My Calendar I - Implementation

Here is the simple solution:

class MyCalendar
 bookings := []

 function book(start, end)
 for each (s, e) in bookings
 if start < e and s < end then
 return false
 bookings.add((start, end))
 return true

Two intervals $[s_1, e_1)$ and $[s_2, e_2)$ overlap if and only if $s_1 < e_2$ and $s_2 < e_1$. This is the standard overlap check.

Time: $O(n)$ per booking. Space: $O(n)$ total.