The complexity of selection and ranking in X + Y and matrices with sorted columns

AbstractThe complexity of selection is analyzed for two sets, X + Y and matrices with sorted columns. Algorithms are presented that run in time which depends nontrivially on the rank k of the element to be selected and which is sublinear with respect to set cardinality. Identical bounds are also shown for the problem of ranking elements in these sets, and all bounds are shown to be optimal to within a constant multiplicative factor

Cache-Oblivious Selection in Sorted X+Y Matrices

Let X[0..n-1] and Y[0..m-1] be two sorted arrays, and define the mxn matrix A by A[j][i]=X[i]+Y[j]. Frederickson and Johnson gave an efficient algorithm for selecting the k-th smallest element from A. We show how to make this algorithm IO-efficient. Our cache-oblivious algorithm performs O((m+n)/B) IOs, where B is the block size of memory transfers