Comparison of data structures
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This is a comparison of the performance of notable data structures, as measured by the complexity of their logical operations. For a more comprehensive listing of data structures, see List of data structures.
The comparisons in this article are organized by abstract data type. As a single concrete data structure may be used to implement many abstract data types, some data structures may appear in multiple comparisons (for example, a hash map can be used to implement an associative array or a set).
Lists
A list or sequence is an abstract data type that represents a finite number of ordered values, where the same value may occur more than once. Lists generally support the following operations:
- peek: access the element at a given index.
- insert: insert a new element at a given index. When the index is zero, this is called prepending; when the index is the last index in the list it is called appending.
- delete: remove the element at a given index.
Peek (index) |
Mutate (insert or delete) at … | Excess space, average | |||
---|---|---|---|---|---|
Beginning | End | Middle | |||
Linked list | Θ(n) | Θ(1) | Θ(1), known end element; Θ(n), unknown end element |
Θ(n) | Θ(n) |
Array | Θ(1) | — | — | — | 0 |
Dynamic array | Θ(1) | Θ(n) | Θ(1) amortized | Θ(n) | Θ(n)[1] |
Balanced tree | Θ(log n) | Θ(log n) | Θ(log n) | Θ(log n) | Θ(n) |
Random-access list | Θ(log n)[2] | Θ(1) | —[2] | —[2] | Θ(n) |
Hashed array tree | Θ(1) | Θ(n) | Θ(1) amortized | Θ(n) | Θ(√n) |
Maps
Summarize
Perspective
Maps store a collection of (key, value) pairs, such that each possible key appears at most once in the collection. They generally support three operations:[3]
- Insert: add a new (key, value) pair to the collection, mapping the key to its new value. Any existing mapping is overwritten. The arguments to this operation are the key and the value.
- Remove: remove a (key, value) pair from the collection, unmapping a given key from its value. The argument to this operation is the key.
- Lookup: find the value (if any) that is bound to a given key. The argument to this operation is the key, and the value is returned from the operation.
Unless otherwise noted, all data structures in this table require O(n) space.
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Data structure | Lookup, removal | Insertion | Ordered | ||
---|---|---|---|---|---|
average | worst case | average | worst case | ||
Association list | O(n) | O(n) | O(1) | O(1) | No |
B-tree[4] | O(log n) | O(log n) | O(log n) | O(log n) | Yes |
Hash table | O(1) | O(n) | O(1) | O(n) | No |
Unbalanced binary search tree | O(log n) | O(n) | O(log n) | O(n) | Yes |
Integer keys
Some map data structures offer superior performance in the case of integer keys. In the following table, let m be the number of bits in the keys.
Data structure | Lookup, removal | Insertion | Space | ||
---|---|---|---|---|---|
average | worst case | average | worst case | ||
Fusion tree | [?] | O(log m n) | [?] | [?] | O(n) |
Van Emde Boas tree | O(log log m) | O(log log m) | O(log log m) | O(log log m) | O(m) |
X-fast trie | O(n log m)[a] | [?] | O(log log m) | O(log log m) | O(n log m) |
Y-fast trie | O(log log m)[a] | [?] | O(log log m)[a] | [?] | O(n) |
Priority queues
Summarize
Perspective
A priority queue is an abstract data-type similar to a regular queue or stack. Each element in a priority queue has an associated priority. In a priority queue, elements with high priority are served before elements with low priority. Priority queues support the following operations:
- insert: add an element to the queue with an associated priority.
- find-max: return the element from the queue that has the highest priority.
- delete-max: remove the element from the queue that has the highest priority.
Priority queues are frequently implemented using heaps.
Heaps
A (max) heap is a tree-based data structure which satisfies the heap property: for any given node C, if P is a parent node of C, then the key (the value) of P is greater than or equal to the key of C.
In addition to the operations of an abstract priority queue, the following table lists the complexity of two additional logical operations:
- increase-key: updating a key.
- meld: joining two heaps to form a valid new heap containing all the elements of both, destroying the original heaps.
Here are time complexities[5] of various heap data structures. The abbreviation am. indicates that the given complexity is amortized, otherwise it is a worst-case complexity. For the meaning of "O(f)" and "Θ(f)" see Big O notation. Names of operations assume a max-heap.
Operation | find-max | delete-max | increase-key | insert | meld | make-heap[b] |
---|---|---|---|---|---|---|
Binary[5] | Θ(1) | Θ(log n) | Θ(log n) | Θ(log n) | Θ(n) | Θ(n) |
Skew[6] | Θ(1) | O(log n) am. | O(log n) am. | O(log n) am. | O(log n) am. | Θ(n) am. |
Leftist[7] | Θ(1) | Θ(log n) | Θ(log n) | Θ(log n) | Θ(log n) | Θ(n) |
Binomial[5][9] | Θ(1) | Θ(log n) | Θ(log n) | Θ(1) am. | Θ(log n)[c] | Θ(n) |
Skew binomial[10] | Θ(1) | Θ(log n) | Θ(log n) | Θ(1) | Θ(log n)[c] | Θ(n) |
2–3 heap[12] | Θ(1) | O(log n) am. | Θ(1) | Θ(1) am. | O(log n)[c] | Θ(n) |
Bottom-up skew[6] | Θ(1) | O(log n) am. | O(log n) am. | Θ(1) am. | Θ(1) am. | Θ(n) am. |
Pairing[13] | Θ(1) | O(log n) am. | o(log n) am.[d] | Θ(1) | Θ(1) | Θ(n) |
Rank-pairing[16] | Θ(1) | O(log n) am. | Θ(1) am. | Θ(1) | Θ(1) | Θ(n) |
Fibonacci[5][17] | Θ(1) | O(log n) am. | Θ(1) am. | Θ(1) | Θ(1) | Θ(n) |
Strict Fibonacci[18][e] | Θ(1) | Θ(log n) | Θ(1) | Θ(1) | Θ(1) | Θ(n) |
Brodal[19][e] | Θ(1) | Θ(log n) | Θ(1) | Θ(1) | Θ(1) | Θ(n)[20] |
- For persistent heaps (not supporting increase-key), a generic transformation reduces the cost of meld to that of insert, while the new cost of delete-max is the sum of the old costs of delete-max and meld.[11] Here, it makes meld run in Θ(1) time (amortized, if the cost of insert is) while delete-max still runs in O(log n). Applied to skew binomial heaps, it yields Brodal-Okasaki queues, persistent heaps with optimal worst-case complexities.[10]
- Brodal queues and strict Fibonacci heaps achieve optimal worst-case complexities for heaps. They were first described as imperative data structures. The Brodal-Okasaki queue is a persistent data structure achieving the same optimum, except that increase-key is not supported.
Notes
References
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