Comparison of data structures
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.
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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[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] | Θ(1) | O(log n)[a] | Θ(1)[a] | Θ(1) | O(log n) |
- ^ 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.