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Shift-Safe EStream-OPF: Reusable Top-K Forgetting Queries over Order-Preserving Index Streams

This paper introduces Shift-Safe EStream-OPF, a reusable query framework for order-preserving index streams that employs a numerically stable shift-safe factorization to efficiently compute repeated global top- $K$ rankings under exponential forgetting, demonstrating significant latency improvements over baseline methods when leveraging index reuse and occurrence-equivalent compression.

Original authors: Khac-Chien Nguyen, Nhut-Minh Bui, Cong-Phat Tran

Published 2026-11-10

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CC BY 4.0

Original authors: Khac-Chien Nguyen, Nhut-Minh Bui, Cong-Phat Tran

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Time series data is the record of how things change over time: the daily temperature, the hourly price of a stock, or the yearly flow of a river. For decades, computer scientists have developed ways to find patterns in these streams of numbers. One powerful approach ignores the exact values and focuses instead on the shape of the movement. If a line goes up, then down, then up again, that shape is the same whether the numbers are small or large. This "order-preserving" view allows computers to match trends that look alike, even if they happen at different scales. However, a new challenge has emerged in how we value these patterns. In the real world, what happened yesterday matters more than what happened ten years ago. To reflect this, researchers use "forgetting" weights, which make recent occurrences count heavily while older ones fade into the background. The problem is that when you try to find the very best patterns in a massive dataset while applying this fading effect, the computer's math can break down. The numbers representing old events become so tiny they vanish, while the numbers representing the fading effect become so huge they explode, leaving the computer with a result that is neither a number nor a valid answer, but a mathematical error.

A team of researchers at the People's Police University in Vietnam has solved this specific mathematical trap and built a system that can repeatedly ask for the best patterns without rebuilding the entire map every time. Their work, titled Shift-Safe EStream-OPF, addresses a scenario where a user wants to explore a fixed archive of data over and over again, perhaps looking for the top fifty most significant trends under different rules for how quickly the past should be forgotten. Instead of scanning the raw data from scratch for every single question, which is slow and wasteful, their method builds a specialized index once. This index groups together patterns that share the exact same history of when they appeared. The researchers then discovered a clever way to calculate the importance of these groups that avoids the mathematical explosion. By anchoring their calculation to the most recent appearance of a pattern, they ensure that every step of the math stays within safe, manageable limits. This allows the system to handle the "fading" effect correctly, keeping the old data from vanishing into nothingness and the new data from blowing up the system.

The researchers tested this new approach against several existing methods using ten different real-world datasets, ranging from daily sunspot counts to household energy usage records containing up to 500,000 data points. They found that their method was consistently faster than the standard way of handling these queries, often cutting the time required by a significant margin. For instance, on a dataset tracking weekly carbon dioxide levels, their system was nearly twice as fast as the next best competitor. On a massive dataset of household energy usage, it performed just as well as the strongest existing method, proving it could handle very large scales without losing speed. However, the researchers were careful to note that their solution is not a magic bullet that works perfectly in every single situation. The speed advantage depends heavily on how the data is structured and how many results the user is asking for. In some cases, where the data patterns are very shallow or the request is for a huge number of results, the new method offered no advantage over the old ways.

The core of their discovery is a two-part strategy. First, they realized that patterns sharing the same history of appearances can be treated as a single unit, allowing the computer to do the heavy math just once for the whole group rather than for every single instance. Second, they applied a "shift-safe" calculation that rearranges the math to keep all the numbers positive and small, preventing the computer from crashing when dealing with very old data. This combination allows the system to reuse the same index for thousands of different questions, making it highly efficient for interactive analysis where a user might want to tweak the rules and see new results instantly. The researchers demonstrated that when a user asks many questions in a row, the time saved per question grows, eventually making the new system nearly forty percent faster than the competition for a thousand queries.

This work does not claim to have invented a new way to build the initial map of the data, nor does it promise to solve every problem in data mining. Instead, it provides a robust and reliable engine for ranking patterns when time matters. The team showed that by carefully managing how the computer handles the fading of old information, it is possible to keep the system running smoothly and quickly, even when the data stretches back far into the past. Their findings offer a practical tool for analysts who need to explore large archives of time-based data repeatedly, ensuring that the most relevant, recent patterns rise to the top without the system getting bogged down by mathematical errors or unnecessary repetition.

Technical Summary: Shift-Safe EStream-OPF

Problem Statement
The paper addresses the problem of performing repeated, global Top-K ranking over a fixed, materialized Order-Preserving (OP) index under endpoint exponential forgetting. While existing OP mining methods (e.g., OPF-Miner) handle forgetting, they are typically threshold-driven and single-shot, requiring candidate generation from raw time series for every query. Conversely, existing Top-K methods do not leverage the structural properties of OP indexes where multiple pattern loci share identical occurrence-start sets.

The core challenge is to enable a "build-once/query-many" model where a pre-computed index supports repeated queries with varying forgetting rates ( $k$ ), result counts ( $K$ ), and length ranges. A specific technical hurdle identified is numerical robustness: a direct product-form factorization of the forgetting score ( $M_k(A) \cdot e^{k(d-1)}$ ) can cause floating-point failure modes where an old-occurrence mass underflows to zero while a positive depth exponential overflows to infinity, resulting in NaN ( $0 \cdot \infty$ ), even though the true score is valid.

Methodology
The proposed solution, Shift-Safe EStream-OPF, introduces a query layer that operates on maximal occurrence-equivalent streams.

Structural Abstraction: The method groups OP loci into streams where every locus in a stream shares the exact same set of occurrence starts ( $A$ ). This allows the computation of a shared "shift-safe mass" for the entire stream rather than per locus.
Shift-Safe Factorization: To resolve the numerical instability, the authors derive a factorization anchored at the rightmost occurrence start ( $a = \max A$ $a = max A$ ) of the stream.
- The forgetting support is rewritten as:
  $fsup_k(A, d) = e^{-k(n - (a + d - 1))} \cdot R_k(A)$
  where $R_k(A) = \sum_{s \in A} e^{-k(a-s)}$ .
- This ensures that the exponential argument for the mass term is always non-positive (since $a \ge s$ ), and the depth-dependent decay term $e^{-k(n - (a + d - 1))}$ is also non-positive. The term for $s=a$ is exactly 1. This prevents the $0 \cdot \infty$ failure mode without requiring arbitrary-precision arithmetic.
Query Processing Pipeline:
- Stream Heads: For each eligible stream, the algorithm identifies the "head" (the best locus under the total comparator) using the shift-safe score.
- K-Head Pruning: Based on the strict total order, only the top $\min(K, E)$ stream heads are retained (where $E$ is the number of eligible streams). Loci in streams whose heads are not in the top $K$ are discarded.
- Multiway Merge: A priority queue performs a comparator-consistent merge of the retained streams to produce the final global Top-K list.

Key Contributions

Formal Query Model: Definition of a reusable indexed Top-K forgetting query over weak-order pattern loci, including canonical weak-order identity, maximal occurrence-equivalent streams, and a deterministic total-order comparator.
Shift-Safe Factorization: A mathematically proven transformation that anchors the exponential decay at the rightmost occurrence start, guaranteeing numerical safety (no NaN or Infinity) in standard IEEE 754 binary64 arithmetic while preserving exact-real semantics.
Comparator-Consistent Stream Ordering: An algorithmic approach to handle rounded-score plateaus deterministically within the Java binary64 contract, ensuring that the machine ordering is consistent and reproducible.
Comprehensive Implementation and Evaluation: A self-contained Java 21 artifact implementing the proposed method (B6) and five baselines/ablations (B1–B5). This includes:
- Ten real-world benchmark instances (up to $n=500,000$ ).
- Controlled synthetic generators for tie-rich scenarios.
- An independent Python Decimal oracle for semantic validation.
- A "build-once/query-many" workload analysis.

Experimental Results
The evaluation compares the proposed method (B6) against baselines including a full sort (B1), a capacity-K heap (B2), and ablations isolating factorization (B4) and pruning (B5).

Performance: On 8 out of 10 real-world datasets, B6 achieved a lower median latency than the competitive capacity-K heap baseline (B2). Notable speedups include:
- Weekly CO2: 1.72 $\times$
- SILSO daily sunspots: 1.69 $\times$
- UCI Household 100k: 1.64 $\times$
- NASDAQ Composite: 1.43 $\times$
Ablation Insights:
- Factorization (B4) was the fastest method on four datasets and remained within 6% of B6 on the four where B6 was fastest, indicating that sharing the shifted mass is the primary source of portable speedup.
- K-Head Pruning (B6) provides additional gains when compression is high and the ratio of requested results to eligible streams ( $K/E$ ) is small.
- Negative Boundaries: On the UCI Household 500k dataset (shallow compression, large $E$ ), B6 and B2 were at practical parity (1.00 $\times$ ), demonstrating that the method's benefits are conditional.
Reuse: In a repeated-query workload (1,000 queries on one index), the amortized benefit of B6 over B2 increased from 1.01 $\times$ (single query) to 1.39 $\times$ , validating the "build-once/query-many" model.

Significance and Claims
The paper modestly claims that its contribution is conditional and specific to the intersection of repeated global Top-K ranking, endpoint exponential forgetting, and weak-order index structures.

No New Index Construction: The authors explicitly state they do not propose a new production OPST construction algorithm; they assume a materialized index exists.
No New Generic Top-K Theorem: The use of heaps and multiway merges is standard; the novelty lies in the OPF-specific transformation (occurrence-equivalent factorization) that makes these standard tools effective in this specific context.
Numerical Robustness: The shift-safe factorization is presented as a necessary engineering fix to prevent machine-arithmetic failures in interactive queries, rather than a change in the underlying mathematical definition of forgetting.
Scope: The benefits depend on occurrence-equivalent compression, the $K/E$ ratio, and index reuse. The method is not universally superior to heap-based baselines, particularly in scenarios with shallow compression or single-shot queries where index construction dominates latency.

Technical Summary: Shift-Safe EStream-OPF

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