Let the speed of the car be v, the speed of a walking person be u, and the distance between cities by L. We want to minimize T, the time t at which all persons are at displacement x=L (city B), when they all start at displacement x=0 (city A) at time t=0.

I'll assume that the solution has everyone starting out from city A at the same time t=0 arriving in city B at the same time t=T so nobody is standing around idly waiting. Let's plot everyone's movements on a graph showing coordinates (t,x). Then at time t just after 0, (N-2) walkers are on the line L0 through (0,0) with slope dx/dt = u, and 2 in the car are on a line through (0,0) with slope v, and at t just before T, (N-2) walkers are on the line L1 through (T,L) with slope u, and 2 in the car are on a line through (T,L) with slope v. Obviously L1 lies "above" L0 (greater x coordinate given the same t coordinate). In between t=0 and t=T, the car zigzags between L0 and L1 along lines of slope v and -v, picking up people from L0 and dropping them off at L1. I will not prove that this is the optimal strategy; in fact you can make an infinite number of variations on it which all come up with the same elapsed time.

Now examine the graph again. Say the car travels distance r between picking someone up and dropping that person off, and distance s back to pick up the next person. The car makes (N-1) trips forward and (N-2) trips back to pick up and ferry everyone, so its total travel is

vT = (N - 1)r + (N - 2)s = (N - 2)(r + s) + r

Moreover the car makes (r-s) net displacement on each round trip, plus r displacement on the extra forward trip, so

L = (N - 2)(r - s) + r

Note that a person walks distance (r-s) in the time it takes the car to go (r+s), so

r - s = (u/v)(r + s)

A little algebraic manipulation of this equation shows us that

r - s = r * 2u/(v + u) r + s = r * 2v/(v + u)

Plug this into the equation for L, and we get the first important piece of information, how far the car should drive before dropping off the passenger (once you know this, you tell it to the driver and this

guarantees the people get to B in minimum time)

L = r + (N - 2) r * 2u/(v + u)

= r * (v + u + (N - 2)*2u)/(v + u)

v + u

r = ------------ L

2uN + v - 3u

We can also find out what the elapsed time T will be

vT = (N - 2)r*2v/(v + u) + r

= r * ((N - 2)*2v + v + u)/(v + u)

1 2vN - 3v + u

T = - * ------------ r

v v + u

Therefore

2vN - (3v - u)

T = -------------- (L/v)

2uN + v - 3u

-- David Karr (karr@cs.cornell.edu)

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