 
Summary: PREPRINT. SIAM Journal of Computing, 24(5):10911103, 1995.
NEW TIGHT BOUNDS ON UNIQUELY REPRESENTED
DICTIONARIES \Lambda
ARNE ANDERSSON y AND THOMAS OTTMANN z
KEYWORDS: analysis of algorithms, data structures, dictionary problem, uniquely
represented dictionaries, binary search trees
Abstract. We present a solution to the dictionary problem where each subset of size n of an
ordered universe is represented by a unique structure, containing a (unique) binary search tree. The
structure permits the execution of search, insert, and delete operations in O(n 1=3 ) time in the worst
case. We also give a general lower bound, stating that for any unique representation of a set in a
graph of bounded outdegree, one of the operations search or update must require a cost of \Omega\Gamma n 1=3 ).
Therefore, our result sheds new light on previously claimed lower bounds for unique representation
of dictionaries.
1. Introduction. A dictionary is a set of items on which search, insert or delete
operations can be performed. The dictionary problem asks for a family of data struc
tures to store the sets of items and for algorithms to carry out the dictionary oper
ations efficiently. We consider a data structure as a graph consisting of nodes linked
together by pointers, one item is stored in each node. The nodes represent the storage
locations. Pointer paths correspond to access paths for the stored items.
In general, there can be many different structures which store the same set of
