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208 lines
7.3 KiB
Markdown
208 lines
7.3 KiB
Markdown
---
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id: breaking
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slug: /hash-tables/breaking
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title: Breaking Hash Table
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description: |
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How to get the linear time complexity in a hash table.
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tags:
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- cpp
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- python
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- hash-tables
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last_update:
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date: 2023-11-28
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---
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We will try to break a hash table and discuss possible ways how to prevent such
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issues to occur.
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## Introduction
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Hash tables are very commonly used to represent sets or dictionaries. Even when
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you look up solution to some problem that requires set or dictionary, it is more
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than likely that you'll find something that references usage of the hash table.
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You might think it's the only possible option[^1], or it's the best one[^2].
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One of the reasons to prefer hash tables over any other representation is the
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fact that they are **supposed** to be faster than the alternatives, but the
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truth lies somewhere in between.
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One of the other possible implementations of the set is a balanced tree. Majorly
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occurring implementations rely on the _red-black tree_, but you may see also
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others like an _AVL tree_[^3] or _B-tree_[^4].
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## Hash Table v. Trees
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The most interesting part are the differences between their implementations. Why
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should you choose hash table, or why should you choose the tree implementation?
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Let's compare the differences one by one.
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### Requirements
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We will start with the fundamentals on which the underlying data structures
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rely. We can also consider them as _requirements_ that must be met to be able to
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use the underlying data structure.
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Hash table relies on the _hash function_ that is supposed to distribute the keys
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in such way that they're evenly spread across the slots where the keys (or
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pairs, for dictionary) are stored, but at the same time they're somewhat unique,
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so no clustering occurs.
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Trees depend on the _ordering_ of the elements. They maintain the elements in
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a sorted fashion, so for any pair of the elements that are used as keys, you
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need to be able to decide which one of them is _smaller or equal to_ the other.
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Hash function can be easily created by using the bits that _uniquely_ identify
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a unique element. On the other hand, ordering may not be as easy to define.
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:::tip Example
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If you are familiar with complex numbers, they are a great example of a key that
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does not have ordering (unless you go element-wise for the sake of storing them
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in a tree; though the ordering **is not** defined on them).
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Hashing them is much easier though, you can just “combine” the hashes of the
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real and imaginary parts of the complex number to get a hash of the complex
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number itself.
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:::
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### Underlying data structure
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The most obvious difference is the _core_ of the idea behind these data
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structures. Hash tables rely on data being stored in one continuous piece of
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memory (the array) where you can “guess” (by using the hash function) the
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location of what you're looking for in a constant time and also access that
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location in the, said, constant time[^5]. In case the hash function is
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_not good enough_[^6], you need to go in _blind_, and if it comes to the worst,
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check everything.
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:::tip tl;dr
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- I know where should I look
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- I can look there instantenously
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- If my guesses are very wrong, I might need to check everything
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:::
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On the other hand, tree implementations rely on the self-balancing trees in
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which you don't get as _amazing_ results as with the hash table, but they're
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**consistent**. Given that we have a self-balancing tree, the height of the tree
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is same for **every** input and therefore checking for any element can take the
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same time even in the worst case.
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:::tip tl;dr
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- I don't know where to look
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- I know how to get there
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- Wherever I look, it takes me about the same time
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:::
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Let's compare side by side:
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| time complexity | hash table | tree |
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| --------------: | :--------------------: | :-------------------: |
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| expected | constant | depends on the height |
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| worst-case | gotta check everything | depends on the height |
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## Major Factors of Hash Tables
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Let's have a look at the major factors that affect the efficiency and
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functioning of a hash table. We have already mentioned the hash function that
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plays a crucial role, but there are also different ways how you can implement
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a hash table, so we will have a look at those too.
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### Hash functions
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:::info
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We will start with a definition of hash function in a mathematical definition
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and type signature in some known language:
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$$
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h : T \rightarrow \mathbb{N}
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$$
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For a type signature we will just take the declaration from C++[^7]:
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```cpp
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std::size_t operator()(const T& key) const;
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```
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If you compare with the mathematical definition, it is very similar, except for
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the fact that the memory is not unlimited, so the _natural number_ turned into
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an _unsigned integer type_ (on majority of platforms it will be a 64-bit
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unsigned integer).
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:::
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As we have already touched above, hash function gives “a guess” where to look
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for the key (either when doing a look up, or for insertion to guess a suitable
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spot for the insertion).
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Hash functions are expected to have a so-called _avalanche effect_ which means
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that the smallest change to the key should result in a massive change of hash.
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Avalanche effect technically guarantees that even when your data are clustered
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together, it should lower the amount of conflicts that can occur.
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:::tip Exercise for the reader
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Try to give an example of a hash function that is not good at all.
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:::
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### Implementation details
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There are different variations of the hash tables. You've more than likely seen
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an implementation that keeps linked lists for buckets. However there are also
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other variations that use probing instead.
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With regards to the implementation details, we need to mention the fact that
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even with the bounded hash (as we could've seen above), you're not likely to
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have all the buckets for different hashes available. Most common approach to
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this is having a smaller set of buckets and modifying the hash to fit within.
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One of the most common approaches is to keep lengths of the hash tables in the
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powers of 2 which allows bit-masking to take place.
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:::tip Example
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Let's say we're given `h = 0xDEADBEEF` and we have `l = 65536=2^16` spots in our
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hash table. What can we do here?
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Well, we definitely have a bigger hash than spots available, so we need to
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“shrink” it somehow. The most common practice is to take the lower bits of the
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hash to represent an index in the table:
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```
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h & (l - 1)
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```
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_Why does this work?_ Firstly we subtract 1 from the length (indices run from
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`⟨0 ; l - 1⟩`, since table is zero-indexed). Therefore if we do _binary and_ on
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any number, we always get a valid index within the table. Let's find the index
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for our hash:
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```
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0xDEADBEEF & 0xFFFF = 0xBEEF
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```
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:::
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[^1]: not true
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[^2]: also not true
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[^3]: actually the first of its kind (the self-balanced trees)
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[^4]:
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Rust chose to implement this instead of the common choice of the red-black
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or AVL tree; main difference lies in the fact that B-trees are not binary
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trees
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[^5]:
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This, of course, does not hold true for the educational implementations of
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the hash tables where conflicts are handled by storing the items in the
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linked lists. In practice linked lists are not that commonly used for
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addressing this issue as it has even worse impact on the efficiency of the
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data structure.
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[^6]: My guess is not very good, or it's really bad…
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[^7]: https://en.cppreference.com/w/cpp/utility/hash
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