Analysis of Memory Efficiency in Sequential Processing

A system is designed to process a continuous stream of data points (e.g., sensor readings) and must maintain an up-to-date average of all points seen so far. Consider two approaches for updating this average after receiving the Nth data point:

Approach 1: Uses a recursive formula that takes the previous average (calculated up to point N-1) and the new Nth data point to compute the new average.

Approach 2: Stores every single data point from 1 to N in a list and recalculates the average of the entire list every time a new point arrives.

As the number of data points (N) grows very large, what is the most significant difference in the memory requirements between these two approaches?

A system uses a recursive formula to update its memory state, where the new state Mem_i is calculated based on the previous state Mem_{i-1} and the current input item_i. For this system to correctly calculate the state at step 1,000,000, it must store all one million individual inputs from item_1 to item_1,000,000 in its memory.