I need help with some algorithm

I have following problem:
Let's f,g be a continous, piece-wise linear functions in range
[0,1], which have finite number of
extreme values, and
(1) f(0)=g(0)=0, f(1)=g(1)=1.
Find functions fi1, fi2 - continous in [0,1] and also fullfil (1)
which satisfy for all t from [0,1] following
equation:
(2) f(fi1(t))=g(fi2(t))

When we'll imagine that f and g describe shapes of two mountains, then
fi1, and fi2 describe walkers' ways for which both of walkers are on
the same level.

I can plot functions f,g and in another figure plot walker's way, which
satisfy condition from exercise (projections of f's argument - say x,
g's argument - say y). From this way I can easy obtain fi1, f2, but
problem is: how to write algorithm for findind this way?

I note that:
1) when both functions f,g increase in some range [a,b] then both
walkers go in the same direction
2) when one function is constant in some range [a,b] then one walker
wait for second until he go this piece of his way,
3) when one function decrease and second increase then one walker have
to return (to be on the same level with second walker)
but I can't any idea for algorithm for this problem.

Any sugestions?

.