Competitive Analysis of Aggregate Max in Windowed Streaming

We consider the problem of maintaining a fixed number

k

of items observed over a data stream, so as to optimize the maximum value over a fixed number

n

of recent observations. Unlike previous approaches, we use the competitive analysis framework and compare the performance of the online streaming algorithm against an optimal adversary that knows the entire sequence in advance. We consider the problem of maximizing the

aggregate max

, i.e., the sum of the values of the largest items in the algorithm’s memory over the entire sequence. For this problem, we prove an asymptotically tight competitive ratio, achieved by a simple heuristic, called partition-greedy, that performs stream updates efficiently and has almost optimal performance. In contrast, we prove that the problem of maximizing, for every time

t

, the value maintained by the online algorithm in memory, is considerably harder: in particular, we show a tight competitive ratio that depends on the maximum value of the stream. We further prove negative results for the closely related problem of maintaining the aggregate minimum and for the generalized version of the aggregate max problem in which every item comes with an individual window.