We show that a context-free grammar of size that produces a single string of length (such a grammar is also called a string straight-line program) can be transformed in linear time into a context-free grammar for of size , whose unique derivation tree has depth . This solves an open problem in the area of grammar-based compression, improves many results in this area, and greatly simplifies many existing constructions. Similar results are shown for two formalisms for grammar-based tree compression: top dags and forest straight-line programs. These balancing results can be all deduced from a single meta-theorem stating that the depth of an algebraic circuit over an algebra with a certain finite base property can be reduced to with the cost of a constant multiplicative size increase. Here, refers to the size of the unfolding (or unravelling) of the circuit. In particular, this results applies to standard arithmetic circuits over (noncommutative) semirings.

中文翻译：

---

平衡直线程序

我们证明了大小的上下文无关文法 产生单个字符串 长度 （这样的文法也称为字符串直线程序）可以在线性时间内转化为上下文无关文法 大小的 ，其唯一派生树有深度 . 这解决了基于语法的压缩领域的一个开放问题，改进了该领域的许多结果，并大大简化了许多现有结构。对于基于语法的树压缩的两种形式，也显示了类似的结果：top dags 和森林直线程序。这些平衡结果都可以从一个单一的元定理推导出来 以恒定乘法大小增加的成本。这里， 指电路展开（或解开）的大小。特别是，该结果适用于（非交换）半环上的标准算术电路。