Binary Search with Distance-Dependent Costs
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We introduce a search problem generalizing the typical setting of Binary Search on the line. Similar to the setting for Binary Search, a target is chosen adversarially on the line, and in response to a query, the algorithm learns whether the query was correct, too high, or too low. Different from the Binary Search setting, the cost of a query is a monotone non-decreasing function of the distance between the query and the correct answer; different functions can be used for queries that are too high vs. those that are too low. The algorithm's goal is to identify an adversarially chosen target with minimum total cost. Note that the algorithm does not even know the cost it incurred until the end, when the target is revealed. This abstraction captures many natural settings in which a principal experiments by setting a quantity (such as an item price, bandwidth, tax rate, medicine dosage, etc.) where the cost or regret increases the further the chosen setting is from the optimal one. First, we show that for arbitrary symmetric cost functions (i.e., overshooting vs. undershooting by the same amount leads to the same cost), the standard Binary Search algorithm is a 4-approximation. We then show that when the cost functions are bounded-degree polynomials of the distance, the problem can be solved optimally using Dynamic Programming; this relies on a careful encoding of the combined cost of past queries (which, recall, will only be revealed in the future). We then generalize the setting to finding a node on a tree; here, the response to a query is the direction on the tree in which the target is located, and the cost is increasing in the distance on the tree from the query to the target. Using the k-cut search tree framework of Berendsohn and Kozma and the ideas we developed for the case of the line, we give a PTAS when the cost function is a bounded-degree polynomial.
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