Worst Case Running Time of Pushing a Value to a Stack Implemented as an Array

When implementing a stack using an array, the performance characteristics of the push operation can vary depending on the current size of the array relative to its allocated capacity. To fully understand the worst-case scenario, we need to explore how each operation contributes to the overall time complexity during a push operation, particularly when the array size equals twice its capacity.

Capacity vs. Size: Defining the Context

Let's assume that the stack is implemented using an array of capacity 2n, where n is the maximum number of elements that can be stored before a potential reallocation is required. If the array currently has 2n elements, each push operation involves several steps:

Increment the end counter: This operation is constant time, O(1), as the end counter simply increases by one. Check for capacity: This check also takes constant time, O(1), to determine if the stack has reached its capacity. Copy the new value: If the check passes, the new value is copied into the next available slot in the array, again a constant time operation, O(1). Reallocation: This is where the worst-case time complexity can arise if the array exceeds its capacity. However, we must consider the possibility of repeated reallocations and the associated overhead.

Reallocation and Its Impact

In cases where the array reaches full capacity, a reallocation is necessary to accommodate the additional element. Originally, the array has a capacity of 2n, but the actual size can be less. For example, if only n elements have been pushed, the array is still of size 2n. Assuming the array is full (size 2n) and a new element needs to be added:

Increment the end counter: O(1). Check for capacity: O(1), identifies that the array has reached its capacity. Allocate a new array: This step involves memory reallocation. Depending on the heap allocator, the reallocation could take up to O(3.2n) time. However, in practice, it often takes O(n) due to optimization techniques employed by modern allocators. Copy all elements: All 2n elements must be copied from the old array to the new one. This step has a time complexity of O(2n). Copy the end counter: O(1), copying the updated end counter to the new array. Free the old array: Depending on the heap allocator, this step has a worst-case time complexity of O(2n). Insert the new value: O(1), placing the new element in the newly allocated array.

Implications and Practical Considerations

The worst-case time complexity for a push operation when the array reaches its capacity is O(n). This is primarily due to the overhead of reallocation and subsequent copying of all existing elements to a new, larger array. While this may seem inefficient, the infrequency of such events in practice can mitigate this concern.

It’s important to note that modern heap allocators are highly optimized, and the reallocation process often involves less overhead than the worst-case scenario suggests. Additionally, efficient stack implementations typically use a doubling strategy for allocating new arrays when the current capacity is reached, which helps to reduce the frequency of such operations.

Despite the potential for a worst-case O(n) performance, it’s often not the best reason to limit the number of elements inserted into the stack. Other factors, such as memory management and resource constraints, may be more relevant to consider.

Conclusion

In summary, while the worst-case running time for pushing a value onto a stack implemented as an array can be O(n) when the array reaches its capacity, this is a rare occurrence in practice. Efficient heap allocators and optimized algorithms help to minimize the impact of such operations. Understanding the time complexity is crucial for selecting the appropriate data structure for specific use cases in software development.