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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2406.06468 |
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| _version_ | 1866911923854901248 |
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| author | Caracci, Agustín Dürr, Christoph Verschae, José |
| author_facet | Caracci, Agustín Dürr, Christoph Verschae, José |
| contents | We study a generalized binary search problem on the line and general trees. On the line (e.g., a sorted array), binary search finds a target node in $O(\log n)$ queries in the worst case, where $n$ is the number of nodes. In situations with limited budget or time, we might only be able to perform a few queries, possibly sub-logarithmic many. In this case, it is impossible to guarantee that the target will be found regardless of its position. Our main result is the construction of a randomized strategy that maximizes the minimum (over the target position) probability of finding the target. Such a strategy provides a natural solution where there is no apriori (stochastic) information of the target's position. As with regular binary search, we can find and run the strategy in $O(\log n)$ time (and using only $O(\log n)$ random bits). Our construction is obtained by reinterpreting the problem as a two-player (\textit{seeker} and \textit{hider}) zero-sum game and exploiting an underlying number theoretical structure.
Furthermore, we generalize the setting to study a search game on trees. In this case, a query returns the edge's endpoint closest to the target. Again, when the number of queries is bounded by some given $k$, we quantify a \emph{the-less-queries-the-better} approach by defining a seeker's profit $p$ depending on the number of queries needed to locate the hider. For the linear programming formulation of the corresponding zero-sum game, we show that computing the best response for the hider (i.e., the separation problem of the underlying dual LP) can be done in time $O(n^2 2^{2k})$, where $n$ is the size of the tree. This result allows to compute a Nash equilibrium in polynomial time whenever $k=O(\log n)$. In contrast, computing the best response for the hider is NP-hard. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_06468 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Randomized Binary and Tree Search under Pressure Caracci, Agustín Dürr, Christoph Verschae, José Data Structures and Algorithms Computer Science and Game Theory We study a generalized binary search problem on the line and general trees. On the line (e.g., a sorted array), binary search finds a target node in $O(\log n)$ queries in the worst case, where $n$ is the number of nodes. In situations with limited budget or time, we might only be able to perform a few queries, possibly sub-logarithmic many. In this case, it is impossible to guarantee that the target will be found regardless of its position. Our main result is the construction of a randomized strategy that maximizes the minimum (over the target position) probability of finding the target. Such a strategy provides a natural solution where there is no apriori (stochastic) information of the target's position. As with regular binary search, we can find and run the strategy in $O(\log n)$ time (and using only $O(\log n)$ random bits). Our construction is obtained by reinterpreting the problem as a two-player (\textit{seeker} and \textit{hider}) zero-sum game and exploiting an underlying number theoretical structure. Furthermore, we generalize the setting to study a search game on trees. In this case, a query returns the edge's endpoint closest to the target. Again, when the number of queries is bounded by some given $k$, we quantify a \emph{the-less-queries-the-better} approach by defining a seeker's profit $p$ depending on the number of queries needed to locate the hider. For the linear programming formulation of the corresponding zero-sum game, we show that computing the best response for the hider (i.e., the separation problem of the underlying dual LP) can be done in time $O(n^2 2^{2k})$, where $n$ is the size of the tree. This result allows to compute a Nash equilibrium in polynomial time whenever $k=O(\log n)$. In contrast, computing the best response for the hider is NP-hard. |
| title | Randomized Binary and Tree Search under Pressure |
| topic | Data Structures and Algorithms Computer Science and Game Theory |
| url | https://arxiv.org/abs/2406.06468 |