Since the problem posed is scale free, so should the solution be, and if there is a solution it must succeed in O(k) steps. Increase step size geometrically instead of linearly, picking an arbitrary distance for the first leg, and the worst-case is O(k), with a ratio of 2 giving the best worst-case value approaching 9k. The adversary chooses k to be much larger than the first leg and just after one of your turning points.
In the non-adversarial case, if log(k) is uniformly distributed between the two turning points in the right direction that enclose k, the optimum ratio is still somewhere close to 2 and the constant is around 4 or 5 (I didn’t do an exact calculation).
ETA: That worst-case ratio of 9k is not right, given that definition of the adversary’s choices. If the adversary is trying to maximise the ratio of distance travelled to k, they can get an unbounded ratio by placing k very close to the starting point and in the opposite direction to the first leg. If we idealise the search path to consist of infinitely many steps in increasing geometric progression, or assume that the adversary is constrained to choose a k at least one quarter of the first step, then the value of 9k holds.
They can’t, because this is all happening in a discrete setting (“Imagine an undirected graph...”) where the minimum possible distance is 1 and your initial distance isn’t going to vary over a very large range.
Since the problem posed is scale free, so should the solution be, and if there is a solution it must succeed in O(k) steps. Increase step size geometrically instead of linearly, picking an arbitrary distance for the first leg, and the worst-case is O(k), with a ratio of 2 giving the best worst-case value approaching 9k. The adversary chooses k to be much larger than the first leg and just after one of your turning points.
In the non-adversarial case, if log(k) is uniformly distributed between the two turning points in the right direction that enclose k, the optimum ratio is still somewhere close to 2 and the constant is around 4 or 5 (I didn’t do an exact calculation).
ETA: That worst-case ratio of 9k is not right, given that definition of the adversary’s choices. If the adversary is trying to maximise the ratio of distance travelled to k, they can get an unbounded ratio by placing k very close to the starting point and in the opposite direction to the first leg. If we idealise the search path to consist of infinitely many steps in increasing geometric progression, or assume that the adversary is constrained to choose a k at least one quarter of the first step, then the value of 9k holds.
They can’t, because this is all happening in a discrete setting (“Imagine an undirected graph...”) where the minimum possible distance is 1 and your initial distance isn’t going to vary over a very large range.