How efficient can an exhaustive search be? Whether the goal is to locate every mushroom in a forest, say, or specific sequences on a DNA strand, the cover time τ—defined as the time it takes for a random walk to find all the targets in a spatial network—can quantifiably answer that question. But despite τ’s relevance to a broad range of situations, from animals foraging for food to diseases spreading through a city, analytical results have been scarce and mostly limited to regular random walks—those involving moves between nearest neighbors in Euclidean geometry.

Not all random trajectories look the same, and in more complex strategies, the random walker’s movement among neighbors differs from diffusive Brownian motion: It can be “persistent,” for instance, weighted toward a particular direction that depends on a previous step; be “intermittent,” stepping between nearest neighbors at one rate and leapfrogging over them at another...

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