(Really) Tight bounds for dispatching binary methods

We consider binary dispatching problem originating from object oriented programming. We want to preprocess a hierarchy of classes and collection of methods so that given a function call in the run-time we are able to retrieve the most specialized implementation which can be invoked with the actual types of the arguments. This problem has been thoroughly studied for the case of mono dispatching [7,4], where the methods take just one argument, resulting in (expected) O(log logm) query time after just linear preprocessing. For the binary dispatching, where the methods take exactly two arguments, logarithmic query time is possible [5], even if the structure is allowed to take linear space [1]. Unfortunately, constructing such structure requires as much as (expected) Θ(m(log logm)) time [1,9]. Using a different idea we are able to construct in (deterministic) linear time and space a structure allowing dispatching binary methods in the same logarithmic time. Then we show how to improve the query time to just O( logm log logm ), which is easily seen to be optimal as a consequence of some already known lower bounds if we want to keep the size of the resulting structure close to linear. Key-words: method dispatching, persistent data structures, rectangle geometry