Stream-order sensitivity in distinct element estimation: HyperLogLog and the shielding effect

Probabilistic cardinality estimators such as HyperLogLog (HLL) assume stream-order independence, yet their transient convergence behaviour under realistic, non-uniform streams has not been systematically studied. This work addresses this gap by analysing how stream ordering affects the convergence dynamics of HLL across four large-scale datasets spanning 0% to 97% redundancy. We introduce the Shielding Effect , a mechanism by which temporally clustered duplicates saturate a subset of HLL registers, delaying convergence while leaving the final estimate unaffected. Analytical bounds are derived showing that expected register activation scales inversely with burst length, and an information-theoretic model links conditional stream entropy to estimation latency. Experiments on the Enron Email Corpus reveal a 1.538 × convergence penalty under natural chronological ordering compared to a randomized stream, requiring 53.8% more data to reach 5% error. Comparative evaluation of k-Minimum Values (KMV) and Theta Sketch demonstrates contrasting behaviour: KMV achieves a 20 × convergence acceleration under grouped ordering (sensitivity factor S = 0.050 ), while Theta Sketch remains near order-independent ( S ≈ 1.0 ). A controlled synthetic sweep identifies the critical failure region at duplication ratios exceeding 90% combined with burst lengths above 100. The results establish that duplication alone is insufficient to trigger sensitivity; rather, the joint presence of high redundancy and strong temporal locality constitutes the critical condition. Mitigation strategies including randomized buffering and strided sampling are evaluated.