Related to: Half-assing it with everything you've got; Wasted motion; Say it Loud.
Once upon a time (true story), I was on my way to a hotel in a new city. I knew the hotel was many miles down this long, branchless road. So I drove for a long while.
After a while, I began to worry I had passed the hotel.
So, instead of proceeding at 60 miles per hour the way I had been, I continued in the same direction for several more minutes at 30 miles per hour, wondering if I should keep going or turn around.
- I wasn't sure if I was a good enough writer to write a given doc myself, or if I should try to outsource it. So, I sat there kind-of-writing it while also fretting about whether the task was correct.
- (Solution: Take a minute out to think through heuristics. Then, either: (1) write the post at full speed; or (2) try to outsource it; or (3) write full force for some fixed time period, and then pause and evaluate.)
- I wasn't sure (back in early 2012) that CFAR was worthwhile. So, I kind-of worked on it.
- An old friend came to my door unexpectedly, and I was tempted to hang out with her, but I also thought I should finish my work. So I kind-of hung out with her while feeling bad and distracted about my work.
- A friend of mine, when teaching me math, seems to mumble specifically those words that he doesn't expect me to understand (in a sort of compromise between saying them and not saying them)...
- Duncan reports that novice Parkour students are unable to safely undertake certain sorts of jumps, because they risk aborting the move mid-stream, after the actual last safe stopping point (apparently kind-of-attempting these jumps is more dangerous than either attempting, or not attempting the jumps)
- It is said that start-up founders need to be irrationally certain that their startup will succeed, lest they be unable to do more than kind-of work on it...
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.