On the Enumeration of all Unique Paths of Recombining Trinomial Trees
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908574379147264 |
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| author | Torres, Ethan Sreenivas, Ramavarapu Sowers, Richard |
| author_facet | Torres, Ethan Sreenivas, Ramavarapu Sowers, Richard |
| contents | 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 $\mathcal{O}\bigl(D^{1/2}1.612^{D}\bigr)$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_02727 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Enumeration of all Unique Paths of Recombining Trinomial Trees Torres, Ethan Sreenivas, Ramavarapu Sowers, Richard Data Structures and Algorithms Discrete Mathematics Combinatorics 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 $\mathcal{O}\bigl(D^{1/2}1.612^{D}\bigr)$. 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. |
| title | On the Enumeration of all Unique Paths of Recombining Trinomial Trees |
| topic | Data Structures and Algorithms Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2510.02727 |