Algorithm to calculate number of intersecting discs

Question

Given an array A of N integers we draw N discs in a 2D plane, such that i-th disc has center in (0,i) and a radius

Answer 1

Thanks to Falk for the great idea! Here is a ruby implementation that takes advantage of sparseness.

def int(a)

    event = Hash.new{|h,k| h[k] = {:start => 0, :stop => 0}}
    a.each_index {|i|
        event[i - a[i]][:start] += 1
        event[i + a[i]][:stop ] += 1
    }
    sorted_events = (event.sort_by {|index, value| index}).map! {|n| n[1]}

    past_start = 0
    intersect = 0

    sorted_events.each {|e|

        intersect += e[:start] * (e[:start]-1) / 2 +
                     e[:start] * past_start

        past_start += e[:start]
        past_start -= e[:stop]
    }
    return intersect
end

puts int [1,1]

puts int [1,5,2,1,4,0]