842. Split Array into Fibonacci Sequence

Problem Description

In this problem, we are provided with a string num which represents a sequence of digits. Our goal is to check if we can split this string into a sequence of non-negative integers such that the sequence forms a Fibonacci-like series. A Fibonacci-like series is one where every number in the series is the sum of the two preceding numbers, just like in the Fibonacci series but it does not necessarily start with 0 and 1.

There are a few constraints we need to consider:

1. Each number in the series must be less than 2**31 and fit within a 32-bit signed integer.
2. The length of the series must be at least 3.
3. There should be no leading zeroes in any number of the series, except if the number itself is zero.

If we can form such a sequence, we should return the sequence. If it's not possible, we should return an empty list.

Intuition

Our solution strategy involves trying to build the Fibonacci-like sequence from the start of the string by making choices at each step and using backtracking to explore other possibilities if a certain choice doesn't lead to a solution.

Here's the intuition behind the steps we take in our approach:

1. We need to keep track of the current position in the string we're trying to convert into the next number in the sequence. This is managed by the index i.
2. For the current position i, we try to extract each possible number (which doesn't have leading zeroes) by iterating through the string from i to the end. Leading zeroes are invalid (except for the number 0 itself), so if we encounter a zero as the first character of a number (and it's not a standalone zero), we stop trying to build further from this position.
3. For each number x we consider, we check if it's valid by making sure it doesn't exceed the 32-bit integer limit and that it fits the Fibonacci-like requirement (if we already have at least two numbers in our current sequence).
4. If x fits as the next number in the series, we add it to our ans (answer) list and then recursively call the function to build on the sequence from the next position j + 1.
5. We're essentially performing a depth-first search (DFS) to exhaust all possibilities, and whenever our recursion finishes processing the string and we have more than two numbers in our sequence (ans), we've found a valid Fibonacci-like sequence.
6. If at any point, a choice of x does not lead to a valid sequence, we backtrack by removing the last added number and trying the next possibility.

This is a backtracking problem where we build partial solutions and backtrack as soon as we realize the current path does not lead to a valid full solution. The combination of iterative exploration of each number that could be added from a certain position with the recursive call to continue the sequence from the next character is essential. The use of backtracking helps in reducing the time complexity since we prune the search tree by stopping the exploration of sequences that can no longer be valid as early as possible.

Learn more about Backtracking patterns.

Solution Approach

The reference solution uses Depth First Search (DFS) and backtracking to find a Fibonacci-like sequence from the given string. The implementation of this approach is written in Python and uses recursion to iterate over the possibilities when constructing the sequence. Here is a rundown of the approach, with reference to the solution code provided:

1. Depth First Search (DFS) and Recursion: The dfs function is a recursive method that is called with the current index i. The base case for the recursion is when i equals the length of the given string, n. If we reach the base case and the current answer list, ans, contains more than two numbers, it means a valid sequence has been formed, and the function returns True.

2. Backtracking: The ans list serves as a partial solution that is constructed incrementally. If at any step the sequence can no longer be continued (either because the next number is too big or does not fit the Fibonacci-like property), the last element is removed from ans, and the function backtracks to try other possibilities.

3. Constructing Numbers: A nested loop is used to construct the next number x in the sequence by adding digits one by one from the current index i to the end of the string n. This is done by multiplying x by 10 and adding the integer value of the current digit. Special care is taken to skip any numbers that would start with a leading zero (except the number 0 itself), as per the rules of the problem.

4. Checking Validity: After each number x is constructed, several checks are performed:

   • The number must be less than 2**31 - 1, otherwise, we stop considering longer numbers.
   • The algorithm checks if the length of ans is less than 2, which means we don't have enough numbers to compare for the Fibonacci-like property and just add x. If ans already has two or more numbers, then x is only added if it's the sum of the last two numbers in ans.
   • If x doesn't satisfy the Fibonacci-like condition, the inner loop breaks to try a different starting point for the next number.

5. Further Recursion: When a valid number x is found and appended to ans, the dfs function is called recursively with the next starting index j + 1. If the recursive call returns True, indicating that a complete and valid sequence has been found, the current function also returns True.

6. Pop and Continue: If the recursive call doesn't yield a solution, the last number is popped from ans (this is the backtracking step), and the search continues with the next possible number x.

7. Returning the Result: The solution defines the list ans to store the numbers of the Fibonacci-like sequence. If the dfs function eventually returns True, ans will be returned, containing a valid sequence. If no sequence is found, the list remains empty, and an empty list is returned.

In summary, the solution leverages recursion, backtracking, and careful construction of candidate numbers while following the constraints of the problem. These elements combined enable the algorithm to explore the space of all possible sequences efficiently and effectively to arrive at the desired result or determine that no solution exists.

Example Walkthrough

Let's apply the solution approach to a small example to illustrate how it works. Consider the string num = "112358" which is a classic Fibonacci-like sequence starting with 1, 1, 2, 3, 5, and 8.

1. We start with an empty answer list ans and at the first character of the string with i = 0.
2. The first loop tries to form a number x by considering each substring starting from the index i. At the start, x is 1 (the first character).
3. Since ans is empty, we don't need to check for the Fibonacci-like property yet. We add x = 1 to ans and call dfs recursively with i = 1.
4. The recursive call starts again from i = 1, trying to form the next number. It takes the next character, which is 1, and since it's still valid, x = 1 is added to ans.
5. Now ans = [1, 1]. With a new recursive call, we move to i = 2 and try to form the next number.
6. x now is 2 which is the sum of the last two numbers in ans. We add x = 2 to ans, making ans = [1, 1, 2] and call dfs recursively with i = 3.
7. The recursive calls continue, with each call attempting to construct the next number following the Fibonacci-like property.
8. We proceed to positions 4, 5, and 6 in the string forming x = 3, x = 5, and x = 8 respectively, with each satisfying the Fibonacci-like property and getting added to ans.
9. When i reaches the end of the string (i.e., i = n), the base case is reached. ans contains more than two numbers (ans = [1, 1, 2, 3, 5, 8]), so we have successfully found a valid sequence.
10. The recursive calls start to unwind, each returning True since a valid sequence has been built.

At the end of this process, the result is the sequence [1, 1, 2, 3, 5, 8] which confirms that 112358 can indeed be split into a Fibonacci-like sequence.

Conversely, if we had started with a string that could not form such a sequence, like num = "123456789":

1. We start with x = 1, then x = 2, but the next number formed would be 3, which does not equal 1 + 2.
2. The process would backtrack, trying different combinations (next x would be 23), but since there is no combination that works, it would eventually fail.
3. After exhausting all possibilities, the recursive calls would not satisfy the condition and would return an empty list, indicating that no valid Fibonacci-like sequence is found.

Solution Implementation

Time and Space Complexity

The provided code snippet is a Python method splitIntoFibonacci which attempts to split a given numeric string into a Fibonacci-like sequence where each number is the sum of the preceding two. Here's the analysis of the time and space complexity:

Time Complexity

The primary driver of the time complexity is the depth-first search (dfs) function which explores all possible combinations to construct a Fibonacci-like sequence:

1. The dfs function is recursive and in the worst case, the depth of the recursion is the length of the string num, let's denote the length as n. Hence, a simple upper-bound on the recursion depth is O(n).

2. Inside each call of dfs, there is a loop that generates the next potential number in the sequence, which can run up to n times. In the worst case, this loop will iterate n times for each level of recursion.

3. Checking whether a number can be added to the sequence (if len(ans) < 2 or ans[-2] + ans[-1] == x:) is done in O(1) time.

Combining these aspects, the worst-case time complexity of the algorithm is O(n^2 * n) = O(n^3) because you may generate up to n numbers for each of the n recursive calls, and for each number, you might iterate through up to n digits to construct it.

Space Complexity

The space complexity depends on the storage required for the recursive call stack and the list ans:

1. The recursive call stack can go up to n deep in the worst case, which gives a space usage of O(n).

2. The space to store the ans list, in the worst case, is also O(n), as the maximum possible length of the list is n when every digit in num is a single-digit number in the Fibonacci sequence.

Thus, the overall space complexity of the algorithm is O(n) due to the recursive call stack and the storage for the sequence ans.

Learn more about how to find time and space complexity quickly using problem constraints.