Word Encoding Tree Connectivity Works

|Experimentally, we demonstrate the practical relevance of the theoretical trick of encoding trees in words. 1 Introduction In 1983, Gabow and Tarjan [3] showed that union- nd can be supported in linear time, if the union operations, but not the order they come in, is known in advance. A dierent way of casting this problem is that of answering on-line connectivity queries for a subforest F of a given tree T while edges from T are added to F in some arbitrary order, ending with F = T . Gabow and Tarjan [3] introduced the idea of encoding small \atomic" subtrees of T size (log n) in a single word. This is based on the standard assumption that the word length ! is at least log n; for otherwise, we cannot even address the nodes of the tree. Having encoded an atomic tree t, in constant time, we can add an edge from t to F , and answer connectivity queries for F \ t. The techniques from [3] were fairly involved, and unlikely to be of practical relevance, particularly because the alterna..