# Hash Table

Hash tables (aka Hash Map or Dictionary) are one of the most commonly used data structures. While Hash Table is a very appropriate name considering their internal workings, Dictionary is a more suitable characterization considering the way they work conceptually. Just like a physical dictionary stores definitions retrievable via a word. Hash Tables store objects that are retrievable via an arbitrary key. What’s extraordinary is that hash table insertions, deletions, and searches are constant time operations (under reasonable assumptions). Many computer science applications can be reduced to repeated lookups; therefore, the ability to perform a search in constant time is invaluable. Consider a Hash Table of people objects using name as a key as depicted below.

people["Donald Knuth"] = { new person object };
people["Edward Dijkstra"] = { new person object };

person Donald = people["Donald Knuth"];


Most mainstream languages have built-in hash table implementations that utilize the array-ese syntax shown above. C however, has no default implementation so the syntax is anomalous by comparison. Please be aware of this while examining the source code.

Consider a standard Array. If the index of any item is known, it can be inserted, deleted, or retrieved in constant time via direct addressing. The innovation of hash tables is using a hash function to convert an arbitrary key into an array index which can then be used for direct addressing. The pseudo code below is a naively simply hash function that converts an integer key to an index. An integer key is used for the sake of simplicity. Other objects, such as strings are also a valid keys. To use a string, it must be first converted to an integer. Languages such as Java and C# provide a hash code method that does this automatically. Converting a string to an integer in C is fairly trivial. Search for djb2 or sbdm for examples.

hash:
key: integer key value
max_index: maximum index to generate; hash table array size

return key mod max_index


mod divides the first number by the second number and returns the remainder. For instance, $26\mod5=1$ because $26/5$ is 5 with a remainder of 1.

This is by no means a valid hash function; however, it’s simple enough to serve as a learning tool and will be used as a reference for the bulk of this section. Depicted below is a hash table of Person objects that has five available slots commonly referred to as buckets. Regardless of the nomenclature, they are nothing more than items in an array. Each Person object has an arbitrary key value that is used to insert and retrieve items from the hash table. In essence, this is how all hash tables work.

#### Simple Hash Table

Although the above conceptually works, there are some fatal shortcomings. For instance, what happens when inserting a Person with a key of 33. Because $33\mod5$ is 3, Edward Dijkstra would be overwritten. This is known as a collision. Collisions occur anytime a hash function generates the same index for two distinct keys. One option is to simply make the array size larger to reduce collisions. For instance, if the array size is 17, Edward Dijkstra is then changed to index 13 ($98\mod17=13$) and the new item to index 16 ($33\mod17=16$). While this is better, it is certainly not adequate. Inserting a key of 81 ($81\mod17=13$) will cause a collision with Edward Dijkstra’s new index. The question becomes, “is it possible to create a hash function that is immune to collisions”.

Unfortunately, the answer to the question above is an unequivocal no. The crux of the issue can be summarized with a single concept: the Pigeonhole Principal. Given x pigeonholes and x + 1 pigeons, it’s impossible to place all the pigeons in pigeonholes without placing two pigeons in a single hole. How does this relate to hash tables? Suppose there are 100 items stored in a hash table using a ten character string as the key. There are $26^{10}$ possible keys. In order for a hash function to produce a unique array index for every possible key, the array would need to be equally sized. Even if the hash table is stored on a industrial grade super computer with virtually unlimited resources, it would be beyond wasteful to use that much memory to store 100 items. This creates a bit of a quandary because it’s theoretically impossible to generate unique indices from keys when the total possible number of keys is larger than the number of possible indices.

The problem is actually even worse than outlined in the previous paragraph. As is already substantiated, collisions are unavoidable. Just how unavoidable is yet to be established. The concept is best illustrated by the Birthday Paradox (technically not a paradox, more of a failing in human intuition). 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 reader is spared the somewhat mundane math behind these facts; however, in the event of unbridled curiosity typing “birthday paradox” into a search engine should satiate. The salient point is that if there is a hash table with 365 possible keys and 23 available storage locations, there is a 50% probability of a collision. It is hopefully obvious that collisions are unavoidable. The only option is to deal with duplicate indices.

One common method for dealing with collisions is chaining1, which is storing a Linked List in each hash table array item. The image below depicts a hash table array with 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.

#### Chaining

The challenge of chaining (and all collision resolution techniques) is that performance degrades as the number of collisions increase. While collisions are a certainty, two techniques for minimizing them are managing load and using an optimal hash function.

Hash Table 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 load gets too high. Obviously, this is an expensive operation that should be avoided when possible. Choosing the starting size of the array is critical.

### Optimal Hash Functions

The quality of the hash algorithm is a major contributing factor to the quantity of collisions. An optimal hash function has two primary attributes: exceptional run time performance and the ability to generate indices that are evenly distributed across the possible values (commonly referred to simple uniform hashing). The naive hash function defined above possesses the first attribute but fails miserably at the second. Imagine if the hash table were storing widgets objects with the price as the key. If the array length is 100, every price that ends with 99 will be stored at the same index. It’s fairly easy to fabricate many data sets that will cause this particular hash function to generate a single index. The question now is, “is it possible to create a hash function that will always generate evenly distributed indices?”.

The unfortunate answer to the question above is no. While it’s true that some hash functions are much better than others, with some advance knowledge of its’ inner workings it is possible to fabricate any number of pathological data sets for any hash function. A pathological data set is a collection of key values that is engineered to prevent even distribution of data. What this means is that any user consuming a hash table can produce a set of keys that forces all items into a single bucket effectively changing the run time from constant to linear. This will enable malicious actors to mount denial of service or similar attacks. Luckily, there is some recourse by using universal hashing.

### Universal Hashing

TO DO: Expand on this - it’s not real obvious what it is based on this description

Universal hashing eliminates the risk of pathological data sets by randomly choosing a hash function independently from the stored keys. Each execution of the algorithm chooses a different hash function, even for the same input. This virtually eliminates the possibility that any one data set can create a single index for all keys. Maintaining several hash functions, randomly assigning them, and tracking which keys used which hash function adds a considerable amount of complexity and overhead. Carefully ensure it is required before attempting to implement it.

### Hash Table Design

The opening paragraph in this section stated: “What’s extraordinary is that hash table insertions, deletions, and searches are constant time operations (under reasonable assumptions)”. Those reasonable assumptions are 1) the keys are not pathological and 2) the hash function is capable of simple uniform hashing. Failing either of these two assumptions results in problems that are rather difficult to troubleshoot. However, a correctly implemented hash table has unparalleled performance. The reward is well worth the risk.

WARNING: The reader may (and probably should) be feeling some anxiety about implementing a hash table at this point. For mission critical applications, it’s highly advisable to use a well known implementation. If that isn’t an option, the recommendation is to further research the topic and scrutinize the code of known experts. The intent of this section is to make the reader understand how hash tables work; it does not represent the state-of-the-art in hash table design. The remainder of this section details how to design a hash table that works well for most non-critical applications.

There are three important decisions when designing a hash table.

1. How to generate hash codes from keys
2. How to compress hash codes into a number between 0 and the count of buckets
3. How many buckets (the length of the array). This decision has implications on maximum load and resizing.

The first decision is likely the most complex. Recall that a good hashing function is both performant and capable of simple uniform hashing. There are many freely available hashing algorithms that have been heavily vetted (e.g. FarmHash, MurmurHash3, SpookyHash). Therefore; it really doesn’t make sense to “roll your own”. One important note is that the design considerations for a cryptographic hash are considerably more complex. The complexity is well warranted but adds non-trivial performance overhead that is ill-suited for hash tables. Non-cryptographic hash functions are the best option.

The second decisions has some implications on the third. A hash is typically a large integer and translating it into an array index has implications on the distribution of the data. The most common reduction method is known as the division method. The division method takes the remainder of dividing the hash ($h$) by the number of available buckets ($m$): $h\mod m$. 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 non-uniform 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 number of buckets is not critical to the distribution of data. The formula for the multiplication method is $\left \lfloor m(hA\mod1) \right \rfloor$ where $A$ is a constant, $h$ is the hash value, and $m$ is the number of buckets. The value of $A$ can be anything; however, the great Donald Knuth suggests that a good general purpose value is $\frac{\sqrt{5}-1}{2}$. The reasons for this are out of scope. The fact that it comes from Donald Knuth should be enough to convince anyone. However, please refer to The Art of Computer Programming v.3 Sorting and Searching for more detail if so desired. As an added bonus, with a bit of ingenuity one can optimize the implementation of the multiplication method for efficiency on most machine architectures.

The final decision is the size of the hash table array. Managing the load of the hash table is very important and avoiding resizing is equally critical. Choose an initial size that will keep the load under 0.7 and also avoid resizing.

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

• De-Duplication: Inserting items into a hash table automatically removes duplicates by nature of the way they work. This has many application such as tracking unique visitors to a web site.
• Symbol Tables: Compilers commonly track symbols using hash tables.

## Asymptotic Complexity

• Insert: $O(1)$
• Delete: $O(1)$
• Find: $O(1)$

Assuming non-pathological data and simple uniform hashing

## Pseudo Code

global variables:
hash_table = array for hash table values

insert:
key = key used to find value in the future
value = value to store in the hash table

index = hash_function(key)

if key exists in hash_table:
update key with new value
return

list_item = new linked list item
list_item->key = key
list_item->value = value
list_item->next = hash_table[index]
hash_table[index] = list_item

delete:
key = key to locate the value to delete

index = hash_function(key)
list_item = hash_table[index]
while list_item is not NULL:
if key = key:
delete list_item from list
return

list_item = list_item->next

find:
key = key to locate the value

index = hash_function(key)
list_item = hash_table[index]
while list_item is not NULL:
if key = key:
return list_item->value

list_item = list_item->next