Introduction to Greedy Algorithms
Authors: Darren Yao, Benjamin Qi
Contributor: Ryan Chou
Problems that can be solved by selecting the choice that seems to be the best at the moment at every step.
Greedy Algorithms
Some USACO Bronze problems that appear to be ad hoc can actually be solved using greedy algorithms. This idea will be covered in a future module, but we'll introduce the general mindset in this section.
Resources | ||||
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CPH | other examples are outside scope of bronze |
Warning!
True "greedy" problems start to show up in silver, though the greedy mindset can be very helpful for bronze problems.
From the above:
A greedy algorithm constructs a solution to the problem by always making a choice that looks the best at the moment. A greedy algorithm never takes back its choices, but directly constructs the final solution. For this reason, greedy algorithms are usually very efficient.
Greedy does not refer to a single algorithm, but rather a way of thinking that is applied to problems; there's no one way to do greedy algorithms. Hence, we use a selection of well-known examples to help you understand the greedy paradigm.
Example - Mad Scientist
Focus Problem – try your best to solve this problem before continuing!
Try to come up with a greedy algorithm for problem above.
Correct Greedy Algorithm
In this problem, the correct greedy solution is to continually flip the longest possible ranges of mismatching cows.
Mad Scientist has an excellent editorial with a video solution and intuitive proof.
It is highly recommended you read it to gain a better understanding of the greedy algorithm.
Solution - Mad Scientist
C++
From the official analysis:
#include <iostream>#include <string>using namespace std;using ll = long long;int main() {freopen("breedflip.in", "r", stdin);freopen("breedflip.out", "w", stdout);ll n;cin >> n;
Java
From the official analysis (with Kattio added):
import java.io.*;import java.util.*;public class Solution {public static void main(String[] args) throws IOException {Kattio io = new Kattio("breedflip");int n = io.nextInt();char[] a = io.next().toCharArray();char[] b = io.next().toCharArray();int ret = 0;while (!new String(a).equals(new String(b))) {
Python
import syssys.stdin = open("breedflip.in", "r")sys.stdout = open("breedflip.out", "w")N, a, b = input(), input(), input()s = [False] + [x != y for x, y in zip(a, b)] # difference list# now count occurrences of [False,True], as the first C++ solution doesprint(sum(1 if not x and y else 0 for x, y in zip(s, s[1:])))
Note that not all greedy problems necessarily require mathematical proofs of correctness. It is often sufficent to intuitively convince yourself your algorithm is correct.
Pro Tip
Sometimes, if the algorithm is easy enough to implement, you don't even need to convince yourself it's correct; just code it and see if it passes. Competitive programmers refer to this as "Proof by AC," or "Proof by Accepted."
Problems
Status | Source | Problem Name | Difficulty | Tags | |
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Bronze | Easy | Show TagsGreedy | |||
Bronze | Normal | Show TagsGreedy | |||
Bronze | Normal | Show TagsGreedy | |||
Bronze | Hard | Show TagsGreedy | |||
Bronze | Hard | Show TagsGreedy | |||
Bronze | Very Hard | Show TagsGreedy |
Quiz
What is a greedy algorithm?
Module Progress:
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