How does a hash table work?

Question

I'm looking for an explanation of how a hash table works - in plain English for a simpleton like me!

For example, I know it takes the key, calculates the hash (I am looking for an explanation how) and then performs some kind of modulo to work out where it lies in the array where the value is stored, but that's where my knowledge stops.

Could anyone clarify the process?

Edit: I'm not asking specifically about how hash codes are calculated, but a general overview of how a hash table works.

Solution 1

Here's an explanation in layman's terms.

Let's assume you want to fill up a library with books and not just stuff them in there, but you want to be able to easily find them again when you need them.

So, you decide that if the person that wants to read a book knows the title of the book and the exact title to boot, then that's all it should take. With the title, the person, with the aid of the librarian, should be able to find the book easily and quickly.

So, how can you do that? Well, obviously you can keep some kind of list of where you put each book, but then you have the same problem as searching the library, you need to search the list. Granted, the list would be smaller and easier to search, but still you don't want to search sequentially from one end of the library (or list) to the other.

You want something that, with the title of the book, can give you the right spot at once, so all you have to do is just stroll over to the right shelf, and pick up the book.

But how can that be done? Well, with a bit of forethought when you fill up the library and a lot of work when you fill up the library.

Instead of just starting to fill up the library from one end to the other, you devise a clever little method. You take the title of the book, run it through a small computer program, which spits out a shelf number and a slot number on that shelf. This is where you place the book.

The beauty of this program is that later on, when a person comes back in to read the book, you feed the title through the program once more, and get back the same shelf number and slot number that you were originally given, and this is where the book is located.

The program, as others have already mentioned, is called a hash algorithm or hash computation and usually works by taking the data fed into it (the title of the book in this case) and calculates a number from it.

For simplicity, let's say that it just converts each letter and symbol into a number and sums them all up. In reality, it's a lot more complicated than that, but let's leave it at that for now.

The beauty of such an algorithm is that if you feed the same input into it again and again, it will keep spitting out the same number each time.

Ok, so that's basically how a hash table works.

Technical stuff follows.

First, there's the size of the number. Usually, the output of such a hash algorithm is inside a range of some large number, typically much larger than the space you have in your table. For instance, let's say that we have room for exactly one million books in the library. The output of the hash calculation could be in the range of 0 to one billion which is a lot higher.

So, what do we do? We use something called modulus calculation, which basically says that if you counted to the number you wanted (i.e. the one billion number) but wanted to stay inside a much smaller range, each time you hit the limit of that smaller range you started back at 0, but you have to keep track of how far in the big sequence you've come.

Say that the output of the hash algorithm is in the range of 0 to 20 and you get the value 17 from a particular title. If the size of the library is only 7 books, you count 1, 2, 3, 4, 5, 6, and when you get to 7, you start back at 0. Since we need to count 17 times, we have 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, and the final number is 3.

Of course modulus calculation isn't done like that, it's done with division and a remainder. The remainder of dividing 17 by 7 is 3 (7 goes 2 times into 17 at 14 and the difference between 17 and 14 is 3).

Thus, you put the book in slot number 3.

This leads to the next problem. Collisions. Since the algorithm has no way to space out the books so that they fill the library exactly (or the hash table if you will), it will invariably end up calculating a number that has been used before. In the library sense, when you get to the shelf and the slot number you wish to put a book in, there's already a book there.

Various collision handling methods exist, including running the data into yet another calculation to get another spot in the table (double hashing), or simply to find a space close to the one you were given (i.e. right next to the previous book assuming the slot was available also known as linear probing). This would mean that you have some digging to do when you try to find the book later, but it's still better than simply starting at one end of the library.

Finally, at some point, you might want to put more books into the library than the library allows. In other words, you need to build a bigger library. Since the exact spot in the library was calculated using the exact and current size of the library, it goes to follow that if you resize the library you might end up having to find new spots for all the books since the calculation done to find their spots has changed.

I hope this explanation was a bit more down to earth than buckets and functions :)

Solution 2

Usage and Lingo:

1. Hash tables are used to quickly store and retrieve data (or records).
2. Records are stored in buckets using hash keys
3. Hash keys are calculated by applying a hashing algorithm to a chosen value (the key value) contained within the record. This chosen value must be a common value to all the records.
4. Each bucket can have multiple records which are organized in a particular order.

Real World Example:

Hash & Co., founded in 1803 and lacking any computer technology had a total of 300 filing cabinets to keep the detailed information (the records) for their approximately 30,000 clients. Each file folder were clearly identified with its client number, a unique number from 0 to 29,999.

The filing clerks of that time had to quickly fetch and store client records for the working staff. The staff had decided that it would be more efficient to use a hashing methodology to store and retrieve their records.

To file a client record, filing clerks would use the unique client number written on the folder. Using this client number, they would modulate the hash key by 300 in order to identify the filing cabinet it is contained in. When they opened the filing cabinet they would discover that it contained many folders ordered by client number. After identifying the correct location, they would simply slip it in.

To retrieve a client record, filing clerks would be given a client number on a slip of paper. Using this unique client number (the hash key), they would modulate it by 300 in order to determine which filing cabinet had the clients folder. When they opened the filing cabinet they would discover that it contained many folders ordered by client number. Searching through the records they would quickly find the client folder and retrieve it.

In our real-world example, our buckets are filing cabinets and our records are file folders.

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An important thing to remember is that computers (and their algorithms) deal with numbers better than with strings. So accessing a large array using an index is significantly much faster than accessing sequentially.

As Simon has mentioned which I believe to be very important is that the hashing part is to transform a large space (of arbitrary length, usually strings, etc) and mapping it to a small space (of known size, usually numbers) for indexing. This if very important to remember!

So in the example above, the 30,000 possible clients or so are mapped to a smaller space.

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The main idea in this is to divide your entire data set into segments as to speed up the actual searching which is usually time consuming. In our example above, each of the 300 filing cabinet would (statistically) contain about 100 records. Searching (regardless the order) through 100 records is much faster than having to deal with 30,000.

You may have noticed that some actually already do this. But instead of devising a hashing methodology to generate a hash key, they will in most cases simply use the first letter of the last name. So if you have 26 filing cabinets each containing a letter from A to Z, you in theory have just segmented your data and enhanced the filing and retrieval process.

Hope this helps,

Jeach!

Solution 3

This turns out to be a pretty deep area of theory, but the basic outline is simple.

Essentially, a hash function is just a function that takes things from one space (say strings of arbitrary length) and maps them to a space useful for indexing (unsigned integers, say).

If you only have a small space of things to hash, you might get away with just interpreting those things as integers, and you're done (e.g. 4 byte strings)

Usually, though, you've got a much larger space. If the space of things you allow as keys is bigger than the space of things you are using to index (your uint32's or whatever) then you can't possibly have a unique value for each one. When two or more things hash to the same result, you'll have to handle the redundancy in an appropriate way (this is usually referred to as a collision, and how you handle it or don't will depend a bit on what you are using the hash for).

This implies you want it to be unlikely to have the same result, and you probably also would really like the hash function to be fast.

Balancing these two properties (and a few others) has kept many people busy!

In practice you usually should be able to find a function that is known to work well for your application and use that.

Now to make this work as a hashtable: Imagine you didn't care about memory usage. Then you can create an array as long as your indexing set (all uint32's, for example). As you add something to the table, you hash it's key and look at the array at that index. If there is nothing there, you put your value there. If there is already something there, you add this new entry to a list of things at that address, along with enough information (your original key, or something clever) to find which entry actually belongs to which key.

So as you go a long, every entry in your hashtable (the array) is either empty, or contains one entry, or a list of entries. Retrieving is a simple as indexing into the array, and either returning the value, or walking the list of values and returning the right one.

Of course in practice you typically can't do this, it wastes too much memory. So you do everything based on a sparse array (where the only entries are the ones you actually use, everything else is implicitly null).

There are lots of schemes and tricks to make this work better, but that's the basics.

Solution 4

Lots of answers, but none of them are very visual, and hash tables can easily "click" when visualised.

Hash tables are often implemented as arrays of linked lists. If we imagine a table storing people's names, after a few insertions it might be laid out in memory as below, where ()-enclosed numbers are hash values of the text/name.

bucket#  bucket content / linked list

[0]      --> "sue"(780) --> null
[1]      null
[2]      --> "fred"(42) --> "bill"(9282) --> "jane"(42) --> null
[3]      --> "mary"(73) --> null
[4]      null
[5]      --> "masayuki"(75) --> "sarwar"(105) --> null
[6]      --> "margaret"(2626) --> null
[7]      null
[8]      --> "bob"(308) --> null
[9]      null

A few points:

• each of the array entries (indices [0], [1]...) is known as a bucket, and starts a - possibly empty - linked list of values (aka elements, in this example - people's names)
• each value (e.g. "fred" with hash 42) is linked from bucket [hash % number_of_buckets] e.g. 42 % 10 == [2]; % is the modulo operator - the remainder when divided by the number of buckets
• multiple data values may collide at and be linked from the same bucket, most often because their hash values collide after the modulo operation (e.g. 42 % 10 == [2], and 9282 % 10 == [2]), but occasionally because the hash values are the same (e.g. "fred" and "jane" both shown with hash 42 above)
  • most hash tables handle collisions - with slightly reduced performance but no functional confusion - by comparing the full value (here text) of a value being sought or inserted to each value already in the linked list at the hashed-to bucket

Linked list lengths relate to load factor, not the number of values

If the table size grows, hash tables implemented as above tend to resize themselves (i.e. create a bigger array of buckets, create new/updated linked lists there-from, delete the old array) to keep the ratio of values to buckets (aka load factor) somewhere in the 0.5 to 1.0 range.

Hans gives the actual formula for other load factors in a comment below, but for indicative values: with load factor 1 and a cryptographic strength hash function, 1/e　(~36.8%) of buckets will tend to be empty, another 1/e (~36.8%) have one element, 1/(2e) or ~18.4% two elements, 1/(3!e) about 6.1% three elements, 1/(4!e) or ~1.5% four elements, 1/(5!e) ~.3% have five etc.. - the average chain length from non-empty buckets is ~1.58 no matter how many elements are in the table (i.e. whether there are 100 elements and 100 buckets, or 100 million elements and 100 million buckets), which is why we say lookup/insert/erase are O(1) constant time operations.

How a hash table can associate keys with values

Given a hash table implementation as described above, we can imagine creating a value type such as `struct Value { string name; int age; };`, and equality comparison and hash functions that only look at the `name` field (ignoring age), and then something wonderful happens: we can store `Value` records like `{"sue", 63}` in the table, then later search for "sue" without knowing her age, find the stored value and recover or even update her age - happy birthday Sue - which interestingly doesn't change the hash value so doesn't require that we move Sue's record to another bucket.

When we do this, we're using the hash table as an associative container aka map, and the values it stores can be deemed to consist of a key (the name) and one or more other fields still termed - confusingly - the value (in my example, just the age). A hash table implementation used as a map is known as a hash map.

This contrasts with the example earlier in this answer where we stored discrete values like "sue", which you could think of as being its own key: that kind of usage is known as a hash set.

There are other ways to implement a hash table

Not all hash tables use linked lists (known as separate chaining), but most general purpose ones do, as the main alternative closed hashing (aka open addressing) - particularly with erase operations supported - has less stable performance properties with collision-prone keys/hash functions.

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A few words on hash functions

Strong hashing...

A general purpose, worst-case collision-minimising hash function's job is to spray the keys around the hash table buckets effectively at random, while always generating the same hash value for the same key. Even one bit changing anywhere in the key would ideally - randomly - flip about half the bits in the resultant hash value.

This is normally orchestrated with maths too complicated for me to grok. I'll mention one easy-to-understand way - not the most scalable or cache friendly but inherently elegant (like encryption with a one-time pad!) - as I think it helps drive home the desirable qualities mentioned above. Say you were hashing 64-bit doubles - you could create 8 tables each of 256 random numbers (code below), then use each 8-bit/1-byte slice of the double's memory representation to index into a different table, XORing the random numbers you look up. With this approach, it's easy to see that a bit (in the binary digit sense) changing anywhere in the double results in a different random number being looked up in one of the tables, and a totally uncorrelated final value.

// note caveats above: cache unfriendly (SLOW) but strong hashing...
std::size_t random[8][256] = { ...random data... };
auto p = (const std::byte*)&my_double;
size_t hash = random[0][p[0]] ^
              random[1][p[1]] ^
              ... ^
              random[7][p[7]];

Weak but oft-fast hashing...

Many libraries' hashing functions pass integers through unchanged (known as a trivial or identity hash function); it's the other extreme from the strong hashing described above. An identity hash is extremely collision prone in the worst cases, but the hope is that in the fairly common case of integer keys that tend to be incrementing (perhaps with some gaps), they'll map into successive buckets leaving fewer empty than random hashing leaves (our ~36.8% at load factor 1 mentioned earlier), thereby having fewer collisions and fewer longer linked lists of colliding elements than is achieved by random mappings. It's also great to save the time it takes to generate a strong hash, and if keys are looked up in order they'll be found in buckets nearby in memory, improving cache hits. When the keys don't increment nicely, the hope is they'll be random enough they won't need a strong hash function to totally randomise their placement into buckets.

Solution 5

You guys are very close to explaining this fully, but missing a couple things. The hashtable is just an array. The array itself will contain something in each slot. At a minimum you will store the hashvalue or the value itself in this slot. In addition to this you could also store a linked/chained list of values that have collided on this slot, or you could use the open addressing method. You can also store a pointer or pointers to other data you want to retrieve out of this slot.

It's important to note that the hashvalue itself generally does not indicate the slot into which to place the value. For example, a hashvalue might be a negative integer value. Obviously a negative number cannot point to an array location. Additionally, hash values will tend to many times be larger numbers than the slots available. Thus another calculation needs to be performed by the hashtable itself to figure out which slot the value should go into. This is done with a modulus math operation like:

uint slotIndex = hashValue % hashTableSize;

This value is the slot the value will go into. In open addressing, if the slot is already filled with another hashvalue and/or other data, the modulus operation will be run once again to find the next slot:

slotIndex = (remainder + 1) % hashTableSize;

I suppose there may be other more advanced methods for determining slot index, but this is the common one I've seen... would be interested in any others that perform better.

With the modulus method, if you have a table of say size 1000, any hashvalue that is between 1 and 1000 will go into the corresponding slot. Any Negative values, and any values greater than 1000 will be potentially colliding slot values. The chances of that happening depend both on your hashing method, as well as how many total items you add to the hash table. Generally, it's best practice to make the size of the hashtable such that the total number of values added to it is only equal to about 70% of its size. If your hash function does a good job of even distribution, you will generally encounter very few to no bucket/slot collisions and it will perform very quickly for both lookup and write operations. If the total number of values to add is not known in advance, make a good guesstimate using whatever means, and then resize your hashtable once the number of elements added to it reaches 70% of capacity.

I hope this has helped.

PS - In C# the GetHashCode() method is pretty slow and results in actual value collisions under a lot of conditions I've tested. For some real fun, build your own hashfunction and try to get it to NEVER collide on the specific data you are hashing, run faster than GetHashCode, and have a fairly even distribution. I've done this using long instead of int size hashcode values and it's worked quite well on up to 32 million entires hashvalues in the hashtable with 0 collisions. Unfortunately I can't share the code as it belongs to my employer... but I can reveal it is possible for certain data domains. When you can achieve this, the hashtable is VERY fast. :)

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