We obtain matching upper and lower bounds for the amount of time to find the predecessor of a given element among the elements of a fixed efficiently stored set. Our algorithms are for the unit-cost word-level RAM with multiplication and extend to give optimal dynamic algorithms. The lower bounds are proved in a much stronger communication game model, but they apply to the cell probe and RAM models and to both static and dynamic predecessor problems.

We obtain matching upper and lower bounds for the amount of time to find the predecessor of a given element among the elements of a fixed compactly stored set. Our algorithms are for the unit-cost word RAM with multiplication and are extended to give dynamic algorithms. The lower bounds are proved for a large class of problems, including both static and dynamic predecessor problems, in a much stronger communication game model, but they apply to the cell probe and RAM models.

We introduce a novel technique for converting static polynomial space search structures for ordered sets into fullydynamic linear space data structures. Based on this we present optimal bounds for dynamic integer searching, including finger search, and exponentially improved bounds for priority queues.

It is shown that a static dictionary that offers constant-time access to n elements with w-bit keys and occupies O(n) words of memory can be constructed deterministically in O(n log n) time on a unit-cost RAM with word length w and a standard instruction set including multiplication. Whereas a randomized construction working in linear expected time was known, the running time of the best previous deterministic algorithm was Ω(n²). Using a standard dynamization technique, the first deterministic dynamic dictionary with constant lookup time and sublinear update time is derived. The new algorithms are weakly nonuniform; i.e., they require access to a fixed number of precomputed constants dependent on w. The main technical tools employed are unit-cost error-correcting codes, word parallelism, and derandomization using conditional expectations.

We introduce exponential search trees as a novel technique for converting static polynomial space search structures for ordered sets into fully-dynamic linear space data structures. This leads to an optimal bound of O ( � log n / log log n) for searching and updating a dynamic set X of n integer keys in linear space. Searching X for an integer y means finding the maximum key in X which is smaller than or equal to y. This problem is equivalent to the standard text book problem of maintaining an ordered set. The best previous deterministic linear space bound was O(log n / log log n) due to Fredman and Willard from STOC 1990. No better deterministic search bound was known using polynomial space. We also get the following worst-case linear space trade-offs between the number n, the word length W, and the maximal key U < 2W: O(min{log log n + log log U log n / log W, log log n · log log log U}). These trade-offs are, however, not likely to be optimal. Our results are generalized to finger searching and string searching, providing optimal results for both in terms of n.

We reexamine fundamental problems from computational geometry in the word RAM model, where input coordinates are integers that fit in a machine word. We develop a new algorithm for offline point location, a two-dimensional analog of sorting where one needs to order points with respect to segments. This result implies, for example, that the convex hull of n points in three dimensions can be constructed in (randomized) time n·2 O( √ lglgn). Similar bounds hold for numerous other geometric problems, such as planar Voronoi diagrams, planar off-line nearest neighbor search, line segment intersection, and triangulation of non-simple polygons. In FOCS’06, we developed a data structure for online point location, which implied a bound of O(n lgn lglgn) for three-dimensional convex hulls and the other problems. Our current bounds are dramatically better, and a convincing improvement over the classic O(nlgn) algorithms. As in the field of integer sorting, the main challenge is to find ways to manipulate information, while avoiding the online problem (in that case, predecessor search).

This thesis is centered around one of the most basic information retrieval problems, namely that of storing and accessing the elements of a set. Each element in the set has some associated information that is returned along with it. The problem is referred to as the dictionary problem, due to the similarity to a bookshelf dictionary, which contains a set of words and has an explanation associated with each word. In the static version of the problem the set is fixed, whereas in the dynamic version, insertions and deletions of elements are possible. The approach

revisits classic problems in computational geometry from the modern algorithmic