82 lines
3.7 KiB
Text
82 lines
3.7 KiB
Text
---
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id: binary-search-sorted
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title: "Binary Search on a Sorted Array"
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author: Siyong Huang, Michael Cao, Nathan Chen
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description: "Using binary search to find elements quickly in a sorted array."
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prerequisites:
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- intro-ds
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frequency: 2
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---
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import { Problem } from "../models";
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export const metadata = {
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problems: {
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bubble: [
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new Problem("HR", "Bubble Sort", "https://www.hackerrank.com/challenges/ctci-bubble-sort/problem", "Easy", false, [], "O(N^2)"),
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new Problem("Silver", "Out of Sorts", "834", "Very Hard", false, []),
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],
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count: [
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new Problem("Silver", "Counting Haybales", "666", "Normal", false, []),
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],
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}
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};
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Suppose that we want to find an element in a sorted array of size $N$ in $O(\log N)$ time. We can do this with [**binary search**](https://en.wikipedia.org/wiki/Binary_search_algorithm); each iteration of the binary search cuts the search space in half, so the algorithm tests $O(\log N)$ values. This is efficient and much better than testing every element in an array.
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<resources>
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<resource source="KA" title="Binary Search" url="https://www.khanacademy.org/computing/computer-science/algorithms/binary-search/a/binary-search" starred>animations!</resource>
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<resource source="CSA" title="Binary Search" url="binary_search" starred>animation!</resource>
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<resource source="CPH" title="3.3 - Binary Search" starred></resource>
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<resource source="TC" title="Binary Search" url="binary-search"></resource>
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<resource source="GFG" title="Binary Search" url="binary-search"></resource>
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</resources>
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## Library Functions
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<LanguageSection>
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<CPPSection>
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<resources>
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<resource source="CPP" url="http://www.cplusplus.com/reference/algorithm/lower_bound/" title="lower_bound"> </resource>
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<resource source="CPP" url="http://www.cplusplus.com/reference/algorithm/upper_bound/" title="upper_bound"> </resource>
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</resources>
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</CPPSection>
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<JavaSection>
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<resources>
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<resource source="JAVA" url="https://docs.oracle.com/javase/7/docs/api/java/util/Arrays.html#binarySearch(int[],%20int)" title="Arrays.binarySearch"> </resource>
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<resource source="JAVA" url="https://docs.oracle.com/javase/7/docs/api/java/util/Collections.html#binarySearch(java.util.List,%20T)" title="Collections.binarySearch"> </resource>
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</resources>
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</JavaSection>
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</LanguageSection>
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## Coordinate Compression
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A related topic is **coordinate compression**, which takes some points and reassigns them to remove wasted space.
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<problems-list problems={metadata.problems.count} />
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> Farmer John has just arranged his $N$ haybales $(1\le N \le 100,000)$ at various points along the one-dimensional road running across his farm. To make sure they are spaced out appropriately, please help him answer $Q$ queries ($1 \le Q \le 100,000$), each asking for the number of haybales within a specific interval along the road.
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However, each of the points are in the range $0 \ldots 1,000,000,000$, meaning you can't store locations of haybales in, for instance, a boolean array.
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<spoiler title="Solution">
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Let's place all of the locations of the haybales into a list and sort it.
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(fix part below so transform to range $1\ldots N$)
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<IncompleteSection />
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<!-- Now, we can map distinct points to smaller integers without gaps. For example, if the haybales existed at positions $[1, 4, 5, 9]$ and queries were $(1, 2)$ and $(4, 6)$, we can place the integers together and map them from $[1, 2, 4, 5, 6, 9] \rightarrow [1, 2, 3, 4, 5, 6]$. This effectively transforms the haybale positions into $[1, 3, 4, 6]$ and the queries into $1, 2$ and $3, 5$.
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By compressing queries and haybale positions, we've transformed the range of points to $0 \ldots N + 2Q$, allowing us to store prefix sums to effectively query for the number of haybales in a range.
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-->
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</spoiler> |