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Search patterns of randomly oriented steps of different lengths have beenobserved on all scales of the biological world, ranging from the microscopic tothe ecological, including in protein motors, bacteria, T-cells, honeybees,marine predators, and more. Through different models, it has been demonstratedthat adopting a variety in the magnitude of the step lengths can greatlyimprove the search efficiency. However, the precise connection between thesearch efficiency and the number of step lengths in the repertoire of thesearcher has not been identified. Motivated by biological examples inone-dimensional terrains, a recent paper studied the best cover time on ann-node cycle that can be achieved by a random walk process that uses k steplengths. By tuning the lengths and corresponding probabilities the authorstherein showed that the best cover time is roughly n 1+$\Theta$(1/k). Whilethis bound is useful for large values of k, it is hardly informative for smallk values, which are of interest in biology. In this paper, we provide a tightbound for the cover time of such a walk, for every integer k > 1. Specifically,up to lower order polylogarithmic factors, the upper bound on the cover time isa polynomial in n of exponent 1+ 1/(2k--1). For k = 2, 3, 4 and 5 the exponentis thus 4/3 , 6/5 , 8/7 , and 10/9 , respectively. Informally, our resultimplies that, as long as the number of step lengths k is not too large,incorporating an additional step length to the repertoire of the processenables to improve the cover time by a polynomial factor, but the extent of theimprovement gradually decreases with k.

Tight Bounds for the Cover Times of Random Walks with Heterogeneous Step Lengths