图中的确定性和概率二叉搜索
Deterministic and Probabilistic Binary Search in Graphs
摘要：
我们考虑以下二叉搜索的自然推广：在给定的无向，正加权图中，一个顶点是目标。该算法的任务是通过自适应查询顶点来识别目标。响应于查询节点$q$，该算法获知$q$是目标，或者被给予位于从$q$到目标的最短路径上的$q$的边。我们研究这个问题在一个一般的噪声模型，其中每个查询独立地接收一个正确的答案概率$p > \frac{1}{2}$（一个已知的常数），和（对抗）不正确的概率$1-p$。我们的主要积极结果是，当$p = 1$（即，所有答案都是正确的），则$\log_2 n$查询总是足够的。对于一般的$p$，我们给出了一个（几乎信息理论上最优的）算法，使用，在预期中，不超过$(1 - δ)\frac{\log_2 n}{1 - H(p)} + o(\log n) + O(\log^2 (1/δ))$查询，并确定目标正确的概率至少为$1-δ$。这里，$H(p) = -(p \log p + (1-p) \log(1-p))$表示熵。第一个界限是通过迭代查询尚未排除的节点的1-中值的算法来实现的;第二个界限是通过仔细地重复调用乘法权重算法来实现的。即使是$p = 1$，我们显示了几个硬度的问题，确定是否可以使用$K$查询找到一个目标的结果。我们的上限$\log_2 n$意味着一个准多项式时间算法无向连通图，我们表明，这是最好的可能下的强指数时间假设（SETH）。此外，对于有向图或具有非均匀节点查询成本的无向图，该问题是PSPACE完全的。对于一个半自适应版本，其中一个可以查询$r$节点，每个在$k$轮，我们显示在$Σ_{2k-1}$在多项式层次的成员资格，和硬度为$Σ_{2k-5}$。
英文摘要：
We consider the following natural generalization of Binary Search: in a given undirected, positively weighted graph, one vertex is a target. The algorithm's task is to identify the target by adaptively querying vertices. In response to querying a node $q$, the algorithm learns either that $q$ is the target, or is given an edge out of $q$ that lies on a shortest path from $q$ to the target. We study this problem in a general noisy model in which each query independently receives a correct answer with probability $p > \frac{1}{2}$ (a known constant), and an (adversarial) incorrect one with probability $1-p$. Our main positive result is that when $p = 1$ (i.e., all answers are correct), $\log_2 n$ queries are always sufficient. For general $p$, we give an (almost information-theoretically optimal) algorithm that uses, in expectation, no more than $(1 - δ)\frac{\log_2 n}{1 - H(p)} + o(\log n) + O(\log^2 (1/δ))$ queries, and identifies the target correctly with probability at leas $1-δ$. Here, $H(p) = -(p \log p + (1-p) \log(1-p))$ denotes the entropy. The first bound is achieved by the algorithm that iteratively queries a 1-median of the nodes not ruled out yet; the second bound by careful repeated invocations of a multiplicative weights algorithm. Even for $p = 1$, we show several hardness results for the problem of determining whether a target can be found using $K$ queries. Our upper bound of $\log_2 n$ implies a quasipolynomial-time algorithm for undirected connected graphs; we show that this is best-possible under the Strong Exponential Time Hypothesis (SETH). Furthermore, for directed graphs, or for undirected graphs with non-uniform node querying costs, the problem is PSPACE-complete. For a semi-adaptive version, in which one may query $r$ nodes each in $k$ rounds, we show membership in $Σ_{2k-1}$ in the polynomial hierarchy, and hardness for $Σ_{2k-5}$.
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