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Merge pull request #5176 from cpinitiative/silver-sorting-reorg
reorg Silver sorting and searching modules, move some modules to Gold
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‎content/3_Silver/Binary_Search.mdx

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title: 'Binary Search'
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author: Darren Yao, Abutalib Namazov, Andrew Wang, Qi Wang, Dustin Miao
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prerequisites:
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- greedy-sorting
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- binary-search-sorted-array
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description:
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'Binary searching on arbitrary monotonic functions and built-in functions for
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binary search.'
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'Binary searching on arbitrary monotonic functions.'
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frequency: 3
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---
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<!-- process numbers from left to right, keep track of the last element of each block and use binary search of lower_bound to find which block the current numbers belongs -->
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## Introduction
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2017
<Resources>
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<YouTube id="GU7DpgHINWQ" />
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When we binary search on an answer, we start with a search space of size $N$
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When we binary search, we start with a search space of size $N$
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which we know the answer lies in. Then each iteration of the binary search cuts
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the search space in half, so the algorithm tests $\mathcal{O}(\log N)$ values.
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This is efficient and much better than testing each possible value in the search
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space.
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## Binary Searching on Monotonic Functions
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## Binary Search on Monotonic Functions
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Let's say we have a boolean function `f(x)`. Usually, in such problems, we want
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to find the maximum or minimum value of $x$ such that `f(x)` is true. Similarly
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`INT_MAX` at any point during execution. If this is an issue, use `long long`s
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instead of `int`s.
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## Library Functions For Binary Search
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<LanguageSection>
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<CPPSection>
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<Resources>
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<Resource
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source="CPP"
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url="http://www.cplusplus.com/reference/algorithm/lower_bound/"
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title="lower_bound, upper_bound"
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starred
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>
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with examples
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</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
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source="JAVA"
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url="https://docs.oracle.com/javase/7/docs/api/java/util/Arrays.html#binarySearch(int[],%20int)"
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title="Arrays.binarySearch"
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/>
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<Resource
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source="JAVA"
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url="https://docs.oracle.com/javase/7/docs/api/java/util/Collections.html#binarySearch(java.util.List,%20T)"
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title="Collections.binarySearch"
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/>
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</Resources>
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</JavaSection>
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<PySection>
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<Resources>
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<Resource
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source="Python"
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url="https://docs.python.org/3/library/bisect.html"
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title="Array bisection algorithm"
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/>
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<Resource
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source="GFG"
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url="https://www.geeksforgeeks.org/binary-search-bisect-in-python/"
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title="Binary Search (bisect) in Python"
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/>
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</Resources>
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</PySection>
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</LanguageSection>
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### Example - Counting Haybales
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<FocusProblem problem="count" />
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As each of the points are in the range $0 \ldots 1\,000\,000\,000$, storing
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locations of haybales in a boolean array and then taking prefix sums of that
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would take too much time and memory.
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Instead, let's place all of the locations of the haybales into a list and sort
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it. Now we can use binary search to count the number of cows in any range
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$[A,B]$ in $\mathcal{O}(\log N)$ time.
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### With Built-in Function
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<LanguageSection>
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<CPPSection>
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We can use the the built-in
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[`lower_bound`](https://en.cppreference.com/w/cpp/algorithm/lower_bound) and
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[`upper_bound`](https://en.cppreference.com/w/cpp/algorithm/upper_bound)
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functions.
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```cpp
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#include <bits/stdc++.h>
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using namespace std;
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void setIO(string name = "") { // name is nonempty for USACO file I/O
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ios_base::sync_with_stdio(0);
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cin.tie(0); // see Fast Input & Output
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// alternatively, cin.tie(0)->sync_with_stdio(0);
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if (!name.empty()) {
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freopen((name + ".in").c_str(), "r", stdin); // see Input & Output
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freopen((name + ".out").c_str(), "w", stdout);
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}
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}
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int main() {
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setIO("haybales");
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int bale_num;
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int query_num;
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cin >> bale_num >> query_num;
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vector<int> bales(bale_num);
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for (int i = 0; i < bale_num; i++) { cin >> bales[i]; }
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sort(begin(bales), end(bales));
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for (int i = 0; i < query_num; i++) {
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int q_start;
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int q_end;
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cin >> q_start >> q_end;
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cout << upper_bound(begin(bales), end(bales), q_end) -
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lower_bound(begin(bales), end(bales), q_start)
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<< "\n";
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}
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}
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```
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</CPPSection>
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<JavaSection>
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We can use the builtin
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[`Arrays.binarySearch`](<https://docs.oracle.com/javase/7/docs/api/java/util/Arrays.html#binarySearch(int[],%20int)>)
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function.
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```java
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import java.io.*;
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import java.util.*;
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public class Haybales {
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public static void main(String[] args) throws IOException {
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BufferedReader br = new BufferedReader(new FileReader(new File("haybales.in")));
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PrintWriter out = new PrintWriter("haybales.out");
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StringTokenizer st = new StringTokenizer(br.readLine());
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int baleNum = Integer.parseInt(st.nextToken());
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int queryNum = Integer.parseInt(st.nextToken());
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int[] bales = new int[baleNum];
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st = new StringTokenizer(br.readLine());
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for (int i = 0; i < baleNum; i++) {
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bales[i] = Integer.parseInt(st.nextToken());
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}
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Arrays.sort(bales);
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for (int i = 0; i < queryNum; i++) {
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st = new StringTokenizer(br.readLine());
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int start = Integer.parseInt(st.nextToken());
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int end = Integer.parseInt(st.nextToken());
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// Get the left-est bale that's still in the interval
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int bi = Arrays.binarySearch(bales, start);
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if (bi < 0) { bi = Math.abs(bi + 1); }
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// And also get the right-est bale that's still in the interval
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int ti = Arrays.binarySearch(bales, end);
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if (ti < 0) { ti = Math.abs(ti + 2); }
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out.println(ti - bi + 1);
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}
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out.close();
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}
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}
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```
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</JavaSection>
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<PySection>
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We can use the builtin
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[`bisect.bisect`](https://docs.python.org/3/library/bisect.html#bisect.bisect)
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function.
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```py
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from bisect import bisect
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inp = open("haybales.in", "r")
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out = open("haybales.out", "w")
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bale_num, query_num = map(int, inp.readline().split())
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bales = sorted(list(map(int, inp.readline().split())))
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for _ in range(query_num):
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start, end = map(int, inp.readline().split())
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print(bisect(bales, end) - bisect(bales, start - 1), file=out)
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```
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</PySection>
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</LanguageSection>
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### Without Using Built-in Functions
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<LanguageSection>
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<CPPSection>
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```cpp
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#include <bits/stdc++.h>
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using namespace std;
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void setIO(string name = "") { // name is nonempty for USACO file I/O
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ios_base::sync_with_stdio(0);
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cin.tie(0); // see Fast Input & Output
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// alternatively, cin.tie(0)->sync_with_stdio(0);
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if (!name.empty()) {
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freopen((name + ".in").c_str(), "r", stdin); // see Input & Output
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freopen((name + ".out").c_str(), "w", stdout);
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}
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}
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int main() {
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setIO("haybales");
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int bale_num;
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int query_num;
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cin >> bale_num >> query_num;
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vector<int> bales(bale_num);
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for (int &i : bales) { cin >> i; }
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sort(begin(bales), end(bales));
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// Returns the number of elements that are at most x
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auto atMost = [&](int x) {
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int lo = 0;
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int hi = bales.size();
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while (lo < hi) {
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int mid = (lo + hi) / 2;
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if (bales[mid] <= x) {
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lo = mid + 1;
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} else {
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hi = mid;
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}
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}
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return lo;
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};
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for (int i = 0; i < query_num; ++i) {
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int q_start;
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int q_end;
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cin >> q_start >> q_end;
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cout << atMost(q_end) - atMost(q_start - 1) << "\n";
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}
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}
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```
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</CPPSection>
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<JavaSection>
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```java
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import java.io.*;
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import java.util.*;
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public class Haybales {
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static int[] bales;
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public static void main(String[] args) throws IOException {
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Kattio io = new Kattio("haybales");
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int baleNum = io.nextInt();
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int queryNum = io.nextInt();
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bales = new int[baleNum];
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for (int i = 0; i < baleNum; i++) { bales[i] = io.nextInt(); }
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Arrays.sort(bales);
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for (int i = 0; i < queryNum; ++i) {
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int start = io.nextInt();
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int end = io.nextInt();
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io.println(atMost(end) - atMost(start - 1));
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}
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io.close();
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}
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// Returns the number of elements that are at most x
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public static int atMost(int x) {
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int lo = 0;
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int hi = bales.length;
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while (lo < hi) {
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int mid = (lo + hi) / 2;
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if (bales[mid] <= x) {
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lo = mid + 1;
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} else {
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hi = mid;
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}
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}
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return lo;
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}
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// CodeSnip{Kattio}
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}
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```
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</JavaSection>
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<PySection>
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```py
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def at_most(x: int) -> int:
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lo = 0
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hi = len(bales)
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while lo < hi:
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mid = (lo + hi) // 2
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if bales[mid] <= x:
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lo = mid + 1
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else:
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hi = mid
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return lo
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inp = open("haybales.in", "r")
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out = open("haybales.out", "w")
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bale_num, query_num = map(int, inp.readline().split())
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bales = sorted(list(map(int, inp.readline().split())))
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for _ in range(query_num):
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start, end = map(int, inp.readline().split())
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print(at_most(end) - at_most(start - 1), file=out)
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```
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</PySection>
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</LanguageSection>
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## Problems
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‎content/3_Silver/Binary_Search.problems.json

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}
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}
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],
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"count": [
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{
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"uniqueId": "usaco-666",
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"name": "Counting Haybales",
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"url": "http://www.usaco.org/index.php?page=viewproblem2&cpid=666",
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"source": "Silver",
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"difficulty": "Easy",
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"isStarred": false,
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"tags": ["Binary Search"],
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"solutionMetadata": {
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"kind": "internal"
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}
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}
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],
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"general": [
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{
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"uniqueId": "cf-702C",
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"name": "Cellular Network",
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"url": "https://codeforces.com/contest/702/problem/C",
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"source": "CF",
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"difficulty": "Easy",
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"isStarred": false,
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"tags": ["Binary Search"],
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"solutionMetadata": {
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"kind": "internal"
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}
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},
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{
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"uniqueId": "cses-1620",
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"name": "Factory Machines",
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"kind": "internal"
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}
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},
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{
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"uniqueId": "cf-862E",
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"name": "Mahmoud & Ehab & Function",
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"url": "https://codeforces.com/contest/862/problem/E",
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"source": "CF",
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"difficulty": "Normal",
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"isStarred": false,
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"tags": ["Binary Search"],
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"solutionMetadata": {
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"kind": "internal"
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}
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},
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{
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"uniqueId": "cc-TRPTSTIC",
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"name": "TripTastic",

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