126 lines
3.4 KiB
Markdown
126 lines
3.4 KiB
Markdown
---
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title: Binary Search
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TARGET DECK: Obsidian::STEM
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FILE TAGS: algorithm
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tags:
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- algorithm
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---
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## Overview
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Property | Value
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----------- | --------
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Best Case | $O(1)$
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Worst Case | $O(\lg{n})$
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Aux. Memory | $O(1)$
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%%ANKI
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Basic
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What precondition must the input of `BINARY_SEARCH` satisfy?
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Back: It must already be sorted.
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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: 1708781334247-->
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END%%
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%%ANKI
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Basic
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What is the best case running time of `BINARY_SEARCH`?
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Back: $\Omega(1)$
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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: 1708117310004-->
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END%%
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%%ANKI
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Basic
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What input does `BINARY_SEARCH` perform best on?
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Back: One in which the value being searched for is already in the middle.
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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: 1708117310011-->
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END%%
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%%ANKI
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Basic
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What is the worst case running time of `BINARY_SEARCH`?
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Back: $O(\lg{n})$
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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: 1708117310015-->
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END%%
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%%ANKI
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Basic
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What input does `BINARY_SEARCH` perform worst on?
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Back: One in which the value does not exist.
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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: 1708117310018-->
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END%%
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%%ANKI
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Basic
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What is the typical output of `BINARY_SEARCH`?
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Back: The index of the element in the array being searched for, if found.
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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: 1708117310021-->
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END%%
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A recursive solution looks as follows:
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```c
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static int aux(const int needle, const int i, const int j, int *A) {
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if (i > j) {
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return -1;
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}
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int mid = (i + j) / 2;
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if (A[mid] == needle) {
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return mid;
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} else if (A[mid] < needle) {
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return aux(needle, mid + 1, j, A);
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} else {
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return aux(needle, i, mid - 1, A);
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}
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}
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int binary_search(const int needle, const int n, int A[static n]) {
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return aux(needle, 0, n - 1, A);
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}
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```
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We can also write this iteratively:
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```c
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int binary_search(const int needle, const int n, int A[static n]) {
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int i = 0;
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int j = n - 1;
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while (i <= j) {
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int mid = (i + j) / 2;
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if (A[mid] == needle) {
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return mid;
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} else if (A[mid] < needle) {
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i = mid + 1;
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} else {
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j = mid - 1;
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}
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}
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return -1;
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}
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```
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%%ANKI
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Basic
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In `BINARY_SEARCH`, when could using floor for midpoint calculations yield different answers than ceiling?
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Back: When there exist duplicate members.
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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: 1708174545522-->
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END%%
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%%ANKI
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Basic
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In `BINARY_SEARCH`, what ensures left pointer `L` and right pointers `R` eventually satisfy `L > R`?
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Back: The found midpoint is always excluded from the next `BINARY_SEARCH` invocation.
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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: 1708174545527-->
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END%%
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## References
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* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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