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).

## Contents

* 1 Lists
* 2 Maps
* 2.1 Integer keys
* 3 Priority queues
* 3.1 Heaps
* 4 Notes
* 5 References

## Lists

Further information: List (abstract data type)

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.

Comparison of list data structures  | Linked list | Array | Dynamic array | Balanced tree | Random access list | Hashed array tree

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Indexing  | Θ(n)  | Θ(1)  | Θ(1)  | Θ(log n)  | Θ(log n)[1] | Θ(1)
Insert/delete at beginning  | Θ(1)  | N/A  | Θ(n)  | Θ(log n)  | Θ(1)  | Θ(n)
Insert/delete at end  | Θ(1) when last element is known;
Θ(n) when last element is unknown | N/A  | Θ(1) amortized | Θ(log n)  | Θ(log n) updating  | Θ(1) amortized
Insert/delete in middle  | search time + Θ(1)[2][3] | N/A  | Θ(n)  | Θ(log n)  | Θ(log n) updating  | Θ(n)
Wasted space (average)  | Θ(n)  | 0  | Θ(n)[4] | Θ(n)  | Θ(n)  | Θ(√n)

## Maps

Further information: Associative array

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 expanding it.

Data structure  | Lookup, removal  | Insertion  | Ordered

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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

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

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average  | worst case  | average  | worst case
Fusion tree | Template:Data missing | O(log m n)  | Template:Data missing | Template:Data missing | 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)[lower-alpha 1] | Template:Data missing | O(log log m)  | O(log log m)  | O(n log m)
Y-fast trie | O(log log m)[lower-alpha 1] | Template:Data missing | O(log log m)[lower-alpha 1] | Template:Data missing | O(n)

## Priority queues

Further information: Priority queue

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

Further information: Heap (data structure)

A (max) heap is a tree-based data structure which satisfies the Template:Dfni: 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 min-heap. For the meaning of "O(f)" and "Θ(f)" see Big O notation.

Operation  | find-min  | delete-min  | insert  | decrease-key  | meld

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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)[lower-alpha 1] | Θ(log n)  | O(log n)[lower-alpha 2]
Fibonacci[7][9] | Θ(1)  | O(log n)[lower-alpha 1] | Θ(1)  | Θ(1)[lower-alpha 1] | Θ(1)
Pairing[10] | Θ(1)  | O(log n)[lower-alpha 1] | Θ(1)  | o(log n)[lower-alpha 1][lower-alpha 3] | Θ(1)
Brodal[13][lower-alpha 4] | Θ(1)  | O(log n)  | Θ(1)  | Θ(1)  | Θ(1)
Rank-pairing[15] | Θ(1)  | O(log n)[lower-alpha 1] | Θ(1)  | Θ(1)[lower-alpha 1] | Θ(1)
Strict Fibonacci[16] | Θ(1)  | O(log n)  | Θ(1)  | Θ(1)  | Θ(1)
2-3 heap[17] | O(log n)  | O(log n)[lower-alpha 1] | O(log n)[lower-alpha 1] | Θ(1)  | ?

1. ↑ 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 Amortized time.
2. ↑ n is the size of the larger heap.
3. ↑ Lower bound of Ω(log⁡log⁡n),[11] upper bound of O(22log⁡log⁡n).[12]
4. ↑ 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).[14]

## Notes

1. ↑ 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.
2. ↑ Day 1 Keynote - Bjarne Stroustrup: C++11 Style at GoingNative 2012 on channel9.msdn.com from minute 45 or foil 44
3. ↑ Number crunching: Why you should never, ever, EVER use linked-list in your code again at kjellkod.wordpress.com
4. ↑ Brodnik, Andrej; Carlsson, Svante; Sedgewick, Robert; Munro, JI; Demaine, ED (1999), Resizable Arrays in Optimal Time and Space (Technical Report CS-99-09), Department of Computer Science, University of Waterloo, http://www.cs.uwaterloo.ca/research/tr/1999/09/CS-99-09.pdf
5. ↑ Mehlhorn, Kurt; Sanders, Peter (2008), "4 Hash Tables and Associative Arrays", Algorithms and Data Structures: The Basic Toolbox, Springer, pp. 81–98, http://people.mpi-inf.mpg.de/~mehlhorn/ftp/Toolbox/HashTables.pdf
6. ↑ Cormen et al. 2022, p. 484.
7. ↑ 7.0 7.1 7.2 7.3 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.
8. ↑ "Binomial Heap | Brilliant Math & Science Wiki" (in en-us). https://brilliant.org/wiki/binomial-heap/.
9. ↑ Fredman, Michael Lawrence; Tarjan, Robert E. (July 1987). "Fibonacci heaps and their uses in improved network optimization algorithms". Journal of the Association for Computing Machinery 34 (3): 596-615. doi:10.1145/28869.28874. http://bioinfo.ict.ac.cn/~dbu/AlgorithmCourses/Lectures/Fibonacci-Heap-Tarjan.pdf.
10. ↑ Iacono, John (2000), "Improved upper bounds for pairing heaps", Proc. 7th Scandinavian Workshop on Algorithm Theory, Lecture Notes in Computer Science, 1851, Springer-Verlag, pp. 63–77, doi:10.1007/3-540-44985-X_5, ISBN 3-540-67690-2, http://john2.poly.edu/papers/swat00/paper.pdf
11. ↑ Fredman, Michael Lawrence (July 1999). "On the Efficiency of Pairing Heaps and Related Data Structures". Journal of the Association for Computing Machinery 46 (4): 473-501. doi:10.1145/320211.320214. http://www.uqac.ca/azinflou/Fichiers840/EfficiencyPairingHeap.pdf.
12. ↑ Pettie, Seth (2005). "Towards a Final Analysis of Pairing Heaps". FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science. pp. 174-183. doi:10.1109/SFCS.2005.75. ISBN 0-7695-2468-0. http://web.eecs.umich.edu/~pettie/papers/focs05.pdf.
13. ↑ Brodal, Gerth S. (1996), "Worst-Case Efficient Priority Queues", Proc. 7th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 52–58, http://www.cs.au.dk/~gerth/papers/soda96.pdf
14. ↑ 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.
15. ↑ Haeupler, Bernhard; Sen, Siddhartha; Tarjan, Robert E. (November 2011). "Rank-pairing heaps". SIAM J. Computing 40 (6): 1463–1485. doi:10.1137/100785351. http://sidsen.org/papers/rp-heaps-journal.pdf.
16. ↑ Brodal, Gerth Stølting; Lagogiannis, George; Tarjan, Robert E. (2012). "Strict Fibonacci heaps". Proceedings of the 44th symposium on Theory of Computing - STOC '12. pp. 1177–1184. doi:10.1145/2213977.2214082. ISBN 978-1-4503-1245-5. http://www.cs.au.dk/~gerth/papers/stoc12.pdf.
17. ↑ Takaoka, Tadao (1999), Theory of 2–3 Heaps, pp. 12, https://ir.canterbury.ac.nz/bitstream/handle/10092/14769/2-3heaps.pdf

## References

* Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2022-04-05) (in en). Introduction to Algorithms, fourth edition. MIT Press. ISBN 978-0-262-36750-9. https://books.google.com/books?id=RSMuEAAAQBAJ&dq=cormen+algorithms&pg=PR13.

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