Backtracking Template

To make backtracking problems easier to solve, we are providing you with a template that can guide you through the process:

function someProblem(candidates) {
  function backtrack(candidates, candidate, result) {
    if (/* <<success condition>> */) {
      result.push([...candidate]);
      return;
    }

    for (let elem of candidates) {
      if (true) {  // replace true with the dead-end condition
        continue;
      }

      candidate.push(elem);  // take
      backtrack(candidates, candidate, result);  // explore
      candidate.pop();  // clean up
    }
  }

  const result = [];
  const candidate = [];
  backtrack(candidates, candidate, result);
  return result;
}

Learning this code by heart is not what we want. Instead, we'll explain the template so you can understand its logic.

Note: Even though we're dealing with individual elements in the loop, the candidate variable represents a potential candidate solution, typically an array or collection of elements.

Template Explanation

1. Initialization and Main Function Call

const result = [];
const candidate = [];
backtrack(candidates, candidate, result);
return result;

The backtracking process begins outside the recursive function. We start by creating empty arrays for the result and the initial candidate. Then, we make the first call to backtrack with the full set of candidates. This sets up the initial state for our backtracking exploration.

2. Success Condition

if (/* <<success condition>> */) {
    result.push([...candidate]);
    return;
}

The success condition is the first check in our backtracking function. It determines when we've found a valid solution that should be added to our results.

If you're using a slicing approach, the success condition might be when the length of the remaining collection is zero. If you're not slicing, it might be when the length of the candidate is equal to some target length.

When the success condition is met, we create a copy of the candidate solution (using the spread operator in JavaScript), add it to the results list, and return from the current function call to backtrack.

3. Exploration Loop

for (let elem of candidates) {
    // ... (dead-end check here)
    candidate.push(elem);  // take
    backtrack(candidates, candidate, result);  // explore
    candidate.pop();  // clean up
}

The main loop in the backtracking function is where we systematically explore all possibilities. For each element in the candidates:

1. We first check the dead-end condition (explained next) to see if we should skip this element.
2. If not skipped, we add the element to our candidate solution (candidate.push(elem)).
3. We then make a recursive call to backtrack, sometimes with a sliced version of the collection to prevent reusing elements.
4. After the recursive call returns (regardless of whether it found a solution), we remove the last added element (candidate.pop()). This crucial step allows us to backtrack and explore other possibilities.

4. Dead-end Condition

if (true) {  // replace true with the dead-end condition
    continue;
}

The dead-end condition, while often overlooked, is crucial for efficient backtracking. It helps us prune the search tree by identifying paths that can't lead to a valid solution early on.

This condition is typically checked at the start of each iteration in the main loop. It might involve checking if a value is valid (e.g., is it a palindrome? Is the number included in the list?). This is where the logic for pruning happens.

If the dead-end condition is met, we use continue to skip to the next iteration, avoiding unnecessary exploration of invalid paths.

It's important to note that this code explicitly checks for the dead-end condition. Sometimes, we can implicitly create a dead-end condition by slicing the array, thus removing some elements as possibilities in further function calls. This approach can efficiently narrow down the search space without explicitly coding a dead-end check. We'll explore that using code examples in the following assignments.

By understanding these components and how they work together, you can adapt this template to solve a wide variety of backtracking problems. Remember, the key is defining the right conditions, recognizing when to prune, and understanding how to build up and backtrack from potential solutions.