Lowest common ancestor
The lowest common ancestor (LCA) is a concept in graph theory and computer science. Let T be a rooted tree with n nodes. The lowest common ancestor between two nodes v and w is defined as the lowest node in T that has both v and w as descendants (where we allow a node to be a descendant of itself).
The LCA of v and w in T is the shared ancestor of v and w that is located farthest from the root. Computation of lowest common ancestors may be useful, for instance, as part of a procedure for determining the distance between pairs of nodes in a tree: the distance from v to w can be computed as the distance from the root to v, plus the distance from the root to w, minus twice the distance from the root to their lowest common ancestor.
In a tree data structure where each node points to its parent, the lowest common ancestor can be easily determined by finding the first intersection of the paths from v and w to the root. In general, the computational time required for this algorithm is O(h) where h is the height of the tree (length of longest path from a leaf to the root). However, there exist several algorithms for processing trees so that lowest common ancestors may be found more quickly.
It is possible to search in constant time per query after a linear time preprocessing stage; Tarjan's off-line lowest common ancestors algorithm is perhaps the most famous example.
Without preprocessing you can improve the naïve algorithm's computation time to O(log h) by storing the paths through the tree using skew-binary random access lists, while still permitting the tree to be extended in constant time (Edward Kmett (2012)).
The lowest common ancestor problem was defined by Alfred Aho, John Hopcroft, and Jeffrey Ullman (1973), but Dov Harel and Robert Tarjan (1984) were the first to develop an optimally efficient lowest common ancestor data structure. Their algorithm processes any tree in linear time, using a heavy path decomposition, so that subsequent lowest common ancestor queries may be answered in constant time per query. However, their data structure is complex and difficult to implement. Tarjan also found a simpler but less efficient algorithm, based on the union-find data structure, for computing lowest common ancestors of an offline batch of pairs of nodes.
Baruch Schieber and Uzi Vishkin (1988) simplified the data structure of Harel and Tarjan, leading to an implementable structure with the same asymptotic preprocessing and query time bounds. Their simplification is based on the principle that, in two special kinds of trees, lowest common ancestors are easy to determine: if the tree is a path, then the lowest common ancestor can be computed simply from the minimum of the levels of the two queried nodes, while if the tree is a complete binary tree, the nodes may be indexed in such a way that lowest common ancestors reduce to simple binary operations on the indices. The structure of Schieber and Vishkin decomposes any tree into a collection of paths, such that the connections between the paths have the structure of a binary tree, and combines both of these two simpler indexing techniques.
Omer Berkman and Uzi Vishkin (1993) discovered a completely new way to answer lowest common ancestor queries, again achieving linear preprocessing time with constant query time. Their method involves forming an Euler tour of a graph formed from the input tree by doubling every edge, and using this tour to write a sequence of level numbers of the nodes in the order the tour visits them; a lowest common ancestor query can then be transformed into a query that seeks the minimum value occurring within some subinterval of this sequence of numbers. They then handle this range minimum query problem by combining two techniques, one technique based on precomputing the answers to large intervals that have sizes that are powers of two, and the other based on table lookup for small-interval queries. This method was later presented in a simplified form by Michael Bender and Martin Farach-Colton (2000). As had been previously observed by Gabow, Bentley & Tarjan (1984), the range minimum problem can in turn be transformed back into a lowest common ancestor problem using the technique of Cartesian trees.
A variant of the problem is the dynamic LCA problem in which the data structure should be prepared to handle LCA queries intermixed with operations that change the tree (that is, rearrange the tree by adding and removing edges) This variant can be solved using O(logN) time for all modifications and queries. This is done by maintaining the forest using the dynamic trees data structure with partitioning by size; this then maintains a heavy-;light decomposition of each tree, and allows LCA queries to be carried out in logarithmic time.
- Aho, Alfred; Hopcroft, John; Ullman, Jeffrey (1973), "On finding lowest common ancestors in trees", Proc. 5th ACM Symp. Theory of Computing (STOC), pp. 253–265, doi:10.1145/800125.804056.
- Alstrup, Stephen; Gavoille, Cyril; Kaplan, Haim; Rauhe, Theis (2004), "Nearest Common Ancestors: A Survey and a New Algorithm for a Distributed Environment", Theory of Computing Systems 37 (3): 441–456, doi:10.1007/s00224-004-1155-5. A preliminary version appeared in SPAA 2002.
- Bender, Michael A.; Farach-Colton, Martin (2000), "The LCA problem revisited", Proceedings of the 4th Latin American Symposium on Theoretical Informatics, Lecture Notes in Computer Science 1776, Springer-Verlag, pp. 88–94, doi:10.1007/10719839_9.
- Berkman, Omer; Vishkin, Uzi (1993), "Recursive Star-Tree Parallel Data Structure", SIAM Journal on Computing 22 (2): 221–242, doi:10.1137/0222017.
- Fischer, Johannes; Heun, Volker (2006), "Theoretical and Practical Improvements on the RMQ-Problem, with Applications to LCA and LCE", Proceedings of the 17th Annual Symposium on Combinatorial Pattern Matching, Lecture Notes in Computer Science 4009, Springer-Verlag, pp. 36–48, doi:10.1007/11780441_5.
- Gabow, Harold N.; Bentley, Jon Louis; Tarjan, Robert E. (1984), "Scaling and related techniques for geometry problems", STOC '84: Proc. 16th ACM Symposium on Theory of Computing, New York, NY, USA: ACM, pp. 135–143, doi:10.1145/800057.808675.
- Harel, Dov; Tarjan, Robert E. (1984), "Fast algorithms for finding nearest common ancestors", SIAM Journal on Computing 13 (2): 338–355, doi:10.1137/0213024.
- Schieber, Baruch; Vishkin, Uzi (1988), "On finding lowest common ancestors: simplification and parallelization", SIAM Journal on Computing 17 (6): 1253–1262, doi:10.1137/0217079.
- Lowest Common Ancestor of a Binary Search Tree, by Kamal Rawat
- Python implementation of the algorithm of Bender and Farach-Colton, by David Eppstein
- Lecture notes on LCAs from a 2003 MIT Data Structures course. Course by Erik Demaine, notes written by Loizos Michael and Christos Kapoutsis. Notes from 2007 offering of same course, written by Alison Cichowlas.
- Lowest Common Ancestor in Binary Trees in C. A simplified version of the Schieber–Vishkin technique that works only for balanced binary trees.
- Video of Donald Knuth explaining the Schieber–Vishkin technique
- Range Minimum Query and Lowest Common Ancestor article in Topcoder
- Documentation for the lca package for Haskell by Edward Kmett, which includes the skew-binary random access list algorithm. Purely functional data structures for on-line LCA slides for the same package.