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) | 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
[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:[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.
 |
| 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
[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[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] |
- ^ a b c Amortized time.
- ^ make-heap is the operation of building a heap from a sequence of n unsorted elements. It can be done in Θ(n) time whenever meld runs in O(log n) time (where both complexities can be amortized).[6][7] Another algorithm achieves Θ(n) for binary heaps.[8]
- ^ a b c 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]
- ^ Lower bound of [14] upper bound of [15]
- ^ a b 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
[edit]- ^ 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.
- ^ a b c Sleator, Daniel Dominic; Tarjan, Robert Endre (February 1986). "Self-Adjusting Heaps". SIAM Journal on Computing. 15 (1): 52–69. CiteSeerX 10.1.1.93.6678. doi:10.1137/0215004. ISSN 0097-5397.
- ^ a b Tarjan, Robert (1983). "3.3. Leftist heaps". Data Structures and Network Algorithms. pp. 38–42. doi:10.1137/1.9781611970265. ISBN 978-0-89871-187-5.
- ^ Hayward, Ryan; McDiarmid, Colin (1991). "Average Case Analysis of Heap Building by Repeated Insertion" (PDF). J. Algorithms. 12: 126–153. CiteSeerX 10.1.1.353.7888. doi:10.1016/0196-6774(91)90027-v. Archived from the original (PDF) on 2016-02-05. Retrieved 2016-01-28.
- ^ "Binomial Heap | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2019-09-30.
- ^ a b Brodal, Gerth Stølting; Okasaki, Chris (November 1996), "Optimal purely functional priority queues", Journal of Functional Programming, 6 (6): 839–857, doi:10.1017/s095679680000201x
- ^ Okasaki, Chris (1998). "10.2. Structural Abstraction". Purely Functional Data Structures (1st ed.). pp. 158–162. ISBN 9780521631242.
- ^ Takaoka, Tadao (1999), Theory of 2–3 Heaps (PDF), p. 12
- ^ 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.
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.