Verified Tail Bounds For Randomized Programs

We mechanize a theorem by Karp, along with several extensions, that provide an easy to use "cookbook" method for verifying tail bounds of randomized algorithms, much like the traditional "Master Theorem" gives bounds for deterministic algorithms. We apply these results to several examples: the number of comparisons performed by Quick-Sort, the span of parallel QuickSort, the height of randomly generated binary search trees, and the number of rounds needed for a distributed leader election protocol. Because the constants involved in our symbolic bounds are concrete, we are able to use them to derive numerical probability bounds for various input sizes for these examples.