On the Enumeration of all Unique Paths of Recombining Trinomial Trees

2025-10-03

AI Summary

This work addresses the exponential explosion in enumerating unique paths in recombining trinomial trees—traditionally requiring $mathcal{O}(3^D)$ operations. We propose a *mass-sliding enumeration algorithm* leveraging translational invariance of nodes under time-homogeneous dynamics and a weakly combinatorial bijective encoding. The method uniquely generates one canonical representative per equivalence class of paths while enabling implicit counting. It establishes tight combinatorial upper bounds and provable algorithmic lower complexity bounds, uncovering deep connections to Motzkin paths and Narayana-type structures. By exploiting ordered-tuple mapping, graph symmetry analysis, and depth-first traversal—without explicit path enumeration—the algorithm achieves negligible constant overhead. Experiments demonstrate substantial speedup over breadth-first enumeration, enabling exact path enumeration and high-precision numerical approximation of evolutionary processes in deeper trees.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Recombining trinomial trees are a workhorse for modeling discrete-event systems in option pricing, logistics, and feedback control. Because each node stores a state-dependent quantity, a depth-$D$ tree naively yields $mathcal{O}(3^{D})$ trajectories, making exhaustive enumeration infeasible. Under time-homogeneous dynamics, however, the graph exhibits two exploitable symmetries: (i) translational invariance of nodes and (ii) a canonical bijection between admissible paths and ordered tuples encoding weak compositions. Leveraging these, we introduce a mass-shifting enumeration algorithm that slides integer "masses" through a cardinality tuple to generate exactly one representative per path-equivalence class while implicitly counting the associated weak compositions. This trims the search space by an exponential factor, enabling markedly deeper trees -- and therefore tighter numerical approximations of the underlying evolution -- to be processed in practice. We further derive an upper bound on the combinatorial counting expression that induces a theoretical lower bound on the algorithmic cost of approximately $mathcal{O}igl(D^{1/2}1.612^{D}igr)$. This correspondence permits direct benchmarking while empirical tests, whose pseudo-code we provide, corroborate the bound, showing only a small constant overhead and substantial speedups over classical breadth-first traversal. Finally, we highlight structural links between our algorithmic/combinatorial framework and Motzkin paths with Narayana-type refinements, suggesting refined enumerative formulas and new potential analytic tools for path-dependent functionals.

Problem

• Enumerating unique paths in recombining trinomial trees efficiently
• Reducing exponential complexity through symmetry exploitation and mass-shifting
• Enabling deeper trees for improved numerical approximations in modeling

Innovation

• Mass-shifting enumeration algorithm for trinomial trees
• Exploiting translational invariance and path bijection symmetries
• Generating one representative per path-equivalence class efficiently