Incremental Edge Orientation in Forests
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For any forest G = (V, E) it is possible to orient the edges E so that no vertex in V has out-degree greater than 1. This paper considers the incremental edge-orientation problem, in which the edges E arrive over time and the algorithm must maintain a low-out-degree edge orientation at all times. We give an algorithm that maintains a maximum out-degree of 3 while flipping at most O(loglog n) edge orientations per edge insertion, with high probability in n. The algorithm requires worst-case time O(log n loglog n) per insertion, and takes amortized time O(1). The previous state of the art required up to O(log n / loglog n) edge flips per insertion. We then apply our edge-orientation results to the problem of dynamic Cuckoo hashing. The problem of designing simple families ℋ of hash functions that are compatible with Cuckoo hashing has received extensive attention. These families ℋ are known to satisfy static guarantees, but do not come typically with dynamic guarantees for the running time of inserts and deletes. We show how to transform static guarantees (for 1-associativity) into near-state-of-the-art dynamic guarantees (for O(1)-associativity) in a black-box fashion. Rather than relying on the family ℋ to supply randomness, as in past work, we instead rely on randomness within our table-maintenance algorithm.
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