Union nd with deletions

In the classical union nd problem we maintain a partition of a universe of n elements into disjoint sets subject to the operations union and nd The operation union A B C replaces sets A and B in the partition by their union given the name C The operation find x returns the name of the set containing the element x In this paper we revisit the union nd problem in a context where the underlying partitioned universe is not xed Speci cally we allow a delete x operation which removes the element x from the set containing it We consider both worst case performance and amortized performance In both settings the challenge is to dynamically keep the size of the structure representing each set proportional to the number of elements in the set which may now decrease as a result of deletions For any xed k we describe a data structure that supports nd and delete in O logk n worst case time and union in O k worst case time This matches the best possible worst case bounds for nd and union in the classical setting Furthermore using an incremental global rebuilding technique we obtain a reduction converting any union nd data structure to a union nd with deletions data structure Our reduction is such that the time bounds for nd and union change only by a constant factor The time it takes to delete an element x is the same as the time it takes to nd the set containing x plus the time it takes to unite a singleton set with this set In an amortized setting a classical data structure of Tarjan supports a sequence of m nds and at most n unions on a universe of n elements in O n m m n n log n time where m n l minfk j Ak bmn c lg and Ai j is Ackermann s function as described in We re ne the analysis of this data structure and show that in fact the cost of each nd is proportional to the size of the corresponding set Speci cally we show that one can pay for a sequence of union and nd operations by charging a constant to each participating element and O m n log l for a nd of an element in a set of size l We also show how keep these amortized costs for each nd and each participating element while allowing deletions The amortized cost of deleting an element from a set of l elements is the same as the amortized cost of nding the element namely O m n log l