Randomized algorithms are often analyzed using average-case complexity, but this can mask worst-case failures. High-probability complexity analysis provides stronger guarantees by bounding the probability of large deviations. This article explores key techniques like Chernoff bounds, Hoeffding's inequality, and union bounds, applied to classic algorithms such as randomized quicksort, hash tables, and skip lists. The author explains why high-probability bounds are essential for mission-critical systems where reliability matters more than average performance. For example, in distributed systems or real-time applications, a single bad run can be catastrophic. The article also contrasts high-probability with expected-case analysis, showing how the former gives tighter guarantees for tail events. This is a must-read for algorithm engineers and researchers who need to design or analyze algorithms with provable reliability. The content is mathematically rigorous yet accessible, with clear notation and step-by-step derivations. It serves as both a tutorial and a reference for advanced complexity analysis.
A deep dive into high-probability complexity analysis, explaining how to bound randomized algorithm performance with rigor.