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You could have invented fractional cascading

Suppose that you have k sorted arrays, each of size n. You would like to search for single element in each of the k arrays (or its predecessor, if it doesn't exist).

Obviously you can binary search each array individually, resulting in a runtime. Consider the case where k = 2; everything would be better if only we could guarantee that the first list contained the right elements to give you useful information about the second array. When I follow a pointer down, I might end up on an element which is not actually a member of the current array (it was one that was cascaded up).

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