Hash Tables

Interview Essentials

Hash tables are a must-know data structure for technical interviews. This data structure is used to store an unordered collection of (key, value) pairs, where the key has to be unique. Item lookup by key, inserting a new item, or deleting an item are all fast operations (approximately O(1)).

Because they give you quick and cheap insertion, deletion, and lookups, you can use hash tables to solve many different types of interview questions. Hash tables are great for things like determining whether an element belongs to a collection, solving problems with invariants such as anagrams, and for solving problems where there is a unique, non-arbitrary identifier that you can use as a key (for instance, a person’s phone number or ID number).

A hash table is implemented with a function h(x), called a hash function or simply the hash. The hash function takes keys as inputs and returns a bucket index where the information is stored. The hash table becomes less efficient as the individual buckets store more information. A hash table is best suited to sparse problems, roughly where the actual number of keys used is smaller than the conceivable set of keys. For example, there are 10-digit phone numbers, only a small fraction of which are actually in use.

Crucial Terms

• Key: A unique identifier for each value.

• Value: The information we want to access (generally)

Note: You don't have to worry about the following terms to simply use a hash table, but you do need to know them to implement one (or to answer interview questions about how they're implemented!).

• Bucket: Depending on the implementation, this is either where the value data is stored, or a pointer to the valuedata is stored.

• Sparseness: The property that the number of actual keys used is much smaller than the possible keys used. (e.g. there are an infinite number of strings that could be email addresses, and only a finite number in use.)

• Hash or hash-function: A function h(key) that takes the key and returns an index for the "bucket" containing the information that we want. The number of buckets is much smaller than the number of possible keys. Note this means that multiple keys are assigned to the same bucket. If there are K possible keys, and B buckets, the average number of possible keys assigned to a given bucket is K/B.

• Collision: When two keys k1 and k2 have the same hash (i.e. h(k1) == h(k2)) we say the keys collide, so they are assigned to the same bucket. There are different ways of storing different elements whose keys have the same hash.

• Good hash: A good hash is a hash function that provides few collisions. Unfortunately there is no universally good hash; the hash function has to be good for the distribution of keys that actually occur in your problem.

Strengths & Weaknesses

Hash tables combine random access ability with quick insertion and deletion, which makes it an extremely flexible and useful data structure. If you’re doing something other than storing keys and values, or if you need to sort elements or iterate efficiently through them, though, you should use a different data structure.

Strengths

Note: These features assume the hash table is sparse.

• Hash tables have extremely fast lookup by key (O(1)).

• Insertion and deletion of data is also quick (O(1)).

• In a hash table, you can use the keys as the data, and use that to check if we've seen an element before. A hash table used this way is usually called a set.

Weaknesses

• Hash tables have no notion of order.

• Hash tables cannot match "nearby" keys, or keys that share the same prefix. So a hash table wouldn't be a good choice for checking for words that began with a certain prefix (a trie would be a better choice in that case).

• Lookup by value (instead of by key) is O(n).

• A hash table loses its strengths when the amount of data in a single bucket becomes large. Lookup becomes O(B), where B is the number of things in the bucket.

In Interviews, Use Hash Tables When...

• There is a unique identifier that you use for lookup. Examples:

  • Caller ID: Phone numbers are unique and can be used as a key to quickly retrieve a person. Note that we would not use a hash table that found a phone number given a person's name, since many people have the same name.

  • Car owner information: Using the license plate as a key, you can look up the owner of the car (the value).

• You need to quickly determine if an element belongs to a collection. In these cases, you can represent the collection as a hash table with the elements as keys. Example:

  • A Scrabble checker: If we want to determine if word is allowed, we could create a hash table valid_words where the keys were the words. We would set valid_words[w] = 1 for all valid words, so checking if word in valid_words is an O(1) operation.

• You're dealing with problems about invariants. Example:

  • To see if a word has an anagram, we can sort the letters of the string, and use it as a key in a hash table. (This method yields the same key for any two words that are anagrams of one another). The value can be a dummy value (if you're interested in the "yes/no" question of whether a word has an anagram) or an array of strings with the same key (if you are looking for the actual words).

Common Operations Cheat Sheet

The O(...) times listed below are the answers typically expected by interviewers. This assumes that the none of the buckets has a large number of items it it. Items listed as O(1) are more accurately O(B), where B is the size of the bucket assigned to by the hash function.

Operation	Description	Time complexity	Mutates structure
hash[key] = value	Adding a new (key,value) pair to the hash table	O(1)	Yes
del hash[key]	Removes (key, value) from the hash table	O(1)	Yes
key in hash	Lookup whether key is in hash table	O(1)	No

Note: Lookup by value is not supported directly in a hash table. Instead, you'd iterate through all the keys and check for the value you were looking for (an O(n) procedure).