Welcome to this dive into Advanced Recursion Techniques. These techniques will broaden your understanding of recursion and equip you with the ability to tackle complex problems effectively. Recursion is a method where the solution to a problem depends on smaller instances of the same problem. Advanced recursion techniques allow us to solve problems involving deep tree structures and backtracking, which are prevalent in certain interview questions.

To provide a taste of what's ahead, let's examine a recursive function that generates all permutations of a slice of numbers. The strategy used here is known as backtracking. Backtracking is a general algorithm for finding all (or some) solutions to computational problems. In our example, we recursively swap elements of the slice, exploring one permutation path at a time until we reach the end. Once there, we append the current state of the slice to our results.

The permute algorithm generates all permutations of a slice of numbers using backtracking. Here's how it works:

1. Initialize Result Storage:

   • A slice result is created to store all the permutations.

2. Recursive Backtracking Function:

   • A recursive function backtrack is defined, which takes an integer first (the starting index for permutations) and the current state of the slice nums.

3. Base Case:

   • If first equals the length of nums, a complete permutation has been formed and is added to result.

4. Permutations Through Swapping:

   • Loop through the slice starting from the first index to the end.
   • Swap the element at the first index with the current index i.
   • Recursively call backtrack with first + 1 to continue forming the next part of the permutation.
   • Swap back the elements to revert the slice to its previous state (backtrack) to explore other permutations.

5. Execution:

   • The initial call to backtrack starts with first set to 0.
   • The generated permutations are stored in the result slice, which is then returned.

By using backtracking, this algorithm methodically explores all possible permutations of the given slice by swapping and recursively building permutations one element at a time.

The Go implementation of the algorithm is:

Go
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1package main
2
3import (
4    "fmt"
5)
6
7func backtrack(nums []int, first int, result *[][]int) {
8    if first == len(nums) {
9        newPerm := make([]int, len(nums))
10        copy(newPerm, nums)
11        *result = append(*result, newPerm)
12        return
13    }
14    for i := first; i < len(nums); i++ {
15        nums[first], nums[i] = nums[i], nums[first] // Swap numbers
16        backtrack(nums, first+1, result)
17        nums[first], nums[i] = nums[i], nums[first] // Swap them back
18    }
19}
20
21func permute(nums []int) [][]int {
22    var result [][]int
23    backtrack(nums, 0, &result)
24    return result
25}
26
27func main() {
28    nums := []int{1, 2, 3}
29    permutations := permute(nums)
30    for _, permutation := range permutations {
31        fmt.Println(permutation)
32    }
33}

Let's walk through the permute algorithm using the slice {1, 2, 3} step-by-step:

1. Initial Setup:

   • Input slice: {1, 2, 3}
   • Initial call: backtrack(nums, 0, &result)

2. First Level of Recursion (backtrack(nums, 0, &result)):

   • first = 0:
     • Swap nums[0] with nums[0] (no change): {1, 2, 3}
     • Call backtrack(nums, 1, &result)

3. Second Level of Recursion (backtrack(nums, 1, &result)):

   • first = 1:
     • Swap nums[1] with nums[1] (no change): {1, 2, 3}
     • Call backtrack(nums, 2, &result)

4. Third Level of Recursion (backtrack(nums, 2, &result)):

   • first = 2:
     • Swap nums[2] with nums[2] (no change): {1, 2, 3}
     • Call backtrack(nums, 3, &result)

5. Base Case Reached (backtrack(nums, 3, &result)):

   • first = 3:
     • Add {1, 2, 3} to result
   • Return to backtrack(nums, 2, &result)

6. Backtrack Step in backtrack(nums, 2, &result):

   • Revert swap nums[2] with nums[2]: {1, 2, 3}
   • No more elements to swap; return to backtrack(nums, 1, &result)

7. Backtrack Step in backtrack(nums, 1, &result):

   • Revert swap nums[1] with nums[1]: {1, 2, 3}
   • Swap nums[1] with nums[2]: {1, 3, 2}
   • Call backtrack(nums, 2, &result)

8. Recursive Call (backtrack(nums, 2, &result) with {1, 3, 2}):

   • first = 2:
     • Swap nums[2] with nums[2] (no change): {1, 3, 2}
     • Call backtrack(nums, 3, &result)

9. Base Case Reached (backtrack(nums, 3, &result)):

   • first = 3:
     • Add {1, 3, 2} to result
   • Return to backtrack(nums, 2, &result)

10. Backtrack Step in backtrack(nums, 2, &result):

    • Revert swap nums[2] with nums[2]: {1, 3, 2}
    • No more elements to swap; return to backtrack(nums, 1, &result)

11. Backtrack Step in backtrack(nums, 1, &result):

    • Revert swap nums[2] with nums[1]: {1, 2, 3}
    • No more elements to swap; return to backtrack(nums, 0, &result)

12. Backtrack Step in backtrack(nums, 0, &result):

    • Revert swap nums[0] with nums[0]: {1, 2, 3}
    • Swap nums[0] with nums[1]: {2, 1, 3}
    • Call backtrack(nums, 1, &result)

Following similar steps as above, the rest of the permutations will be generated by backtracking further. After backtracking through all levels and swapping appropriately, the final result slice will include all permutations of {1, 2, 3}:

• {1, 2, 3}
• {1, 3, 2}
• {2, 1, 3}
• {2, 3, 1}
• {3, 1, 2}
• {3, 2, 1}

Now that we've briefly touched on what advanced recursion techniques are about, it's time to practice. Through exercises, you'll gain a solid understanding of how to apply these techniques to real-world problems, preparing you for interviews. The key to mastering recursion is practice and understanding. Let's get started!