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Greedy Algorithm

  • Greedy algorithms are applied to optimization algorithms only.
  • It picks always the current best solution; no backtracking.
  • Not necessarily optimal --> need to prove it.

Optimization Problem

  • find a set of items that can
    • meet some constraints, and
    • optimize some objective function
  • examples: shortest path, minimum spanning tree
  • not an optimization problem: sorting

Prove the Optimality

To prove the optimality of a greedy strategy, we need to prove the following two parts.
  • Greedy Choice
    • find the choice that is part of SOME optimal solution
    • there may exist multiple optimal solutions; no need to prove it for ANY optimal solution
  • Optimal Substructure
    • After making the first choice, the final best solution is first choice + best solution for the rest of (compatible) input
    • We can solve the same optimization problem recursively!
    • Otherwise, we find a contradiction!

Example problems

  • activity selection (pick the earliest-finish task)
  • minimum spanning tree
  • Huffman code
  • Dijkastra's shortest path
  • graph coloring
  • traveling salesman

Greedy vs. Dynamic Programming

  • Greedy: always pick the current best
  • DP: pick the best based on history
  • You can always use DP to solve Activity Selection problem, but it is much costly (since you have to try all other branches)

Example Code: Easy Swaps to Sort 123

#include <iostream>
#include <vector>
#include <algorithm>
int main(){
int n;
std::cin >> n;
std::vector<int> candies(n);
for (int k = 0; k < n; ++k)
std::cin >> candies[k];
// count total number of 1s, 2s, 3s
int sum1 = 0, sum2 = 0, sum3 = 0;
for (int k = 0; k < n; ++k){
if (candies[k] == 1)
else if (candies[k] == 2)
// all 3s should be placed in the range [sum1+sum2, n-1]
// and then leave a subproblem of 1s and 2s in the range [0, sum1+sum2-1]
int swap = 0;
std::vector<int> temp_queue;
for (int k = sum1 + sum2; k < n; ++k){
if (candies[k] != 3){
// move all 1s to the front and 2s to the end
std::sort(temp_queue.begin(), temp_queue.end());
// in [0, sum1+sum2-1], we want to put 1s to the front and 2s to the end
for (int k = sum1 + sum2 - 1; k >= 0; --k){
if (candies[k] == 3){
candies[k] = temp_queue.back();
// finally, count how many 2s are still there in the range [0, sum1-1]
// which is supposed to be all 1s
for (int k = 0; k < sum1; ++k){
if (candies[k] == 2)
std::cout << swap << std::endl;
return 0;