Question

If we are looking for line intersections (horizontal and vertical lines only) and we have n lines with half of them vertical and no intersections then

Sorting the list of line end points on y value will take N log N using mergesort

Each insert delete and search of our data structue (assuming its a b-tree) will be < log n

so the total search time will be N log N

What am i missing here, if the time to sort using mergesort takes a time of N log N and insert and delete takes a time of < log n are we dropping the constant factor to give an overal time of N log N. If not then how comes < log n goes missing in total ONotation run time?

Thanks

Answer 1

The big-O notation describes the asymptotic behavior of the algorithm. That is, it describes the behavior of the algorithm as N trends towards infinity. The portion of the algorithm that takes N log N time will dwarf the portion of the algorithm that takes log N time. The significance of the log N portion diminishes to relatively nothing as N becomes large.