Reference machines require non-linear time to maintain disjoint sets
STOC '77 Proceedings of the ninth annual ACM symposium on Theory of computing
Data Structures for Range Searching
ACM Computing Surveys (CSUR)
Multidimensional divide-and-conquer
Communications of the ACM
Kraft storage and access for list implementations(Extended Abstract)
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
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Let G denote the set of elements of a commutative group whose addition operations is denoted by +, let N be a positive integer, and let A(1) ,..., A(N) denote an array with values in G. We will be concerned with designing data structures for representing the array A, which facilitate efficient implementation of the following two on-line tasks: (1) Update(j,x); replace A(j) by A(j) +x. (j and x are inputs, 1≤j≤N and x&egr;G) (2) Retrieve(j); returns the value of A(1) +...+ A(j). (j is an input, 1≤j≤N) As a motivating example, let G be the group of integers with + denoting the usual addition operation. Imagine a standardized examination given to large numbers of individuals over an indefinite period of time. Assume that each examinee will attain an integer score in the interval [1,N]. If an individual gets j points, this fact is recorded by executing Update(j,1). so that A(j) represents the number of individuals to date having scored j points. In order to compute the percentile currently associated with a particular score k, we need the cumulative sum provided by executing Retrieve(k).