Hiroshi Narushima

Principle of inclusion-exclusion on semilattices☆

Abstract We introduce an enumeration theorem under lattice action. Let L be a finite semilattice and Ω be a nonempty set. Let f: L → P(Ω) be a map satisfying f(x ∇ y) ⊇ f(x) ∩ f(y), where ∇ and P (Ω) mean “join” and the power set of Ω, respectively. Then m ∪ xϵL ƒ(x) = Σ cϵC (−1) l(c) m ∩ xϵc ƒ(x) , where C is the set of all chains in L and l(c) denotes the length of a chain c. Also the theorem can be dualized. Furthermore, we describe two applications of the theorem to a Boolean lattice of sets and a partition lattice of a set.

Generating most parsimonious reconstructions on a tree: a generalization of the Farris-Swofford-Maddison method

Abstract A combinatorial optimization problem regarding the assignments (called reconstructions) for a tree has been discussed in phylogenetic analysis. Farris, Swofford and Maddison have solved the problem of finding the most parsimonious reconstructions on a completely bifurcating phylogenetic tree. We formulate mathematically the problem with its generalization to the case of any tree and call it the MPR problem. We present a solution for the generalized problem by introducing the concept of median interval obtained from sorting the endpoints of some closed intervals. The state set operation which plays an important role in the Farris-Swofford-Mad-dison method, is clarified by the concept of median interval. And then, with an explicit recursive formulation we generalize smoothly their method. Also, the computational complexity of our method is discussed. In the discussion, the PICK algorithm by Blum-Floyd-Pratt-Rivest-Tarjan is essential.

A more efficient algorithm for MPR problems in phylogeny

Abstract A discrete optimization problem of assigning linearly ordered character-states to the hypothetical ancestors of an evolutionary tree under the principle of maximum parsimony has been discussed. Under the transformation relation of linearly ordered character-states, Farris (1970) and Swofford and Maddison (1987) have dealt with the problem on completely bifurcating phylogenetic trees and presented a solution. Hanazawa et al. (1995) have mathematically formulated the problem with its generalization to any tree and called it the MPR (most-parsimonious reconstruction) problem. Then they have presented clear algorithms for the MPR problem and the related problems. We present a more efficient algorithm for one of the problems, the problem of obtaining the MPR sets. The complexity of the previous algorithm for this problem is O(n2) for the number n of nodes in a given tree, but that of the new algorithm is O(n).

Lattice-theoretic properties of MPR-posets in phylogeny

With biological sciences such as taxonomy, cladistics and phylogeny as a background, the principle of maximum parsimony also called Wagner Parsimony has been mathematically formulated and then a mathematical and algorithmic theory has been developing. Recently, a clear method for the character-state minimization problem called the First Most-Parsimonious Reconstruction (MPR) Problem under linearly ordered character-states has been presented by Hanazawa et al. (Appl. Math. 56 (1995) 245-265), Narushima and Hanazawa (Discrete Appl. Math. 80 (1997) 231-238). From a phylogenetic point of view, Minaka (Forma 8 (1993) 277-296) has introduced two partial orderings on the set of MPRs to investigate the relationships among the MPRs. One is the usual ordering, and the other is a partial ordering that depends on a state of a specified root of a given el-tree, which is called a σ(r)-version ordering. In this paper, the following three theorems on MPR-posets induced by these orderings are shown: (1) a usual MPR-poset is a complete distributive lattice, (2) a σ(r)-version MPR-poset is a lower-complete semi-lattice, (3) any interval poser of a σ(r)-version MPR-poset is a complete distributive lattice. Some possible applications and meanings of the theorems are also mentioned.

On characteristics of ancestral character-state reconstructions under the accelerated transformation optimization

A combinatorial optimization problem regarding assignments of real numbers (called reconstructions) on a tree has been discussed in phylogenetic analysis. Recently, a clear method for finding most-parsimonious reconstructions (MPRs) on a given end-labeled-tree (phylogenetic tree) has been presented by Hanazawa et at. (Discrete Appl. Math. 56 (1995) 245-265, Narushima and Hanazawa, Discrete Appl. Math. 80 (1997) 231-238). In the framework based on the method, we refine and generalize the accelerated transformation (ACCTRAN) reconstruction which originated with Farris (Syst. Zool. 19 (1970) 92) and was defined more explicitly by Swofford and Maddison (Math. Biosci. 87 (1987) 229). This is considered one of the more meaningful and useful of the possible MPRs. We also generalize the MPR-poset of MPRs, which is introduced by Minaka (Forma 8 (1993) 296). Then two theorems on characteristics of ACCTRANs are given. One shows that the ACCTRAN on a rooted e.l.tree T is the unique MPR on T for which the lengths of all subtrees are minimized, that is, the completeness in most-parsimonious properties of ACCTRANs. Another states some conditions for the ACCTRAN to be the greatest element in the MPR-poset.

A variant of inclusion-exclusion on semilattices☆

Abstract Let ( S ,∪) be a finite join-semilattice and ( D , ∨, ∧) be a distributive lattice. Let ⨍:S→D be a map satisfying ⨍(x ∪ y) ⩾ ⨍(x) ∧ ⨍(y) for each x and y in S . Then for any valuation v on D the following identity holds. v ⋁ xυS f(x) = ∑ cυC (−1) l(c) v ⋀ xυc f(x) where C is the set of all chains in S and l ( c ) denotes the length of a chain c. Also the theorem can be dualized.

A graph-theoretical characterization of the order complexes on the 2-sphere

Publisher Summary This chapter presents a graph-theoretical characterization of the order complexes on the 2-sphere. Recently, some remarkable connections between commutative algebra and combinatorics have been discovered. Among the main topics in this area are the concepts of Cohen-Macaulay and Gorenstein complexes. The geometric realization of the order complex A(P) is a triangulation of the 2-sphere. The graph G (P) is a 2-connected plane graph and is regarded as a complex, where vertices, edges and regions of G (P) are faces of the complex. The faces of G(P) ordered by inclusion, form a poset. This poset is called the face poset of G(P).