We study the heavy hitters and related sparse recovery problems in the low-failure probability regime. This regime is not well-understood, and has only been studied for non-adaptive schemes. The main previous work is one on sparse recovery by Gilbert et al.(ICALP13). We recognize an error in their analysis, improve their results, and contribute new non-adaptive and adaptive sparse recovery algorithms, as well as provide upper and lower bounds for the heavy hitters problem with low failure probability.
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