usaco-guide/content/3_Bronze/Intro_Sorting.mdx

---
id: intro-sorting
title: "Introduction to Sorting"
author: Siyong Huang, Michael Cao, Nathan Chen
description: Introduces sorting and binary searching on a sorted array.
frequency: 0
---

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", "Apartments", "1084", "Normal", false, [], "Sort applicants and apartments, then greedily assign applicants"),
      new Problem("CSES", "Ferris Wheel", "1090", "Normal", false, [], "Sort children, keep a left pointer and a right pointer. Each gondola either is one child from the right pointer or two children, one left and one right."),
      new Problem("CSES", "Restaurant Customers", "1619", "Normal", false, [], ""),
      new Problem("CSES", "Stick Lengths", "1074", "Normal", false, [], "Spoiler: Optimal length is median"),
    ],
  }
};

**Sorting** is exactly what it sounds like: arranging items in some particular order. 

<info-block title="Pro Tip">
No bronze problem should require sorting, but it can be an alternate solution that is sometimes much easier to implement.
</info-block>

## Additional Resources

<resources>
  <resource source="CPH" title="3 - Sorting" starred>types of sorting, binary search, C++ code</resource>
</resources>

## Sorting Algorithms

(why are these important?)

There are many sorting algorithms, here are some sources to learn about the popular ones:

### Tutorial

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

  - [HackerEarth Quicksort](https://www.hackerearth.com/practice/algorithms/sorting/quick-sort/tutorial/)
    - expected $O(N\log N)$
  - [HackerEarth Mergesort](https://www.hackerearth.com/practice/algorithms/sorting/merge-sort/tutorial/)
    - $O(N\log N)$

## Library Functions - Sorting

 - C++
    - [std::sort](https://en.cppreference.com/w/cpp/algorithm/sort)
    - [std::stable\_sort](http://www.cplusplus.com/reference/algorithm/stable_sort/)
    - [Golovanov399 - C++ Tricks](https://codeforces.com/blog/entry/74684)
      - first two related to sorting
 - Java
    - [Arrays.sort](https://docs.oracle.com/javase/7/docs/api/java/util/Arrays.html#sort(java.lang.Object[]))
    - [Collections.sort](https://docs.oracle.com/javase/7/docs/api/java/util/Collections.html#sort(java.util.List))
 - Python
    - [Sorting Basics](https://docs.python.org/3/howto/sorting.html)

<info-block title="For Users of Java">

The `Arrays.sort()` function uses quicksort on primitive data types such as `long`s. This is fine for USACO, but in other contests such as CodeForces, it may time out on test cases specifically engineered to trigger worst-case $O(N^2)$ behavior in quicksort. See [here](https://codeforces.com/contest/1324/hacks/625031/) for an example of a solution that was hacked on CF. 

Two ways to avoid this:

  - Declare the underlying array as an array of objects, for example `Long` instead of `long`. This forces the `Arrays.sort()` function to use mergesort, which is always $O(N \log N)$.
  - [Shuffle](https://pastebin.com/k6gCRJDv) the array beforehand.

</info-block>


## Binary Search

[Binary search](https://en.wikipedia.org/wiki/Binary_search_algorithm) can be used on monotonic (\*what's that?) functions for a logarithmic runtime.

\*monotonic means *nondecreasing* or *nonincreasing*

Here is a very basic form of binary search:

> Find an element in a sorted array of size $N$ in $O(\log N)$ time.

Other variations are similar, such as the following:

> Given $K$, find the largest element less than $K$ in a sorted array.

### Tutorial

<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 - Binary Search

#### 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

Another useful application of sorting 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} />