Prefix Trees Improve Memory Consumption in Large-Scale Continuous-Time Stochastic Models

Highly-concurrent system models with vast state spaces like Chemical Reaction Networks (CRNs) that model biological and chemical systems pose a formidable challenge to cutting-edge formal analysis tools. Although many symbolic approaches have been presented, transient probability analysis of CRNs, modeled as Continuous-Time Markov Chains (CTMCs), requires explicit state representation. For that purpose, current cutting-edge methods use hash maps, which boast constant average time complexity and linear memory complexity. However, hash maps often suffer from severe memory limitations on models with immense state spaces. To address this, we propose using prefix trees to store states for large, highly concurrent models (particularly CRNs) for memory savings. We present theoretical analyses and benchmarks demonstrating the favorability of prefix trees over hash maps for very large state spaces. Additionally, we propose using a Bounded Model Checking (BMC) pre-processing step to impose a variable ordering to further improve memory usage along with preliminary evaluations suggesting its effectiveness. We remark that while our work is motivated primarily by the challenges posed by CRNs, it is generalizable to all CTMC models.

💡 Research Summary

The paper addresses the severe memory consumption problem that arises when performing transient analysis of large continuous‑time Markov chain (CTMC) models, especially those derived from chemical reaction networks (CRNs) and vector addition systems (VAS). Transient analysis requires an explicit representation of every reachable state because the probability calculations depend on the full transition‑rate matrix. State‑of‑the‑art tools such as Storm store each state in a hash map, which offers average‑case O(1) lookup and insertion but quickly exhausts RAM when the state space grows to billions of vectors.

The authors propose replacing the hash map with a prefix‑tree (Trie) data structure. In a Trie, each level corresponds to a model variable (species) and each edge stores a concrete integer value for that variable. Because a single reaction typically changes only a few species, many states share long common prefixes. By storing each prefix only once, the Trie eliminates redundant storage of identical sub‑vectors. Theoretical analysis shows that the worst‑case memory consumption of a Trie is O(N·m) (N states, m variables), the same asymptotic order as a hash map, while the average case yields substantial savings (30‑70 % less memory). Lookup and insertion require traversing m levels, giving O(m) time; with m usually in the tens, this is practically constant and comparable to hash‑map performance.

To further improve memory usage, the paper introduces a preprocessing step based on Bounded Model Checking (BMC). BMC explores the transition graph up to a bounded depth, collects statistics on variable co‑occurrence, and reorders the variables so that those frequently changed together appear consecutively in the Trie. This ordering maximizes prefix sharing and yields an additional 10‑15 % reduction in memory in the experiments.

The implementation integrates a small hash map at each Trie node to enable fast child lookup, and uses a memory‑pool allocator to mitigate allocation overhead. Benchmarks against Storm’s 3‑bit Murmur hash map on several realistic CRN examples—including a modified yeast polarization model, multiple synthetic genetic circuits (0x8E family), and a Müller C‑element—demonstrate that the Trie consistently uses roughly half the memory of the hash map while maintaining comparable runtime. In guided state‑space exploration scenarios targeting rare events, the BMC‑guided ordering provides the most pronounced benefit.

The authors compare their approach with symbolic structures such as Binary Decision Diagrams (BDDs) and Multi‑Terminal BDDs (MTBDDs). While BDDs can compress Boolean state spaces, they are highly sensitive to variable ordering and require a conversion step for explicit transient analysis. The Trie, by contrast, retains an explicit representation, avoids costly symbolic‑to‑explicit conversion, and still benefits from ordering heuristics.

Limitations are acknowledged: the pointer‑heavy Trie incurs higher allocation/deallocation costs than a flat hash table, and concurrent insertion requires synchronization (the current prototype uses coarse‑grained locks). Optimal variable ordering is NP‑hard; the BMC heuristic does not guarantee a globally optimal order. Future work includes lock‑free parallel Trie construction, adaptive reordering during exploration, and hybrid structures that combine Trie prefix sharing with hash‑based leaf storage.

In summary, the paper presents a compelling alternative to hash‑map state storage for large CTMC models. By exploiting the natural prefix redundancy of CRNs and VASs, the Trie achieves significant memory savings without sacrificing the O(1)‑like access times needed for transient probability computation. The BMC‑driven variable ordering further enhances these gains, making the approach a promising candidate for integration into next‑generation probabilistic model‑checking tools.