Gems from the world of probability and randomized algorithms

Approximate Distance Oracles

Difficulty level: high

Recall 3-approximate distance oracles we discussed in the class. For any undirected graphon n vertices, this data structure takes O(n3/2) space and can report 3-approximate distancebetween any pair of vertices in O(1) time. Interestingly, there is a whole spectrum of theseoracles for various stretches. More stretch we can tolerate, more compact (less space) willbe taken by the oracle. For example, 5-approximate distance oracle will take O(n4/3) space,7-approximate distance oracle will take O(n5/4) space. In general for stretch 2k 1, theoracle will take O(n1+1/k) space. We gave a sketch of 5-approximate distance oracles in theclass. See Figure below.

Generalize the 3-approximate distance oracle and provide the design and analysis of (2k1)-approximate distance oracle.

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