In a non-expansive hashing scheme, similar inputs are stored in memory locations which are close. We develop a non-expansive hashing scheme wherein any set of size O(R 1−ε) from a large universe may be stored in a memory of size R (any ε>0, and R>R 0(ɛ)), and where retrieval takes O(1) operations. We explain how to use non-expansive hashing schemes for efficient storage and retrieval of noisy data. A dynamic version of this hashing scheme is presented as well. 1.

The following two computational problems are studied: Duplicate grouping: Assume that n items are given, each of which is labeled by an integer key from the set 0,..., U � 1 4. Store the items in an array of size n such that items with the same key occupy a contiguous segment of the array. Closest pair: Assume that a multiset of n points in the d-dimensional Euclidean space is given, where d � 1 is a fixed integer. Each point is represented as a d-tuple of integers in the range 0,..., U � 14 Ž or of arbitrary real numbers.. Find a closest pair, i.e., a pair of points whose distance is minimal over all such pairs.

We describe a linear-time algorithm for 4-coloring planar graphs. We indeed give an O(V + E + |χ | + 1)-time algorithm to C-color V-vertex E-edge graphs embeddable on a 2-manifold M of Euler characteristic χ where C(M) is given by Heawood’s (minimax optimal) formula. Also we show how, in O(V + E) time, to find the exact chromatic number of a maximal planar graph (one with E = 3V − 6) and a coloring achieving it. Finally, there is a linear-time algorithm to 5-color a graph embedded on any fixed surface M except that an M-dependent constant number of vertices are left uncolored. All the algorithms are simple and practical and run on a deterministic pointer machine, except for planar graph 4-coloring which involves enormous constant factors and requires an integer RAM with a random number generator. All of the algorithms mentioned so far are in the ultra-parallelizable deterministic computational complexity class “NC. ” We also have more practical planar 4-coloring algorithms that can run on pointer machines in O(V log V) randomized time and O(V) space, and a very simple deterministic O(V)-time coloring algorithm for planar graphs which conjecturally uses 4 colors.