The Algorithm

Here is the algorithm:

$1.$ Sort all activities by end time (ascending).

$2.$ Select the first activity (it ends earliest).

$3.$ For each remaining activity: if its start time >= the end time of the last selected activity, select it.

$4.$ Return the count of selected activities. The key condition: no overlap means start >= previous end. If an activity starts exactly when the previous one ends, they do not conflict. Time: $O(n \log n)$. Space: $O(1)$.