C++ Coding Test Course, Understanding Combinations

Hello, everyone! In this session, we will talk about preparing for coding tests using C++. Specifically, we will solve a problem to deepen our understanding of the Combination algorithm. Combinations are a very useful algorithm for finding ways to select items that meet specific conditions.

Definition of Combination

Combination refers to the way of selecting a specific number of elements from a given set. For example, when selecting 2 elements from the set {A, B, C}, the possible selections are {A, B}, {A, C}, and {B, C}. The important point here is that the order of the selected elements does not matter. This is a significant distinction from permutations.

Problem Description

Problem: Sum of Combinations

Given an integer array nums and an integer target, find and output all combinations that can sum up to target. Each number in the combination can be used multiple times, and combinations with the same numbers should not be included multiple times.

Input Example

nums = [2, 3, 6, 7]
target = 7

Output Example

[[2, 2, 3], [7]]

Problem Solving Process

1. Understanding the Problem

To solve the problem, we will use the Backtracking algorithm. Backtracking explores possible choices during the problem-solving process and, if a choice is invalid or does not lead to an optimal solution, goes back to a previous step to try another choice.

2. Algorithm Design

The algorithm for this problem consists of the following steps.

  1. Add the current combination to the array.
  2. If the sum of the combination equals target, add that combination to the result array.
  3. If the current combination exceeds target, terminate the recursive call.
  4. Add possible numbers to the combination and make a recursive call to include the same number.
  5. Finally, remove the last selected number to return to the previous state.

3. Implementing the Code

Now, let’s look at the code implemented in C++.

#include <iostream>
#include <vector>
using namespace std;

void backtrack(vector<int> &combination, vector<vector<int>> &result, vector<int> &nums, int target, int start) {
    if (target == 0) {
        result.push_back(combination);
        return;
    }
    
    for (int i = start; i < nums.size(); i++) {
        if (nums[i] > target) continue; // Skip if the current number exceeds target
        combination.push_back(nums[i]); // Add number to the combination
        backtrack(combination, result, nums, target - nums[i], i); // Recursive call
        combination.pop_back(); // Remove the last number to backtrack
    }
}

vector<vector<int>> combinationSum(vector<int> &nums, int target) {
    vector<vector<int>> result;
    vector<int> combination;
    backtrack(combination, result, nums, target, 0);
    return result;
}

int main() {
    vector<int> nums = {2, 3, 6, 7};
    int target = 7;
    vector<vector<int>> result = combinationSum(nums, target);
    
    for (const auto &comb : result) {
        cout << "[ ";
        for (int num : comb) {
            cout << num << " ";
        }
        cout << "]" << endl;
    }
    
    return 0;
}

4. Explanation of the Code

The above code is a function called combinationSum that takes a given array of numbers and a target value, returning an array of all possible combinations. Inside the function, a recursive function called backtrack is called to generate combinations.

  • backtrack function: This function stores the current combination in an array, updates the target value, and recursively explores new combinations.
  • Conditional Statement: When the target is 0, the current combination is added to the result array, and if the target is less than the current number, the exploration is terminated.
  • Adding Combinations: After adding the current number to the combination, the target value is reduced, and a recursive call is made.
  • Removing Combinations: To backtrack, the last number is removed from the combination to return to the previous state.

5. Time Complexity Analysis

The time complexity of this problem is proportional to the number of combinations in the worst case. Therefore, it can be quite inefficient depending on the input size and number of possibilities. However, such algorithms can be very useful in many situations, so it is important to understand and practice them well.

6. Conclusion

Through this lecture, we learned to understand and solve combination problems in C++. Since combination problems can be utilized in various situations, it is important to understand the given context well and choose the appropriate algorithm. I hope you can further enhance your coding skills through various practice problems.

Additional Examples and Exercises

Try solving the additional examples below to practice more diverse combination problems:

  • Example 1: What are the possible combinations for nums = [1, 2, 3] and target = 4?
  • Example 2: What are the possible combinations for nums = [3, 5, 7] and target = 8?

It is important to continuously practice to enhance your coding skills. I hope you gain experience by solving various problems!

References

If you want more detailed algorithms and in-depth content about C++, please refer to the materials below:

We wish you the best in your coding studies!