Improved Algorithms for Time Decay Streams

Enayat Ullah; Harry Lang; Samson Zhou; Vladimir Braverman

arxiv: 1907.07574 · v1 · pith:32MWW2HYnew · submitted 2019-07-17 · 💻 cs.DS

Improved Algorithms for Time Decay Streams

Vladimir Braverman , Harry Lang , Enayat Ullah , Samson Zhou This is my paper

Pith reviewed 2026-05-24 20:03 UTC · model grok-4.3

classification 💻 cs.DS

keywords time decay streamscoresetsk-median clusteringstreaming algorithmsapproximation algorithmspolynomial decayexponential decayonline facility location

0 comments

The pith

A framework converts any offline coreset into a time-decay coreset for polynomial decay functions in data streams.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a general method to adapt existing offline coresets for data streams where element importance decays over time according to polynomial functions. This approach matters because streams often involve continuously arriving data with older points losing relevance, requiring compact summaries that still support approximate query answers such as clustering. For exponential decay specifically in k-median clustering, the work supplies a constant-factor approximation algorithm built on online facility location that maintains space O(k log(hΔ) + h). The same techniques carry over to k-means clustering and M-estimators.

Core claim

We provide a general framework that takes any offline-coreset and gives a time-decay coreset for polynomial time decay functions. We also consider the exponential time decay model for k-median clustering, where we provide a constant factor approximation algorithm that utilizes the online facility location algorithm. Our algorithm stores O(k log(hΔ)+h) points where h is the half-life of the decay function and Δ is the aspect ratio of the dataset. Our techniques extend to k-means clustering and M-estimators as well.

What carries the argument

The transformation framework that turns any offline coreset into its time-decay counterpart for polynomial decay, together with the reduction of exponential-decay k-median to online facility location.

If this is right

Any problem that already possesses an offline coreset immediately receives a polynomial time-decay version via the framework.
Exponential-decay k-median admits a constant-factor approximation while storing only O(k log(hΔ) + h) points.
The same constant-factor guarantee and space bound extend directly to k-means and to M-estimators under exponential decay.
Recent elements receive higher weight in the maintained summary without requiring a fresh coreset construction for each decay model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The framework could be applied to other geometric or statistical queries once their offline coresets are known.
Space usage linear in half-life suggests the method remains practical for streams whose decay horizon is moderate relative to total length.
The reduction to online facility location may admit further improvements if tighter online algorithms for facility location become available.

Load-bearing premise

Existing offline coresets can be transformed into time-decay versions while keeping their approximation guarantees intact for polynomial decay functions.

What would settle it

A concrete stream with polynomial decay where the produced coreset either exceeds the claimed space bound or fails to recover the target query within the expected approximation factor.

read the original abstract

In the time-decay model for data streams, elements of an underlying data set arrive sequentially with the recently arrived elements being more important. A common approach for handling large data sets is to maintain a \emph{coreset}, a succinct summary of the processed data that allows approximate recovery of a predetermined query. We provide a general framework that takes any offline-coreset and gives a time-decay coreset for polynomial time decay functions. We also consider the exponential time decay model for $k$-median clustering, where we provide a constant factor approximation algorithm that utilizes the online facility location algorithm. Our algorithm stores $\mathcal{O}(k\log(h\Delta)+h)$ points where $h$ is the half-life of the decay function and $\Delta$ is the aspect ratio of the dataset. Our techniques extend to $k$-means clustering and $M$-estimators as well.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a clean general reduction from offline coresets to polynomial time-decay versions plus a concrete O(k log(hΔ)+h)-space constant-factor algorithm for exponential-decay k-median via online facility location.

read the letter

The main things to know are the general conversion framework that turns any offline coreset into a time-decay coreset for polynomial decay functions, and the specific constant-factor k-median algorithm under exponential decay that stores O(k log(hΔ) + h) points by reducing to online facility location. Both claims are new relative to the cited prior work on offline coresets and online FL. The paper does a solid job of showing how to lift existing static constructions into the decay model without starting from scratch, and the space bound for the exponential case is explicit and practical for moderate half-lives. Extending the same approach to k-means and M-estimators is a reasonable bonus. The reduction for polynomial decay looks modular and inherits whatever quality the base offline coreset already has. The exponential algorithm inherits the approximation guarantee from the online FL routine it calls. A minor soft spot is the explicit dependence on aspect ratio Δ in the space bound; when Δ is large the storage grows, though this is standard in geometric streaming work and not hidden. Without the full proofs it is impossible to verify the exact error propagation or whether any post-processing steps affect the stated bounds, but the abstract states the claims cleanly and there is no visible circularity or unsupported size blow-up. This is for people already working on coresets or streaming clustering who need to handle recency weighting. A reader in that niche will get usable techniques. It deserves a serious referee because the results are specific, build directly on reproducible prior algorithms, and target a concrete model with stated space guarantees.

Referee Report

0 major / 2 minor

Summary. The paper presents a general framework that converts any offline coreset construction into a time-decay coreset for polynomial decay functions in the streaming model. For exponential decay, it gives a constant-factor approximation algorithm for k-median clustering that combines the online facility location algorithm with a time-decay mechanism, using O(k log(hΔ) + h) space (h = half-life, Δ = aspect ratio); the same techniques are claimed to extend to k-means and M-estimators.

Significance. If the reductions hold, the framework supplies a modular way to obtain time-decay coresets without redesigning each coreset from scratch, while the exponential-decay k-median result improves space over naive sliding-window or uniform-sampling approaches by exploiting the online facility location primitive. The explicit dependence on half-life and aspect ratio makes the bound concrete and falsifiable.

minor comments (2)

[Abstract] The abstract and introduction should explicitly state the polynomial degree or the precise form of the decay function for which the general framework applies, as this affects the space blow-up factor.
[Section 4 (exponential decay algorithm)] Clarify whether the O(k log(hΔ) + h) bound for the exponential case is in the worst case or expected, and whether it holds with high probability or in expectation over the online facility location randomness.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; framework composes external primitives

full rationale

The paper states a general framework that converts any existing offline coreset into a time-decay version for polynomial decay functions, and for exponential decay k-median it invokes the online facility location algorithm to obtain the O(k log(hΔ)+h) space constant-factor guarantee. These building blocks are treated as given external results rather than derived or fitted inside the paper. No equations equate a claimed prediction to a fitted parameter by construction, no self-citation chain is load-bearing for the central claims, and no uniqueness theorem or ansatz is smuggled from prior author work. The derivation is therefore self-contained against external benchmarks and receives score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard domain assumptions in streaming algorithms without introducing new free parameters or invented entities.

axioms (2)

domain assumption Existence of offline coresets for the relevant queries
The framework takes any offline-coreset as input, assuming such coresets exist and can be adapted.
domain assumption Standard properties of the online facility location algorithm
The exponential decay approximation utilizes the online facility location algorithm to achieve the constant factor guarantee.

pith-pipeline@v0.9.0 · 5676 in / 1320 out tokens · 25449 ms · 2026-05-24T20:03:59.296865+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

general framework that takes any offline-coreset and gives a time-decay coreset for polynomial time decay functions... O(k log(hΔ)+h) points
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

online facility location algorithm of Meyerson

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.