每个决策树都有一个影响变量
Every decision tree has an influential variable
摘要：
我们证明了对于任何决策树，计算一个布尔函数$f:\{-1,1\}^n\to\{-1,1\}$，\[ \Var[f] \le \sum_{i=1}^n δ_i \Inf_i(f), \]，其中$δ_i$是$i$第th个输入变量被读取的概率，$\Inf_i(f)$是$i$第th个变量对$f$的影响。在$\{-1,1\}^n$上对任意乘积测度取方差、影响和概率。因此，计算给定平衡函数的决策树的最小深度至少是任何输入变量的最大影响的倒数。类似地，具有深度为$d$的决策树的任何平衡布尔函数具有影响至少为$\frac{1}{d}$的变量。以前唯一已知的非平凡下界是$Ω(d 2^{-d})$。我们的不等式有许多推广，使我们能够证明随机决策树的影响下界，任意产品概率空间上的决策树，以及具有非布尔输出的决策树。作为我们的结果的应用，我们给出了一个非常简单的证明，非平凡单调图属性的随机查询复杂性至少是$Ω(v^{4/3}/p^{1/3})$，其中$v$是顶点的数量和$p \leq \half$是临界阈值概率。这取代了Hajnal的里程碑式的$Ω(v^{4/3})$界，有时优于Chakrabarti Khot和Friedgut Kahn Wigderson的最知名的下界。
英文摘要：
We prove that for any decision tree calculating a boolean function $f:\{-1,1\}^n\to\{-1,1\}$, \[ \Var[f] \le \sum_{i=1}^n δ_i \Inf_i(f), \] where $δ_i$ is the probability that the $i$th input variable is read and $\Inf_i(f)$ is the influence of the $i$th variable on $f$. The variance, influence and probability are taken with respect to an arbitrary product measure on $\{-1,1\}^n$. It follows that the minimum depth of a decision tree calculating a given balanced function is at least the reciprocal of the largest influence of any input variable. Likewise, any balanced boolean function with a decision tree of depth $d$ has a variable with influence at least $\frac{1}{d}$. The only previous nontrivial lower bound known was $Ω(d 2^{-d})$. Our inequality has many generalizations, allowing us to prove influence lower bounds for randomized decision trees, decision trees on arbitrary product probability spaces, and decision trees with non-boolean outputs. As an application of our results we give a very easy proof that the randomized query complexity of nontrivial monotone graph properties is at least $Ω(v^{4/3}/p^{1/3})$, where $v$ is the number of vertices and $p \leq \half$ is the critical threshold probability. This supersedes the milestone $Ω(v^{4/3})$ bound of Hajnal and is sometimes superior to the best known lower bounds of Chakrabarti-Khot and Friedgut-Kahn-Wigderson.
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