here for build and run instructions
Exercises

Indicate true of false for each of the following statements.
a. Simple uniform hashing eliminates collisions.
b. Simple uniform hashing makes collisions improbable.
c. Collisions are avoidable.Answers (click to expand)
 False  Simple uniform hashing reduces the likelihood of collisions.
 False  The birthday paradox demonstrates that with 365 possible array indices there is a 50% probability of a collision after just 23 `put` operations.

Either response is acceptable given the perspective.
True  Strictly speaking, this is theoretically true if you have advanced knowledge of all possible keys and a large enough array. This isn't feasible in practice.
False  In practice, collisions are not avoidable. The pigeonhole principal provides a simple proof.
 What is the worst case asymptotic complexity for a hash table
get
operation given the following details: Load $\leq 0.5$
 Chaining collision management strategy
 Hash function capable of simple uniform hashing
Answers (click to expand)
Linear $O(n)$  Even with the best designed hash table, it's possible to create a pathological data set that hashes all keys to the same array index. In this case, all items will be stored in a single linked list. The asymptotic complexity of a search operation on a linked list is $O(n)$.
As a matter of note, this scenario is highly unlikely. Such a data set must be deliberately engineered using intimate knowledge of the hash table implementation details. 
Assume you have a hash table backed by an array of size 50 (indices 0  49). Given the hash values below, calculate an array index using both the division and multiplication methods. For the multiplication method, use the constant value of $A = \frac{\sqrt{5}1}{2}$.
a. 123456787654321
b. 243798023461879
c. 987654321123490
d. 241378973452560
e. 246897659028600Answers (click to expand)

division = 21
multiplication = 20 
division = 29
multiplication = 39 
division = 40
multiplication = 7 
division = 10
multiplication = 43 
division = 0
multiplication = 20

division = 21
 Specify detailed pseudo code for a hash table implementation with the
following characteristics:
 Consumer specifiable hash function
 Chaining collision management strategy
 Multiplication method hash compression
For formatting purposes, the answers to question 4 are located here.
 Using the programming language of your choice, create a Least Recently Used
(LRU) cache. The abstract interface for the cache is listed below. Assume
that all hash table operations are constant ($\Theta(1)$). The requirements
are:
 Asymptotic complexity = $O(\log{n})$
 Use a hash table
 Use one of the other data structures covered to this point (stack, queue, or priority queue)
 Every item must maintain its last accessed time.
 Every
get
operation must update the last accessed time of the target item  When the cache size exceeds 50, purge the least recently accessed item from the cache
Answers (click to expand)
See the C implementation provided in the links below. Your implementation may vary significantly and still be correct.
 cache.h
 cache.c
 cache_test.c
\begin{algorithm}
\caption{LRU Cache Abstract Interface}
\begin{algorithmic}
\REQUIRE key = arbirary key value
\REQUIRE func = function that returns cacheable value
\OUTPUT value associated with key
\FUNCTION{get}{key, func}
\IF{key $\notin$ C}
\STATE C[key] $\gets$ func(key)
\ENDIF
\STATE C[key].lastAccess $\gets$ now
\RETURN{C[key]}
\ENDFUNCTION
\end{algorithmic}
\end{algorithm}

The term, as applied to computer science, has multiple meanings. It is however common to see the terms hash table and associative array used synonymously. ↩

This is generally correct under reasonable assumptions. It is possible, in deliberately extreme circumstances, to construct pathological data that changes all hash table operations from constant to linear. ↩

Wikipedia reports that a few other people simultaneously converged on the concept. However, authoritative sources of corroborating information are sparse. ↩

See the IEE article Hans Peter Luhn and the Birth of the Hashing Algorithm for a fascinating historical account. ↩

Just like most data structures, while the functionally is fairly standard, the naming conventions are not.
add
,delete
, andsearch
are also common. ↩ 
upsert
is a portmanteau ofupdate
andinsert
: if the key exists its value is updated otherwise the key value pair is inserted. ↩ 
Hashing is also commonly employed in cryptography. Cryptographic hashing is considerably more complex and beyond the scope of this work. ↩

Function as in software function, not math function. As will be explained later, a single hash function domain value may map to multiple range values. ↩

See the bitwise logic prerequisite for a full treatment of the concept. ↩

Also known as Bernstein’s hash. ↩

See the Wikipedia article for additional details. ↩

It is possible to mitigate pathological data with universal hashing. In as concise terms as possible, it’s a technique whereas the hash functions are chosen at random and independently from the stored keys. It is not commonly used in conjunction with for hash tables; however, it’s a good general purpose approach to avoiding pathological data. ↩

It’s technically not a paradox, it’s more of a failing in human intuition. There is an excellent description available here ↩

There are other collision resolution strategies such as open addressing. All are bound by the same constraints and the concepts are sufficiently transferable. ↩

The same is true of all collision resolution techniques. ↩

It can also produce a 32 bit hash value. ↩

The Art of Computer Programming Volume 3 Sorting and Searching p. 516 ↩