The interval-merging problem

Abstract

A closed interval is an ordered pair of real numbers [ x, y], with x ⩽ y. The interval [ x, y] represents the set { i ∈ R∣ x ⩽ i ⩽ y}. Given a set of closed intervals I = { [ a 1 , b 1 ] , [ a 2 , b 2 ] , … , [ a k , b k ] } , the Interval-Merging Problem is to find a minimum-cardinality set of intervals M ( I ) = { [ x 1 , y 1 ] , [ x 2 , y 2 ] , … , [ x j , y j ] } , j ⩽ k, such that the real numbers represented by I = ⋃ i = 1 k [ a i , b i ] equal those represented by M ( I ) = ⋃ i = 1 j [ x i , y i ] . In this paper, we show the problem can be solved in O( d log d) sequential time, and in O(log d) parallel time using O( d) processors on an EREW PRAM, where d is the number of the endpoints of I . Moreover, if the input is given as a set of sorted endpoints, then the problem can be solved in O( d) sequential time, and in O(log d) parallel time using O( d/log d) processors on an EREW PRAM.