On computing the nearest neighbor interchange distance

Abstract

In the practice of molecular evolution, different phylogenetic trees for the same group of species are often produced either by procedures that use diverse optimality criteria [24] or from different genes [15,16,17,18,14]. Comparing these trees to find their similarities (e.g. agreement or consensus) and dissimilarities, i.e. distance, is thus an important issue in computational molecular biology. The nearest neighbor interchange (nni) distance [29,28,34, 3, 6,2,19,20,23,33,22,21,26] is a natural distance metric that has been extensively studied. Despite its many appealing aspects such as simplicity and sensitivity to tree topologies, computing this distance has remained very challenging, and many algorithmic and complexity issues about computing this distance have remained unresolved. This paper studies the complexity and efficient approximation algorithms for computing the nni distance and a natural extension of this distance on weighted phylogenies. The following results answer many open questions about the nni distance posed in the literature.1. Computing the nni distance between two labeled trees is NP-complete. This solves a 25 year old open question appearing again and again in, for example, [29, 34, 3, 6, 2, 19, 20, 23, 22, 21, 26].2. Computing the nni distance between two unlabeled trees is also NPcomplete. This answers an open question in [3] for which an erroneous proof appeared in [23].3. Biological applications motivate us to extend the nni distance to weighted phylogenies, where edge weights indicate the time-span of evolution along each edge. We present an O(n 2 ) time approximation algorithm for computing the nni distance on weighted phylogenies with a performance ratio of 4 log n + 4, where n is the number of leaves in the phylogenies.We also observe that the nni distance is in fact identical to the linear-cost subtree-transfer distance on unweighted phylogenies discussed in [4, 5]. Some consequences of this observation are also discussed.1991 Mathematics Subject Classification. Primary 68Q17, 68W40; Secondary 68Q25. The results reported here also form a subset of the results that appeared in

Walks on SPR Neighborhoods

“…The nearest-neighbor-interchange (NNI) distance was introduced independently by DasGupta et al [8] and Li et al [14]. Roughly, an NNI operation swaps two subtrees that are separated by an internal edge.…”

Boolean Property Encoding for Local Set Pattern Discovery: An Application to Gene Expression Data Analysis

“…The nni (nearest neighbor interchange) and the mast (maximum agreement subtree) are two of the most used metrics. nni has been introduced independently in [18] and [19] and its NP-completeness has been recently proved [20,21]. mast has been proposed in [22], and [23] describes an efficient algorithm for computing this metrics on binary trees.…”

Evolutionary Similarity Among Genes

“…Subtree transfer is most appropriate when considering such events as recombination or gene conversion and the triples distance is not uniquely de ned for unrooted trees. The branch score metric can be calculated straightforwardly and is less sensitive to small differences in topology than NNI (DasGupta et al 1997(DasGupta et al , 1999. We use the branch score metric because we expect small differences to occur and do not wish to overweight them.…”