74 lines
No EOL
2.9 KiB
Markdown
74 lines
No EOL
2.9 KiB
Markdown
---
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id: binary-search
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title: "Binary Search"
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author: Nathan Chen
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prerequisites:
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-
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- Silver - Sorting
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---
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[Binary search](https://en.wikipedia.org/wiki/Binary_search_algorithm) can be used on monotonic functions for a logarithmic runtime.
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<!-- END DESCRIPTION -->
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## Basic Application
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Here is a very basic form of binary search:
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> Find an element in a sorted array of size $N$ in $O(\log N)$ time.
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Other variations are similar, such as the following:
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> Given $K$, find the largest element less than $K$ in a sorted array.
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## Tutorial
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- CSES 3.3
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- [Topcoder Binary Search](https://www.topcoder.com/community/data-science/data-science-tutorials/binary-search/)
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- [CSAcademy Binary Search](https://csacademy.com/lesson/binary_search)
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- [KA Binary Search](https://www.khanacademy.org/computing/computer-science/algorithms/binary-search/a/binary-search)
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- [GeeksForGeeks](https://www.geeksforgeeks.org/binary-search/)
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### Library Functions to do Binary Search
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#### Java
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- [Arrays.binarySearch](https://docs.oracle.com/javase/7/docs/api/java/util/Arrays.html)
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- [Collections.binarySearch](https://docs.oracle.com/javase/7/docs/api/java/util/Collections.html)
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#### C++
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- [lower_bound](http://www.cplusplus.com/reference/algorithm/lower_bound/)
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- [upper_bound](http://www.cplusplus.com/reference/algorithm/upper_bound/)
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### Problems
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- [USACO Silver Counting Haybales](http://www.usaco.org/index.php?page=viewproblem2&cpid=666)
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## Binary Searching on the Answer
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Oftentimes used when you need to find the minimum or maximum of some quantity such that it satisfies some property.
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### Tutorial
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- Intro to USACO 12.1
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### Problems
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- USACO
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- [USACO Silver - Cow Dance](http://www.usaco.org/index.php?page=viewproblem2&cpid=690)
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- binary search on $K$ and simulate
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- [USACO Silver - Convention](http://www.usaco.org/index.php?page=viewproblem2&cpid=858)
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- determine whether $M$ buses suffice if every cow waits at most $T$ minutes
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- use a greedy strategy (applies to next two problems as well)
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- [USACO Silver - Angry Cows](http://usaco.org/index.php?page=viewproblem2&cpid=594)
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- check in $O(N)$ how many haybales you can destroy with fixed radius $R$
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- [USACO Silver - Social Distancing](http://www.usaco.org/index.php?page=viewproblem2&cpid=1038)
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- check in $O(N+M)$ how many cows you can place with distance $D$
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- [USACO Silver - Loan Repayment](http://www.usaco.org/index.php?page=viewproblem2&cpid=991)
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- requires some rather tricky analysis to speed up naive $O(N\log N)$ solution
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- CF
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- [The Meeting Place Cannot Be Changed](http://codeforces.com/contest/782/problem/B) [](48)
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- [Preparing for Merge Sort](http://codeforces.com/contest/847/problem/B) [](53)
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- [Level Generation](http://codeforces.com/problemset/problem/818/F) [](54)
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- [Packmen](http://codeforces.com/contest/847/problem/E) [](57)
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- [Office Keys](http://codeforces.com/problemset/problem/830/A) [](60) |