Backtracking Problems

In backtracking, we explore all possible solutions to a problem by systematically generating and evaluating all potential candidates. This technique is especially useful for solving problems with combinatorial nature, where the solution itself is a combination of choices or a sequence of decisions.

Backtracking problems can often be formulated as a search problem on a graph or a tree, where each node represents a potential solution and the edges represent the choices or decisions that lead to the next potential solutions.

To apply the backtracking technique, we need to define:

1. A solution space or a search space that represents all potential solutions.
2. A constraint function that defines the conditions or rules that a valid solution must satisfy.
3. A backtracking function that systematically explores the solution space by making a series of choices and backtracking when necessary.

Let's take a look at an example to better understand backtracking problems.

Example: Subset Sum Problem

The subset sum problem is a classic example of a backtracking problem. Given a set of integers and a target sum, we need to find all possible subsets of the set that sum up to the target.

1#include <iostream>
2#include <vector>
3using namespace std;
4
5void backtrack(vector<int>& nums, vector<int>& subset, int target, int start, int sum) {
6  if (sum == target) {
7    // Found a valid subset
8    for (int num : subset) {
9      cout << num << " ";
10    }
11    cout << endl;
12  }
13
14  for (int i = start; i < nums.size(); i++) {
15    if (sum + nums[i] <= target) {
16      subset.push_back(nums[i]);
17      backtrack(nums, subset, target, i + 1, sum + nums[i]);
18      subset.pop_back();
19    }
20  }
21}
22
23void subsetSum(vector<int>& nums, int target) {
24  vector<int> subset;
25  backtrack(nums, subset, target, 0, 0);
26}
27
28int main() {
29  vector<int> nums = {1, 2, 3, 4, 5};
30  int target = 7;
31  subsetSum(nums, target);
32  return 0;
33}

In this example, we define a backtrack function subsetSum that takes in a set of integers nums and a target sum target. It initializes an empty subset vector and starts the backtracking process by calling the backtrack function.

The backtrack function takes in the nums vector, the subset vector, the target, the start index, and the current sum. It checks if the current sum equals the target and if so, it prints the valid subset. Otherwise, it iterates through the remaining numbers in num and checks if adding the number to the sum is less than or equal to the target. If it is, the number is added to the subset and the backtrack function is called recursively for the next index.

By backtracking and exploring all possible combinations, we can find all subsets of the set that sum up to the target.

To summarize, backtracking is a powerful technique for solving problems with combinatorial nature. By systematically generating and evaluating potential solutions, we can find all valid solutions that satisfy the given constraints. The key to success is defining the solution space, the constraints, and the backtracking function appropriately for each problem.

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#include <iostream>

using namespace std;

​

int main() {

  // code goes here

}