Summary
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 edit
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)[1][2] | Θ(n) |
| Array | Θ(1) | — | — | — | 0 |
| Dynamic array | Θ(1) | Θ(n) | Θ(1) amortized | Θ(n) | Θ(n)[3] |
| Balanced tree | Θ(log n) | Θ(log n) | Θ(log n) | Θ(log n) | Θ(n) |
| Random-access list | Θ(log n)[4] | Θ(1) | —[4] | —[4] | Θ(n) |
| Hashed array tree | Θ(1) | Θ(n) | Θ(1) amortized | Θ(n) | Θ(√n) |
Maps edit
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:[5]
- 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.
This list is incomplete; you can help by adding missing items. (March 2023) |
| 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[6] | 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 edit
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 edit
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 edit
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[7] of various heap data structures. Function names assume a max-heap. For the meaning of "O(f)" and "Θ(f)" see Big O notation.
| Operation | find-max | delete-max | insert | increase-key | meld |
|---|---|---|---|---|---|
| Binary[7] | Θ(1) | Θ(log n) | O(log n) | O(log n) | Θ(n) |
| Leftist | Θ(1) | Θ(log n) | Θ(log n) | O(log n) | Θ(log n) |
| Binomial[7][8] | Θ(1) | Θ(log n) | Θ(1)[a] | Θ(log n) | O(log n) |
| Skew binomial[9] | Θ(1) | Θ(log n) | Θ(1) | Θ(log n) | O(log n)[b] |
| Pairing[10] | Θ(1) | O(log n)[a] | Θ(1) | o(log n)[a][c] | Θ(1) |
| Rank-pairing[13] | Θ(1) | O(log n)[a] | Θ(1) | Θ(1)[a] | Θ(1) |
| Fibonacci[7][14] | Θ(1) | O(log n)[a] | Θ(1) | Θ(1)[a] | Θ(1) |
| Strict Fibonacci[15] | Θ(1) | O(log n) | Θ(1) | Θ(1) | Θ(1) |
| Brodal[16][d] | Θ(1) | O(log n) | Θ(1) | Θ(1) | Θ(1) |
| 2–3 heap[18] | O(log n) | O(log n)[a] | O(log n)[a] | Θ(1) | ? |
- ^ a b c d e f g h i j k l Amortized time.
- ^ Brodal and Okasaki describe a technique to reduce the worst-case complexity of meld to Θ(1); this technique applies to any heap datastructure that has insert in Θ(1) and find-max, delete-max, meld in O(log n).
- ^ Lower bound of [11] upper bound of [12]
- ^ Brodal and Okasaki later describe a persistent variant with the same bounds except for decrease-key, which is not supported. Heaps with n elements can be constructed bottom-up in O(n).[17]
Notes edit
- ^ Day 1 Keynote - Bjarne Stroustrup: C++11 Style at GoingNative 2012 on channel9.msdn.com from minute 45 or foil 44
- ^ Number crunching: Why you should never, ever, EVER use linked-list in your code again at kjellkod.wordpress.com
- ^ Brodnik, Andrej; Carlsson, Svante; Sedgewick, Robert; Munro, JI; Demaine, ED (1999), Resizable Arrays in Optimal Time and Space (Technical Report CS-99-09) (PDF), Department of Computer Science, University of Waterloo
- ^ a b c Chris Okasaki (1995). "Purely Functional Random-Access Lists". Proceedings of the Seventh International Conference on Functional Programming Languages and Computer Architecture: 86–95. doi:10.1145/224164.224187.
- ^ Mehlhorn, Kurt; Sanders, Peter (2008), "4 Hash Tables and Associative Arrays", Algorithms and Data Structures: The Basic Toolbox (PDF), Springer, pp. 81–98, archived (PDF) from the original on 2014-08-02
- ^ Cormen et al. 2022, p. 484.
- ^ a b c d Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L. (1990). Introduction to Algorithms (1st ed.). MIT Press and McGraw-Hill. ISBN 0-262-03141-8.
- ^ "Binomial Heap | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2019-09-30.
- ^ Brodal, Gerth Stølting; Okasaki, Chris (November 1996), "Optimal purely functional priority queues", Journal of Functional Programming, 6 (6): 839–857, doi:10.1017/s095679680000201x
- ^ Iacono, John (2000), "Improved upper bounds for pairing heaps", Proc. 7th Scandinavian Workshop on Algorithm Theory (PDF), Lecture Notes in Computer Science, vol. 1851, Springer-Verlag, pp. 63–77, arXiv:1110.4428, CiteSeerX 10.1.1.748.7812, doi:10.1007/3-540-44985-X_5, ISBN 3-540-67690-2
- ^ Fredman, Michael Lawrence (July 1999). "On the Efficiency of Pairing Heaps and Related Data Structures" (PDF). Journal of the Association for Computing Machinery. 46 (4): 473–501. doi:10.1145/320211.320214.
- ^ Pettie, Seth (2005). Towards a Final Analysis of Pairing Heaps (PDF). FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science. pp. 174–183. CiteSeerX 10.1.1.549.471. doi:10.1109/SFCS.2005.75. ISBN 0-7695-2468-0.
- ^ Haeupler, Bernhard; Sen, Siddhartha; Tarjan, Robert E. (November 2011). "Rank-pairing heaps" (PDF). SIAM J. Computing. 40 (6): 1463–1485. doi:10.1137/100785351.
- ^ Fredman, Michael Lawrence; Tarjan, Robert E. (July 1987). "Fibonacci heaps and their uses in improved network optimization algorithms" (PDF). Journal of the Association for Computing Machinery. 34 (3): 596–615. CiteSeerX 10.1.1.309.8927. doi:10.1145/28869.28874.
- ^ Brodal, Gerth Stølting; Lagogiannis, George; Tarjan, Robert E. (2012). Strict Fibonacci heaps (PDF). Proceedings of the 44th symposium on Theory of Computing - STOC '12. pp. 1177–1184. CiteSeerX 10.1.1.233.1740. doi:10.1145/2213977.2214082. ISBN 978-1-4503-1245-5.
- ^ Brodal, Gerth S. (1996), "Worst-Case Efficient Priority Queues" (PDF), Proc. 7th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 52–58
- ^ Goodrich, Michael T.; Tamassia, Roberto (2004). "7.3.6. Bottom-Up Heap Construction". Data Structures and Algorithms in Java (3rd ed.). pp. 338–341. ISBN 0-471-46983-1.
- ^ Takaoka, Tadao (1999), Theory of 2–3 Heaps (PDF), p. 12
References edit
- Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2022-04-05). Introduction to Algorithms, fourth edition. MIT Press. ISBN 978-0-262-36750-9.