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{{Short description|Variant of heap data structure}}
{{Infobox data structure
| name = Binary (min) heap
| type = binary tree/heap
| invented_by = [[J. W. J. Williams]]
| invented_year = 1964
<!-- NOTE:
For the purposes of "Big O" notation, all bases are equivalent, because changing the base of a log only introduces a constant factor.
Since the base of logarithms doesn't matter, please do not write complexity expressions that indicate base-2 (or any other base).
DON'T: O(lg n), O(log2 n), O(log_2 n), O(ln n), O(log10 n), etc.
|insert_avg=O(1)▼
-->|
▲| insert_avg = O(1)
|delete_min_worst=O(log ''n'')▼
|
|
|find_min_avg=O(1)▼
| decrease_key_worst = O(log ''n'')
|find_min_worst=O(1)▼
▲| find_min_avg = O(1)
|merge_avg=O(''n'')▼
▲| find_min_worst = O(1)
|merge_worst=O(''n'')}}▼
▲| merge_avg = O(''n'')
| space_avg = O(n)
| space_worst = O(n)
}}
[[File:Max-Heap.svg|thumb|right|Example of a complete binary max-heap]]
[[File:Min-heap.png|thumb|right|Example of a complete binary min heap]]
A '''binary heap''' is a [[heap (data structure)|heap]] [[data structure]] that takes the form of a [[binary tree]]. Binary heaps are a common way of implementing [[priority queue]]s.{{r|clrs|pp=162–163}} The binary heap was introduced by [[J. W. J. Williams]] in 1964
A binary heap is defined as a binary tree with two additional constraints:<ref>{{citation | author=Y Narahari | title=Data Structures and Algorithms | chapter=Binary Heaps | url=https://gtl.csa.iisc.ac.in/dsa/ | chapter-url=http://lcm.csa.iisc.ernet.in/dsa/node137.html}}</ref>
*Shape property: a binary heap is a ''[[complete binary tree]]''; that is, all levels of the tree, except possibly the last one (deepest) are fully filled, and, if the last level of the tree is not complete, the nodes of that level are filled from left to right.
*Heap property: the key stored in each node is either greater than or equal to (≥) or less than or equal to (≤) the keys in the node's children, according to some [[total order]].
Heaps where the parent key is greater than or equal to (≥) the child keys are called ''max-heaps''; those where it is less than or equal to (≤) are called ''min-heaps''. Efficient (that is, [[logarithmic time]]) algorithms are known for the two operations needed to implement a priority queue on a binary heap:
*Inserting an element *Removing the smallest or largest element from (respectively) a min-heap or max-heap Binary heaps are also commonly employed in the [[heapsort]] [[sorting algorithm]], which is an in-place algorithm ==Heap operations==
Both the insert and remove operations modify the heap to
=== Insert ===
To
# Add the element to the bottom level of the heap at the leftmost open space.▼
# Compare the added element with its parent; if they are in the correct order, stop.▼
# If not, swap the element with its parent and return to the previous step.▼
Steps 2 and 3, which restore the heap property by comparing and possibly swapping a node with its parent, are called ''the up-heap'' operation (also known as ''bubble-up'', ''percolate-up'', <!-- not a typo: -->''sift-up'', ''trickle-up'', ''swim-up'', ''heapify-up'', ''cascade-up'', or ''fix-up'').▼
▲#Add the element to the bottom level of the heap at the leftmost open space.
▲#Compare the added element with its parent; if they are in the correct order, stop.
▲#If not, swap the element with its parent and return to the previous step.
▲Steps 2 and 3, which restore the heap property by comparing and possibly swapping a node with its parent, are called ''the up-heap'' operation (also known as ''bubble-up'', ''percolate-up'', ''sift-up'', ''trickle-up'', ''swim-up'', ''heapify-up'', ''cascade-up'', or ''fix-up'').
The number of operations required depends only on the number of levels the new element must rise to satisfy the heap property. Thus, the insertion operation has a worst-case time complexity of {{nowrap|O(log ''n'')}}. For a random heap, and for repeated insertions, the insertion operation has an average-case complexity of O(1).<ref>{{Cite journal|last1=Porter|first1=Thomas|last2=Simon|first2=Istvan|date=Sep 1975|title=Random insertion into a priority queue structure|journal=IEEE Transactions on Software Engineering|volume=SE-1|issue=3|pages=292–298|doi=10.1109/TSE.1975.6312854|s2cid=18907513|issn=1939-3520}}</ref><ref>{{Cite journal |last1=Mehlhorn|first1=Kurt|last2=Tsakalidis|first2=A.|date=Feb 1989| title=Data structures |website=Universität des Saarlandes |url=https://publikationen.sulb.uni-saarland.de/handle/20.500.11880/26179 |language=en|page=27|doi=10.22028/D291-26123 |quote=Porter and Simon [171] analyzed the average cost of inserting a random element into a random heap in terms of exchanges. They proved that this average is bounded by the constant 1.61. Their proof docs not generalize to sequences of insertions since random insertions into random heaps do not create random heaps. The repeated insertion problem was solved by Bollobas and Simon [27]; they show that the expected number of exchanges is bounded by 1.7645. The worst-case cost of inserts and deletemins was studied by Gonnet and Munro [84]; they give log log n + O(1) and log n + log n* + O(1) bounds for the number of comparisons respectively.}}</ref>
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As an example of binary heap insertion, say we have a max-heap
and we want to add the number 15 to the heap. We first place the 15 in the position marked by the X. However, the heap property is violated since {{nowrap|15 > 8}}, so we need to swap the 15 and the 8. So, we have the heap looking as follows after the first swap:
However the heap property is still violated since {{nowrap|15 > 11}}, so we need to swap again:
which is a valid max-heap. There is no need to check the left child after this final step: at the start, the max-heap was valid, meaning the root was already greater than its left child, so replacing the root with an even greater value will maintain the property that each node is greater than its children ({{nowrap|11 > 5}}; if {{nowrap|15 > 11}}, and {{nowrap|11 > 5}}, then {{nowrap|15 > 5}}, because of the [[transitive relation]]).
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#Compare the new root with its children; if they are in the correct order, stop.
#If not, swap the element with one of its children and return to the previous step. (Swap with its smaller child in a min-heap and its larger child in a max-heap.)
Steps 2 and 3, which restore the heap property by comparing and possibly swapping a node with one of its children, are called the ''down-heap'' (also known as ''bubble-down'', ''percolate-down'', <!-- not a typo: -->''sift-down'', ''sink-down'', ''trickle down'', ''heapify-down'', ''cascade-down'', ''fix-down'', ''extract-min'' or ''extract-max'', or simply ''heapify'') operation.
So, if we have the same max-heap as before
We remove the 11 and replace it with the 4.
Now the heap property is violated since 8 is greater than 4. In this case, swapping the two elements, 4 and 8, is enough to restore the heap property and we need not swap elements further:
The downward-moving node is swapped with the ''larger'' of its children in a max-heap (in a min-heap it would be swapped with its smaller child), until it satisfies the heap property in its new position. This functionality is achieved by the '''Max-Heapify''' function as defined below in [[pseudocode]] for an [[Array data structure|array]]-backed heap ''A'' of length ''length''(''A''). ''A'' is indexed starting at 1.
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# Find the index <math>i</math> of the element we want to delete
# Swap this element with the last element. Remove the last element after the swap.
# Down-heapify or up-heapify to restore the heap property. In a max-heap (min-heap), up-heapify is only required when the new key of element <math>i</math> is greater (smaller) than the previous one because only the heap-property of the parent element might be violated. Assuming that the heap-property was valid between element <math>i</math> and its children before the element swap, it can't be violated by a now larger (smaller) key value. When the new key is less (greater) than the previous one then only a down-heapify is required because the heap-property might only be violated in the child elements.
=== Decrease or increase key ===
<!-- section linked from [[Reheapification]] -->
The decrease key operation replaces the value of a node with a given value with a lower value, and the increase key operation does the same but with a higher value. This involves finding the node with the given value, changing the value, and then down-heapifying or up-heapifying to restore the heap property.
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==Building a heap==
Building a heap from an array of {{mvar|n}} input elements can be done by starting with an empty heap, then successively inserting each element. This approach, called Williams' method after the inventor of binary heaps, is easily seen to run in {{math|''O''(''n'' log ''n'')}} time: it performs {{mvar|n}} insertions at {{math|''O''(log ''n'')}} cost each.
However, Williams' method is suboptimal. A faster method (due to [[Robert W. Floyd|Floyd]]{{r|heapbuildjalg}}) starts by arbitrarily putting the elements on a binary tree, respecting the shape property (the tree could be represented by an array, see below). Then starting from the lowest level and moving upwards, sift the root of each subtree downward as in the deletion algorithm until the heap property is restored. More specifically if all the subtrees starting at some height <math>h</math> have already been "heapified" (the bottommost level corresponding to <math>h=0</math>), the trees at height <math>h+1</math> can be heapified by sending their root down along the path of maximum valued children when building a max-heap, or minimum valued children when building a min-heap. This process takes <math>O(h)</math> operations (swaps) per node. In this method most of the heapification takes place in the lower levels. Since the height of the heap is <math> \lfloor \log n \rfloor</math>, the number of nodes at height <math>h</math> is <math>\le \frac{2^{\lfloor \log n \rfloor}}{2^h} \le \frac{n}{2^h}</math>. Therefore, the cost of heapifying all subtrees is:
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</math>
The exact value of the above (the worst-case number of comparisons during the heap construction) is known to be equal to:
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The operation of merging two binary heaps takes Θ(''n'') for equal-sized heaps. The best you can do is (in case of array implementation) simply concatenating the two heap arrays and build a heap of the result.<ref>Chris L. Kuszmaul.
[http://nist.gov/dads/HTML/binaryheap.html "binary heap"] {{Webarchive| url=https://web.archive.org/web/20080808141408/http://www.nist.gov/dads/HTML/binaryheap.html |date=2008-08-08 }}.
Dictionary of Algorithms and Data Structures, Paul E. Black, ed., U.S. National Institute of Standards and Technology. 16 November 2009.</ref> A heap on ''n'' elements can be merged with a heap on ''k'' elements using O(log ''n'' log ''k'') key comparisons, or, in case of a pointer-based implementation, in O(log ''n'' log ''k'') time.<ref>[[Jörg-Rüdiger Sack|J.-R. Sack]] and T. Strothotte
[https://doi.org/10.1007%2FBF00264229 "An Algorithm for Merging Heaps"],
Acta Informatica 22, 171-186 (1985).</ref> An algorithm for splitting a heap on ''n'' elements into two heaps on ''k'' and ''n-k'' elements, respectively, based on a new view
of heaps as an ordered collections of subheaps was presented in.<ref>{{Cite journal |doi = 10.1016/0890-5401(90)90026-E|title = A characterization of heaps and its applications|journal = Information and Computation|volume = 86|pages = 69–86|year = 1990|last1 = Sack|first1 = Jörg-Rüdiger|author1-link = Jörg-Rüdiger Sack| last2 = Strothotte|first2 = Thomas|doi-access = free}}</ref> The algorithm requires O(log ''n'' * log ''n'') comparisons. The view also presents a new and conceptually simple algorithm for merging heaps. When merging is a common task, a different heap implementation is recommended, such as [[binomial heap]]s, which can be merged in O(log ''n'').
Additionally, a binary heap can be implemented with a traditional binary tree data structure, but there is an issue with finding the adjacent element on the last level on the binary heap when adding an element. This element can be determined algorithmically or by adding extra data to the nodes, called "threading" the tree—instead of merely storing references to the children, we store the [[inorder]] successor of the node as well.
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It is possible to modify the heap structure to make the extraction of both the smallest and largest element possible in [[Big O notation|<math>O</math>]]<math>(\log n)</math> time.<ref name="sym">{{cite web
| url = http://cg.scs.carleton.ca/~morin/teaching/5408/refs/minmax.pdf
|
| author1-link = Michael D. Atkinson
| author2 = J.-R. Sack
| author2-link = Jörg-Rüdiger Sack
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In an array-based heap, the children and parent of a node can be located via simple arithmetic on the node's index. This section derives the relevant equations for heaps with their root at index 0, with additional notes on heaps with their root at index 1.
To avoid confusion, we
=== Child nodes ===
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\end{alignat}
</math>
Noting that the left child of any node is always 1 place before its right child, we get <math>\text{left} = 2i + 1</math>.
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=== Parent node ===
Every non-root node is either the left or right child of its parent, so
Hence,
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==Summary of running times==
{{Heap Running Times}}
==References==
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