We define a mountain range via its peaks and valleys. Let
(x_{1}, y_{1}),
(x_{2}, y_{2}), ...
(x_{n}, y_{n})
be a sequence of points in the upper half-plane satisfying the following
conditions:
- (x_{1}, y_{1}) = (a,0) and
(x_{n}, y_{n}) = (b,0)
- x_{i} < x_{i+1}
for i = 1 to n-1
- if y_{i-1} < y_{i} then
y_{i} > y_{i+1}
for i = 2 to n-1;
conversely, if y_{i-1} > y_{i} then
y_{i} < y_{i+1}
Then, we call the set of line segments formed by joining the points
(x_{i}, y_{i}) and
(x_{i+1}, y_{i+1})
for i = 1 to n-1
a "mountain range".
Back to the Parallel Mountain Climbers problem