First-Hitting Times for Finite State Spaces

Abstract

One of the most important aspects of a randomized algorithm is bounding its expected run time on various problems. Formally speaking, this means bounding the expected first-hitting time of a random process. The two arguably most popular tools to do so are the fitness level method and drift theory. The fitness level method considers arbitrary transition probabilities but only allows the process to move toward the goal. On the other hand, drift theory allows the process to move into any direction as long as it move closer to the goal in expectation; however, this tendency has to be monotone and, thus, the transition probabilities cannot be arbitrary.

We provide a result that combines the benefit of these two approaches: our result gives a lower and an upper bound for the expected first-hitting time of a random process over $\{0, \ldots , n\}$ that is allowed to move forward and backward by 1 and can use arbitrary transition probabilities. In case that the transition probabilities are known, our bounds coincide and yield the exact value of the expected first-hitting time. Further, we also state the stationary distribution as well as the mixing time of a special case of our scenario.