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You are standing on the middle of a East-West street. A row of closed doors in front of you, and you have a key on hand which can only open one specific door. Compose a strategy so that X/N value is smallest under the worst situation. X is the distance you covered when arriving at the correct door. N is the distance between the door and the original point.

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Compose a strategy for what? –  Henning Makholm Jan 29 '12 at 16:27
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Your problem is known as the cow-path problem. The answer is an example of an online algorithm. The best $X/N$ is known as the competitive ratio. The competitive ratio in your case is~$9$ for deterministic algorithms, and $4.6$ for randomized algorithms.

See Wikipedia for references.

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That Wikipedia article is almost worthless, I'm afraid. –  TonyK Jan 29 '12 at 16:44
    
Google "cow path problem" instead. –  Yuval Filmus Jan 29 '12 at 17:04
    
Link to that instead. –  TonyK Jan 29 '12 at 19:49
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