A Probabilistic Study on Combinatorial Expanders and Hashing

This paper gives a new way of showing that certain constant degree graphs are graph expanders. This is done by giving new proofs of expansion for three permutations of the Gabber–Galil expander. Our results give an expansion factor of 3 16 for subgraphs of these three-regular graphs with (p− 1)2 inputs for p prime. The proofs are not based on eigenvalue methods or higher algebra. The same methods show the expected number of probes for unsuccessful search in double hashing is bounded by 1 1−α , where α is the load factor. This assumes a double hashing scheme in which two hash functions are randomly and independently chosen from a specified uniform distribution. The result is valid regardless of the distribution of the inputs. This is analogous to Carter and Wegman’s result for hashing with chaining. This paper concludes by elaborating on how any sufficiently sized subset of inputs in any distribution expands in the subgraph of the Gabber–Galil graph expander of focus. This is related to any key distribution having expected 1 1−α probes for unsuccessful search for double hashing given the initial random, independent, and uniform choice of two universal hash functions.