usaco-guide/content/4_Silver/4_Silver_BinSearch.md

slug	title	author	order	prerequisites
/silver/binary-search	Binary Search	Nathan Chen	4	Silver - Sorting

Binary search can be used on monotonic functions for a logarithmic runtime.

Basic Application

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

• CSES 3.3
• Topcoder Binary Search
• CSAcademy Binary Search
• KA Binary Search
• GeeksForGeeks

Library Functions to do Binary Search

Java

• Arrays.binarySearch
• Collections.binarySearch

C++

• lower_bound
• upper_bound

Problems

• USACO Silver Counting Haybales

Binary Searching on the Answer

Oftentimes used when you need to find the minimum or maximum of some quantity such that it satisfies some property.

Tutorial

• Intro to USACO 12.1

Problems

• USACO
  • USACO Silver - Cow Dance
    • binary search on K and simulate
  • USACO Silver - Convention
    • determine whether M buses suffice if every cow waits at most T minutes
    • use a greedy strategy (applies to next two problems as well)
  • USACO Silver - Angry Cows
    • check in O(N) how many haybales you can destroy with fixed radius R
  • USACO Silver - Social Distancing
    • check in O(N+M) how many cows you can place with distance D
  • USACO Silver - Loan Repayment
    • requires some rather tricky analysis to speed up naive O(N\log N) solution
• CF
  • The Meeting Place Cannot Be Changed
  • Preparing for Merge Sort
  • Level Generation
  • Packmen
  • Office Keys