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While solving This algorithm problem I was unable to come up with condition for the collision to occur ( other than the naive O(n^2) algorithm ) on reading the explanation they say

Let’s deepen the observation. For a set of people with the same y coordinate, let’s consider “in what condition does a collision occurs?” Then we can see that a collision occurs if and only if the following two people collides: the leftmost person facing right, and the rightmost person facing left. In other words, among the people with the same y coordinate, all the people except for the two mentioned above is unimportant.

I am unable to understand how the two people who are at extremes of some y=k satisfy the condition it is quite possible that the two at extremes of this y=k face the same direction and there could exist some pair in between these extremes that satisfy the condition, can you help me understand if I'm wrong in understanding the solution?

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Your concern is legitimate.

However,

  • "the leftmost person facing right" means the leftmost person among all persons who face right.
  • "the rightmost person facing left" means the rightmost person among all persons who face left.
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    $\begingroup$ Yes I came to realize that and it worked, thanks for the clarification. $\endgroup$ Mar 19, 2022 at 12:37

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