Additional heap/heapsort flashcards.

notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json

@@ -116,7 +116,9 @@
     "perfect-tree.png",
     "non-complete-tree.png",
     "max-heap-tree.png",
-    "max-heap-array.png"
+    "max-heap-array.png",
+    "max-heapify-1.png",
+    "max-heapify-2.png"
   ],
   "File Hashes": {
     "algorithms/index.md": "3ac071354e55242919cc574eb43de6f8",
@@ -373,7 +375,7 @@
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     "_journal/2024-04/2024-04-24.md": "4cb04e0dea56e0b471fc0e428471a390",
-    "algorithms/heaps.md": "b12c70ec85e514ce912821d133d116d4",
+    "algorithms/heaps.md": "ed37002a7600a05794f668e092265522",
     "_journal/2024-04-26.md": "3ce37236a9e09e74b547a4f7231df5f0",
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     "_journal/2024-04-28.md": "46726bf76a594b987c63ba8b9b6d13d3",

notes/algorithms/heaps.md

@@ -11,6 +11,8 @@ tags:
 
 The **binary heap** data structure is an array object that can be viewed as a [[trees#Positional Trees|complete binary tree]].
 
+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`.
+
 %%ANKI
 Cloze
 A binary heap is an {array} that can be viewed as a {binary tree}.
@@ -147,6 +149,294 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
 <!--ID: 1714356546616-->
 END%%
 
+%%ANKI
+Basic
+What preconditions must hold before invoking `MAX_HEAPIFY_DOWN` on a node?
+Back: The node's left and right subtrees must be max-heaps.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714399155389-->
+END%%
+
+%%ANKI
+Basic
+When is `MAX_HEAPIFY_DOWN` a no-op?
+Back: When the current node is already larger than both its children.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714399155419-->
+END%%
+
+%%ANKI
+Basic
+If not a no-op, which child should `MAX_HEAPIFY_DOWN` swap its current value with?
+Back: The larger of its two children.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714399155425-->
+END%%
+
+%%ANKI
+Basic
+Given a heap of height $h$, *why* is `MAX_HEAPIFY_DOWN`'s worst case runtime $O(h)$?
+Back: Each invocation may violate the max-heap property of a child node.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714399155432-->
+END%%
+
+%%ANKI
+Basic
+What is the runtime of `MAX_HEAPIFY_DOWN`?
+Back: $O(h)$ where $h$ is the height of the heap.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425256-->
+END%%
+
+%%ANKI
+Basic
+What is the result of calling `MAX_HEAPIFY_DOWN` on the highlighted node?
+![[max-heapify-1.png]]
+Back:
+![[max-heapify-2.png]]
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714399155438-->
+END%%
+
+%%ANKI
+Basic
+What is the runtime of `MIN_HEAPIFY_DOWN`?
+Back: $O(h)$ where $h$ is the height of the heap.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425286-->
+END%%
+
+%%ANKI
+Basic
+What preconditions must hold before invoking `MIN_HEAPIFY_DOWN` on a node?
+Back: The node's left and right subtrees must be min-heaps.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714399155443-->
+END%%
+
+%%ANKI
+Basic
+When is `Min_HEAPIFY_DOWN` a no-op?
+Back: When the current node is already smaller than both its children.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714399155448-->
+END%%
+
+%%ANKI
+Basic
+If not a no-op, which child should `MIN_HEAPIFY_DOWN` swap its current value with?
+Back: The smaller of its two children.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714399155453-->
+END%%
+
+%%ANKI
+Basic
+Given a heap of height $h$, *why* is `MIN_HEAPIFY_DOWN`'s worst case runtime $O(h)$?
+Back: Each invocation may violate the min-heap property of a child node.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714399155459-->
+END%%
+
+%%ANKI
+Basic
+What does the "heapify" operation of a heap refer to?
+Back: Repeatedly swapping a node's value with a child until the heap property is achieved.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714399155469-->
+END%%
+
+%%ANKI
+Basic
+How many internal nodes does a binary heap of size $n$ have?
+Back: $\lfloor n / 2 \rfloor$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425292-->
+END%%
+
+%%ANKI
+Basic
+How many internal nodes precede the first external node of a heap of size $n$?
+Back: $\lfloor n / 2 \rfloor$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425296-->
+END%%
+
+%%ANKI
+Basic
+What is the height of a binary heap?
+Back: The height of the heap's root when viewed as a complete binary tree.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425300-->
+END%%
+
+%%ANKI
+Basic
+What is the input of `MAX_HEAPIFY_DOWN`?
+Back: The index of a node in the target heap.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425304-->
+END%%
+
+%%ANKI
+Basic
+What is the input of `BUILD_MAX_HEAP`?
+Back: An array.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425309-->
+END%%
+
+%%ANKI
+Basic
+What is the runtime of `BUILD_MAX_HEAP` on an array of $n$ elements?
+Back: $O(n)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425314-->
+END%%
+
+%%ANKI
+Basic
+How is the `BUILD_MAX_HEAP` function usually implemented?
+Back: As calling heapify on each internal node.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425320-->
+END%%
+
+%%ANKI
+Basic
+Which node does `BUILD_MAX_HEAP` start iterating on?
+Back: The last internal node.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425326-->
+END%%
+
+%%ANKI
+Basic
+Why does `BUILD_MAX_HEAP` "ignore" the external nodes of a heap?
+Back: Because they are already max-heaps of size $1$.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425331-->
+END%%
+
+%%ANKI
+Basic
+Given heap $H[0{..}n{-}1]$, what is `BUILD_MAX_HEAP`'s loop invariant?
+Back: Each node in $H[i{+}1{..}n{-}1]$ is the root of a max-heap.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425336-->
+END%%
+
+%%ANKI
+Basic
+What is initialization of `BUILD_MAX_HEAP`'s loop invariant?
+Back: Every external node is the root of a max-heap.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425340-->
+END%%
+
+%%ANKI
+Basic
+What is maintenance of `BUILD_MAX_HEAP`'s loop invariant?
+Back: Calling `MAX_HEAPIFY_DOWN` maintains the max-heap property of the current node.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425344-->
+END%%
+
+%%ANKI
+Basic
+In pseudocode, how is `BUILD_MAX_HEAP` implemented?
+Back:
+```c
+void BUILD_MAX_HEAP(int n, int H[static n]) {
+  for (int i = (n / 2) - 1; i >= 0; --i) {
+    MAX_HEAPIFY_DOWN(i, H);
+  }
+}
+```
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425348-->
+END%%
+
+%%ANKI
+Basic
+What is the input of `BUILD_MIN_HEAP`?
+Back: An array.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425351-->
+END%%
+
+%%ANKI
+Basic
+What is the runtime of `BUILD_MIN_HEAP` on an array of $n$ elements?
+Back: $O(n)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425355-->
+END%%
+
+%%ANKI
+Basic
+How is the `BUILD_MIN_HEAP` function usually implemented?
+Back: As calling heapify on each internal node.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425359-->
+END%%
+
+%%ANKI
+Basic
+Which node does `BUILD_MIN_HEAP` start iterating on?
+Back: The last internal node.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425363-->
+END%%
+
+%%ANKI
+Basic
+Why does `BUILD_MIN_HEAP` "ignore" the external nodes of a heap?
+Back: Because they are already max-heaps of size $1$.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425367-->
+END%%
+
+%%ANKI
+Basic
+Given heap $H[0{..}n{-}1]$, what is `BUILD_MIN_HEAP`'s loop invariant?
+Back: Each node in $H[i{+}1{..}n{-}1]$ is the root of a min-heap.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425372-->
+END%%
+
+%%ANKI
+Basic
+What is initialization of `BUILD_MIN_HEAP`'s loop invariant?
+Back: Every external node is the root of a min-heap.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425376-->
+END%%
+
+%%ANKI
+Basic
+What is maintenance of `BUILD_MIN_HEAP`'s loop invariant?
+Back: Calling `MIN_HEAPIFY_DOWN` maintains the min-heap property of the current node.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425381-->
+END%%
+
+%%ANKI
+Basic
+In pseudocode, how is `BUILD_MIN_HEAP` implemented?
+Back:
+```c
+void BUILD_MIN_HEAP(int n, int H[static n]) {
+  for (int i = (n / 2) - 1; i >= 0; --i) {
+    MIN_HEAPIFY_DOWN(i, H);
+  }
+}
+```
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1714403425386-->
+END%%
+
 ## Bibliography
 
 * Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).