Min-Wise Independent Permutations

We define and study the notion of min-wise independent families of permutations. We say that F ⊆ Sn is min-wise independent if for any set X ⊆ [n] and any x ∈ X, when π is chosen at random in F we have Pr(min{π(X)} = π(x)) = 1 |X| . In other words we require that all the elements of any fixed set X have an equal chance to become the minimum element of the image of X under π. Our research was motivated by the fact that such a family (under some relaxations) is essential to the algorithm used in practice by the AltaVista web index software to detect and filter near-duplicate documents. However, in the course of our investigation we have discovered interesting and challenging theoretical questions related to this concept – we present the solutions to some of them and we list the rest as open problems. ∗Digital SRC, 130 Lytton Avenue, Palo Alto, CA 94301, USA. E-mail: [email protected]. †Computer Science Department, Stanford University, CA 94305, USA. E-mail: [email protected]. Part of this work was done while this author was a summer intern at Digital SRC. Supported by the Pierre and Christine Lamond Fellowship and in part by an ARO MURI Grant DAAH04-96-1-0007 and NSF Award CCR-9357849, with matching funds from IBM, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. ‡Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA. Part of this work was done while this author was visiting Digital SRC. Supported in part by NSF grant CCR9530974. E-mail: [email protected] §Digital SRC, 130 Lytton Avenue, Palo Alto, CA 94301, USA. E-mail: [email protected].