482 lines
15 KiB
Markdown
482 lines
15 KiB
Markdown
---
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title: Heaps
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TARGET DECK: Obsidian::STEM
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FILE TAGS: data_structure::heap
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tags:
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- algorithm
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- data_structure
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- heap
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---
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## Overview
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The **binary heap** data structure is an array object that can be viewed as a [[trees#Positional Trees|complete binary tree]].
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The primary function used to maintain the max-heap property is `MAX_HEAPIFY_DOWN`. This function assumes the left and right- subtrees at a given node are max heaps but that the current node may be smaller than its children. An analagous function and assumptions exist for `MIN_HEAPIFY_DOWN`.
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```c
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inline int left_child(int i) { return (i << 1) + 1; }
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inline int right_child(int i) { return (i << 1) + 2; }
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void max_heapify_down(int n, int H[static n], int i) {
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while (true) {
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int lc = left_child(i);
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int rc = right_child(i);
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int next = i;
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if (lc < n && H[next] < H[lc]) {
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next = lc;
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}
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if (rc < n && H[next] < H[rc]) {
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next = rc;
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}
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if (next == i) {
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return;
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}
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swap(H, i, next);
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i = next;
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}
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}
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void build_max_heap(int n, int H[static n]) {
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for (int i = n / 2 - 1; i >= 0; --i) {
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max_heapify_down(n, H, i);
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}
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}
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```
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%%ANKI
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Cloze
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A binary heap is an {array} that can be viewed as a {binary tree}.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714356379014-->
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END%%
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%%ANKI
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Basic
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Is the following a valid binary heap?
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![[perfect-tree.png]]
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Back: Yes.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714356379021-->
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END%%
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%%ANKI
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Basic
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Is the following a valid binary heap?
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![[complete-tree.png]]
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Back: Yes.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714356379024-->
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END%%
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%%ANKI
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Basic
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Is the following a valid binary heap?
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![[non-complete-tree.png]]
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Back: No.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714356379030-->
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END%%
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%%ANKI
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Basic
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Why can't the following be a binary heap?
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![[non-complete-tree.png]]
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Back: A heap is equivalently viewed as a *complete* binary tree.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714356379034-->
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END%%
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%%ANKI
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Basic
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What two sizes are associated with binary heaps?
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Back: The size of the heap and the capacity of the underlying array.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714356379038-->
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END%%
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%%ANKI
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Basic
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What are the two kinds of binary heaps?
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Back: Max-heaps and min-heaps.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714356379042-->
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END%%
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%%ANKI
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Basic
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What is the max-heap property?
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Back: Every parent node is greater than or equal to its children in value.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714356379046-->
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END%%
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%%ANKI
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Basic
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Where is the largest element of a max-heap?
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Back: At the root.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714356379052-->
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END%%
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%%ANKI
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Basic
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Where is the smallest element of a max-heap?
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Back: At the leaves.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714356379059-->
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END%%
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%%ANKI
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Basic
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What is the min-heap property?
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Back: Every parent node is less than or equal to its children in value.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714356379072-->
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END%%
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%%ANKI
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Basic
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Where is the smallest element of a min-heap?
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Back: At the root.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714356379077-->
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END%%
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%%ANKI
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Basic
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Where is the largest element of a min-heap?
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Back: At the leaves.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714356379083-->
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END%%
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%%ANKI
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Basic
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How is the following binary heap viewed as an array?
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![[max-heap-tree.png]]
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Back:
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![[max-heap-array.png]]
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714356379065-->
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END%%
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%%ANKI
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Basic
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How is the following binary heap instead viewed as a binary tree?
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![[max-heap-array.png]]
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Back:
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![[max-heap-tree.png]]
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714356442370-->
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END%%
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%%ANKI
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Basic
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What kind of binary heap is the following?
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![[max-heap-array.png]]
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Back: A max-heap.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714356546616-->
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END%%
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%%ANKI
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Basic
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What preconditions must hold before invoking `MAX_HEAPIFY_DOWN` on a node?
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Back: The node's left and right subtrees must be max-heaps.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714399155389-->
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END%%
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%%ANKI
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Basic
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When is `MAX_HEAPIFY_DOWN` a no-op?
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Back: When the current node is already larger than both its children.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714399155419-->
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END%%
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%%ANKI
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Basic
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If not a no-op, which child should `MAX_HEAPIFY_DOWN` swap its current value with?
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Back: The larger of its two children.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714399155425-->
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END%%
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%%ANKI
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Basic
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Given a heap of height $h$, *why* is `MAX_HEAPIFY_DOWN`'s worst case runtime $O(h)$?
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Back: Each invocation may violate the max-heap property of a child node.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714399155432-->
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END%%
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%%ANKI
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Basic
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What is the runtime of `MAX_HEAPIFY_DOWN`?
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Back: $O(h)$ where $h$ is the height of the heap.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425256-->
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END%%
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%%ANKI
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Basic
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What is the result of calling `MAX_HEAPIFY_DOWN` on the highlighted node?
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![[max-heapify-1.png]]
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Back:
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![[max-heapify-2.png]]
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714399155438-->
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END%%
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%%ANKI
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Basic
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What is the runtime of `MIN_HEAPIFY_DOWN`?
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Back: $O(h)$ where $h$ is the height of the heap.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425286-->
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END%%
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%%ANKI
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Basic
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What preconditions must hold before invoking `MIN_HEAPIFY_DOWN` on a node?
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Back: The node's left and right subtrees must be min-heaps.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714399155443-->
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END%%
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%%ANKI
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Basic
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When is `MIN_HEAPIFY_DOWN` a no-op?
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Back: When the current node is already smaller than both its children.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714399155448-->
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END%%
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%%ANKI
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Basic
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If not a no-op, which child should `MIN_HEAPIFY_DOWN` swap its current value with?
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Back: The smaller of its two children.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714399155453-->
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END%%
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%%ANKI
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Basic
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Given a heap of height $h$, *why* is `MIN_HEAPIFY_DOWN`'s worst case runtime $O(h)$?
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Back: Each invocation may violate the min-heap property of a child node.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714399155459-->
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END%%
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%%ANKI
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Basic
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What does the "heapify" operation of a heap refer to?
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Back: Repeatedly swapping a node's value with a child until the heap property is achieved.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714399155469-->
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END%%
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%%ANKI
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Basic
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How many internal nodes does a binary heap of size $n$ have?
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Back: $\lfloor n / 2 \rfloor$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425292-->
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END%%
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%%ANKI
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Basic
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How many internal nodes precede the first external node of a binary heap of size $n$?
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Back: $\lfloor n / 2 \rfloor$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425296-->
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END%%
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%%ANKI
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Basic
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How is the height of a binary heap defined?
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Back: The height of the heap's root when viewed as a complete binary tree.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425300-->
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END%%
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%%ANKI
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Basic
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What is the input of `MAX_HEAPIFY_DOWN`?
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Back: The index of a node in the target heap.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425304-->
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END%%
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%%ANKI
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Basic
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What is the input of `BUILD_MAX_HEAP`?
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Back: An array.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425309-->
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END%%
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%%ANKI
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Basic
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What is the runtime of `BUILD_MAX_HEAP` on an array of $n$ elements?
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Back: $O(n)$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425314-->
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END%%
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%%ANKI
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Basic
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How is the `BUILD_MAX_HEAP` function usually implemented?
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Back: As calling heapify on each internal node.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425320-->
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END%%
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%%ANKI
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Basic
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Which node does `BUILD_MAX_HEAP` start iterating on?
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Back: The last internal node.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425326-->
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END%%
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%%ANKI
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Basic
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Why does `BUILD_MAX_HEAP` "ignore" the external nodes of a heap?
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Back: Because they are already max-heaps of size $1$.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425331-->
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END%%
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%%ANKI
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Basic
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Given heap `H[0:n-1]`, what is `BUILD_MAX_HEAP`'s loop invariant?
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Back: Each node in `H[i+1:n-1]` is the root of a max-heap.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425336-->
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END%%
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%%ANKI
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Basic
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What is initialization of `BUILD_MAX_HEAP`'s loop invariant?
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Back: Every external node is the root of a max-heap.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425340-->
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END%%
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%%ANKI
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Basic
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What is maintenance of `BUILD_MAX_HEAP`'s loop invariant?
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Back: Calling `MAX_HEAPIFY_DOWN` maintains the max-heap property of the current node.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425344-->
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END%%
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%%ANKI
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Basic
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In pseudocode, how is `BUILD_MAX_HEAP` implemented?
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Back:
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```c
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void BUILD_MAX_HEAP(int n, int H[static n]) {
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for (int i = (n / 2) - 1; i >= 0; --i) {
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MAX_HEAPIFY_DOWN(i, H);
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}
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}
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```
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425348-->
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END%%
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%%ANKI
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Basic
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What is the input of `BUILD_MIN_HEAP`?
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Back: An array.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425351-->
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END%%
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%%ANKI
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Basic
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What is the runtime of `BUILD_MIN_HEAP` on an array of $n$ elements?
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Back: $O(n)$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425355-->
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END%%
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%%ANKI
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Basic
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How is the `BUILD_MIN_HEAP` function usually implemented?
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Back: As calling heapify on each internal node.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425359-->
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END%%
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%%ANKI
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Basic
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Which node does `BUILD_MIN_HEAP` start iterating on?
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Back: The last internal node.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425363-->
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END%%
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%%ANKI
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Basic
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Why does `BUILD_MIN_HEAP` "ignore" the external nodes of a heap?
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Back: Because they are already min-heaps of size $1$.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425367-->
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END%%
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%%ANKI
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Basic
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Given heap `H[0:n-1]`, what is `BUILD_MIN_HEAP`'s loop invariant?
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Back: Each node in `H[i+1:n-1]` is the root of a min-heap.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425372-->
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END%%
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%%ANKI
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Basic
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What is initialization of `BUILD_MIN_HEAP`'s loop invariant?
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Back: Every external node is the root of a min-heap.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425376-->
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END%%
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%%ANKI
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Basic
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What is maintenance of `BUILD_MIN_HEAP`'s loop invariant?
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Back: Calling `MIN_HEAPIFY_DOWN` maintains the min-heap property of the current node.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425381-->
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END%%
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%%ANKI
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Basic
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In pseudocode, how is `BUILD_MIN_HEAP` implemented?
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Back:
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```c
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void BUILD_MIN_HEAP(int n, int H[static n]) {
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for (int i = (n / 2) - 1; i >= 0; --i) {
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MIN_HEAPIFY_DOWN(i, H);
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}
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}
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```
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714403425386-->
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END%%
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%%ANKI
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Basic
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`BUILD_MIN_HEAP` can sort arrays of what sizes?
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Back: $\leq 2$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1714410566848-->
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END%%
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## Bibliography
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* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). |