Approximate Aggregation for Tracking Quantiles in Wireless Sensor Networks

Proof. As presented in Section IV.A, the querying process for r(x) in T is a traversal process over T . Assuming for each interval Int of T , there exists a parameter μ that satisfies |I| − μ ≤ |A| ≤ |I|. Thus, the total error introduced by the traversal process from the sink to the leaf node is O(hμ) where h is the height of T . Thus, the total error of querying any r(x) in T is O(hμ+ |Mj |/β), where |Mj |/β resulted from the interval size corresponded by each leaf node of T is less than |Mj |/β. Letting hμ+ |Mj |/β = |Mj |, one can drive the value of μ as presented in Theorem 1.