A Theoretical Study on Variable Ordering of Zero-Suppressed BDDs for Representing Frequent Itemsets

Abstract

Recently, an efficient method of database analysis using Zero-suppressed Binary Decision Diagrams (ZBDDs) has been proposed. BDDs are a graph-based representation of Boolean functions, now widely used in system design and verification. Here we focus on ZBDDs, a special type of BDDs, which are suitable for handling large-scale combinatorial itemsets in frequent itemset mining. In general, it is well-known that the size of ZBDDs greatly depends on variable ordering; however, in the specific cases of applying ZBDDs to data mining, the effect of variable ordering has not been studied well. In this paper, we present a theoretical study on ZBDD variable ordering for representing frequent itemsets. We show two instances of databases we composed, where the ZBDD sizes are exponentially sensitive to the variable ordering. We also show that there is a case where the ZBDD size must be exponential in any variable ordering. Our theoretical results are helpful for developing a good heuristic method of variable ordering.