Several papers describe linear time algorithms to preprocess a tree, such that one can answer subsequent nearest common ancestor queries in constant time. Here, we survey these algorithms and related results. A common idea used by all the algorithms for the problem is that a solution for complete balanced binary trees is straightforward. Furthermore, for complete balanced binary trees we can easily solve the problem in a distributed way by labeling the nodes of the tree such that from the labels of two nodes alone one can compute the label of their nearest common ancestor. Whether it is possible to distribute the data structure into short labels associated with the nodes is important for several applications such as routing. Therefore, related labeling problems have received a lot of attention recently.

A strategy S solving a navigation task T is called competitive with ratio r if the cost of solving any instance t of T does not exceed r times the cost of solving t optimally. The competitive complexity of task T is the smallest possible value r any strategy S can achieve. We discuss this notion, and survey some tasks whose competitive complexities are known.

lar biology, which suddenly faces huge DNA string processing tasks, that were beyond imaginationand certainly beyond feasibilityonly decades ago. Computer Science, namely the study of ecient algorithms, provides the tools to analyze and solve these problems. A concrete proof of the inter-scientic relevance of this type of research is the importance of the concept of NP-completeness, which arose in the study of ecient algorithms only 25 years ago. Since then, the concept has spread to more than 20 dierent scientic disciplines, including Physics, Biology, and Economics. Each year, about 6,000 scientic articles refer to the term; this is more than each of the terms compiler, expert, neural network, and operating system, some of which represent areas that enjoy generous funding. Technologically, the importance of ecient algorithms is beyond dispute. This ranges from engineering tasks like ecient computing of the discrete Fourier transform, via optimizati