---
id: binary-search-sorted
title: "Binary Search on a Sorted Array"
author: Siyong Huang, Michael Cao, Nathan Chen
description: Introduces sorting and binary searching on a sorted array.
prerequisites:
 - Bronze - Introduction to Data Structures
frequency: 4
---

import { Problem } from "../models";

export const metadata = {
  problems: {
    bubble: [
      new Problem("HR", "Bubble Sort", "https://www.hackerrank.com/challenges/ctci-bubble-sort/problem", "Easy", false, [], "O(N^2)"),
      new Problem("Silver", "Out of Sorts", "834", "Very Hard", false, []),
    ],
    count: [
      new Problem("Silver", "Counting Haybales", "666", "Normal", false, []),
    ],
    cses: [
      new Problem("CSES", "Stick Lengths", "1074", "Normal", false, [], "Spoiler: Optimal length is median"),
    ],
  }
};

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.

<resources>
  <resource source="KA" title="Binary Search" url="https://www.khanacademy.org/computing/computer-science/algorithms/binary-search/a/binary-search" starred>animations!</resource>
  <resource source="CSA" title="Binary Search" url="binary_search" starred>animation!</resource>
  <resource source="CPH" title="3.3 - Binary Search" starred></resource>
  <resource source="TC" title="Binary Search" url="binary-search"></resource>
  <resource source="GFG" title="Binary Search" url="binary-search"></resource>
</resources>

## Library Functions

### C++

 - [lower_bound](http://www.cplusplus.com/reference/algorithm/lower_bound/)
 - [upper_bound](http://www.cplusplus.com/reference/algorithm/upper_bound/)

### Java

 - [Arrays.binarySearch](https://docs.oracle.com/javase/7/docs/api/java/util/Arrays.html)
 - [Collections.binarySearch](https://docs.oracle.com/javase/7/docs/api/java/util/Collections.html)

## Coordinate Compression

A related topic is **coordinate compression**, which takes some points and reassigns them to remove wasted space. 

<problems-list problems={metadata.problems.count} />

> 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.

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. 

### Solution

Let's place all of the locations of the haybales into a list and sort it.

(fix part below so transform to range $1\ldots N$)

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 $3, 5$.

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.

## Problems

<problems-list problems={metadata.problems.cses} />