Scapegoat Trees

We present an algorithm for maintaining binary search trees. The amortized complexity per INSERT or DELETE is O(log n) while the worst-case cost of a SEARCH is O(log n). Scapegoat trees, unlike most balanced-tree schemes, do not require keeping extra data (e.g. “colors” or “weights”) in the tree nodes. Each node in the tree contains only a key value and pointers. to its two children. Associated with the root of the whole tree are the only two extra values needed by the scapegoat scheme: the number of nodes in the whole tree, and the maximum number of nodes in the tree since the tree was last completely rebuilt. In a scapegoat tree a typical rebalancing operation begins at a leaf, and successively examines higher ancestors until a node (the “scapegoat”) is found that is so unbalanced that the entire subtree rooted at the scapegoat can be rebuilt at zero cost, in an amortized sense. Hence the name.

• Type: Conference
• Year: 1993
• Where: SODA
• Authors: Igal Galperin, Ronald L. Rivest
• Keywords: Algorithms, Scapegoat Tree, SODA 1993, Tree, Tree Nodes