![]() ![]() Yn of the free monoid M∗ are congruent modulo the Thue congruence generated by T over M∗ if and only if X1∘⋯∘Xm=Y1∘⋯∘Yn.Therefore, we give a finite Thue system over M, where M is now considered as a seven-letter alphabet, such that two words X1 ![]() generating system of the equations of the form X1∘⋯∘Xm=Y1∘⋯Yn, where Xi, YjϵM for 1⩽i⩽m, 1⩽j⩽n and ∘ is the composition of tree transformation classes. In this paper, we present an algebraic scheme to generate ordered trees with $n$ vertices with utmost efficiency whereby our approach uses $\mathcal of which the elements are the class of deterministic bottom-up tree transformations (DB), the linear, the nondeleting and the linear, nondeleting subclasses of DB (LDB, NDB and LNDB) the class of homomorphism tree transformations (H), the linear and the nondeleting subclasses of H (LH and NH).The main aim of this paper is to obtain a finite. In combinatorial optimization, generating ordered trees is relevant to evaluate candidate combinatorial objects. An ordered tree is a rooted tree where the order of the subtrees (children) of a node is significant. Trees are useful entities allowing to model data structures and hierarchical relationships in networked decision systems ubiquitously. The run-times for generating adjacency lists and matrices are somewhat longer than those for weight sequences, but are still over three times as fast as the corresponding implementations of the WROM algorithm. The implementation of our algorithm is over four times as fast as the implementation of the WROM algorithm. We compared the run-times of our Python implementation for generating free trees with the Python implementation of the well-known WROM algorithm taken from NetworkX. We further show how the algorithm can be modifed to generate adjacency list and adjacency matrix representations for free trees. Python implementations of the algorithms incorporate further improvements by using generators to avoid having to store the long lists of trees returned by the recursive calls, as well as caching the lists for rooted trees of small order, thereby eliminating many of the recursive calls. We construct algorithms for generating the weight sequence representations for all rooted and free trees of order n, and then add a number of modifications to improve the efficiency of the algorithms. We then use this to construct new representations for both rooted trees and free trees, namely the canonical weight sequence representation. In this paper, we introduce a new representation for ordered trees, the weight sequence representation. ![]()
0 Comments
Leave a Reply. |