Problem Description

The task involves simulating a stack operation to construct a specific sequence of numbers, given as the target array, using a stream of numbers from 1 to n. We can only use two operations: "Push", to add the next number in the stream on top of the stack, and "Pop", to remove the top element of the stack.

The goal is to ensure that the numbers in the stack, from bottom to top, match the sequence described in the target array. We need to do this by adhering to specific rules:

• We can only take the next integer from the stream if the stream isn't empty and push it onto the stack.
• We can pop the top integer from the stack if the stack isn't empty.
• We must stop taking new integers from the stream and stop performing operations on the stack once it matches the target.

The expected output is a list of the stack operations ("Push" and "Pop") needed to attain the target sequence.

Intuition

To solve this problem, we need to simulate the process of either pushing or popping elements to match the target sequence. We iterate over the target array and compare its elements with the next expected integer from the stream, which we represent with a variable (let's say, cur). For every number in the target array, we do the following:

• If cur is less than the current target value, it implies that there are numbers in the stream that we need to skip to match the target. For each of these numbers, we "Push" (to simulate taking the number from the stream) and then "Pop" (to remove it).
• We increment cur until it matches the current target value, then perform one "Push" operation without a corresponding "Pop" since this number should stay in the stack.

We continue this process for each element in the target array. The sequence of "Push" and "Pop" operations that we accumulate forms the solution to rebuild the target array from the stream. It is worth mentioning that multiple valid sequences of operations can exist, and we can return any of them.

This approach works because it effectively simulates the steps needed to recreate the target array using the given stack operations, mirroring the actual manual process one might use if doing this with a physical stack and stream of numbers.

Learn more about Stack patterns.

Solution Approach

The solution takes a simple iterative approach to build the required sequence of stack operations. Here is a step-by-step explanation of the implementation details:

1. Initialize two variables: cur, which represents the next expected number from the stream starting from 1, and ans, which holds the list of stack operations performed.

2. Loop through each value v in the target array:

   • Inside this loop, increment cur by 1 - it means we are considering the push of the next number in the stream.

   • While cur is less than the current target value v, we realize that we are not interested in having cur in our final stack. Therefore, we perform a "Push" followed immediately by a "Pop". Add these two operations to the ans list, and then increment cur to consider the next number in the stream.

   • By the time cur matches v, the current value of the target array, we only need to "Push" as we want to keep this number in our stack. Hence, we add a "Push" operation to ans.

3. After iterating through each element in the target array, we have a list of operations that leads us to a stack that, if evaluated from bottom to top, will match target.

4. Return the ans list, which contains the sequence of "Push" and "Pop" operations.

This algorithm effectively uses a simulation pattern, where we simulate each step needed to construct the target sequence from the available input numbers (1 to n). The while loop inside the iteration is essential as it allows us to skip over the numbers that are not included in the target sequence by simulating a push/pop pair for each.

Here is how the pattern works with the code:

class Solution:
    def buildArray(self, target: List[int], n: int) -> List[str]:
        cur, ans = 0, []
        for v in target:
            cur += 1
            while cur < v:
                ans.extend(['Push', 'Pop'])
                cur += 1
            ans.append('Push')
        return ans

In this code:

• The for loop iterates over the target.
• The while loop handles skipping numbers not in target.
• The extend(['Push', 'Pop']) method is used to simulate the push/pop operations for numbers we wish to skip.
• When the condition of the while loop is not met (meaning cur equals v), it appends a Push to ans.

By the end of the iteration, ans represents the sequence of stack operations that would build target from the stream of integers within the range [1, n].

Example Walkthrough

Consider the target array [2, 4] and n = 4. We want to simulate stack operations to reconstruct this sequence where the numbers in the stream range from 1 to n.

Step 1: We initialize cur to 0 and ans to an empty list. These will keep track of the next expected number from the stream and the list of operations, respectively.

Step 2: We iterate through each value v in the target array. The first value is 2.

• We increment cur to 1 and start a while loop that will run since cur < v.
• While cur < 2, we do a "Push" (as if we took 1 from the stream and added it to the stack) followed by a "Pop" (we remove 1 since it's not in the target). We add these operations to ans, resulting in ['Push', 'Pop'].
• We increment cur again, now cur is 2 which matches v.
• We perform a "Push" as cur is equal to v to keep 2 on the stack, updating ans to ['Push', 'Pop', 'Push'].

Step 3: We move on to the next value in the target array, which is 4.

• We increment cur to 3 since it is still less than v and perform another "Push" followed by "Pop", updating ans to ['Push', 'Pop', 'Push', 'Push', 'Pop'].
• Incrementing cur once more, now cur is 4 which matches v.
• We perform a "Push" to keep 4 on the stack, updating ans to ['Push', 'Pop', 'Push', 'Push', 'Pop', 'Push'].

Step 4: After processing each element in the target array, we have a complete list of operations that forms the output. Thus, the array ['Push', 'Pop', 'Push', 'Push', 'Pop', 'Push'] is the sequence of operations needed to simulate the stack to achieve the target sequence [2, 4].

Following the above steps ensures that we only push numbers onto the stack that are present in the target and pop those that are not, using the simulation approach described in the solution.

This example clearly demonstrates the iterative process of the algorithm, incrementing the cur variable, and conditionally adding 'Push' and 'Pop' operations to the ans list to match the target array.

Solution Implementation

Time and Space Complexity

The time complexity of the given code can be analyzed based on the number of operations performed relative to the length of target list and n. Assume the length of target is m.

For every element in target, the code performs a series of "Push" and "Pop" operations until it reaches the current target value. In the worst case, this could happen for each target value (i.e., if the target list has all consecutive numbers starting from 1 up to m). Since at each step of iterating through the target list you can have at most two operations (one "Push", and if the number is not in the target one "Pop") per number until you reach the target value, the number of these operations is linear with respect to the last value in target, denoted as target[-1].

Therefore, the worst-case time complexity is O(target[-1]), which in the worst case is O(n) if the last value in target is equal to n.

The space complexity of the code is determined by the size of the ans list, which holds a sequence of strings representing the operations. In the worst-case scenario as described above, the ans list will contain a number of elements equal to twice the value of target[-1] minus the length of target (since each number not in target adds two operations, and each number in target adds one operation). Thus, the space complexity is also O(target[-1]), which in the worst case is O(n).

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