arXiv subjects

Runtian Ren

Publications and source records attributed to Runtian Ren.

9 recordsLinked to original sources

arXiv2026

Paging with Per-Replacement Maximum Delay

Classical paging couples every miss to an immediate replacement. We ask what remains of its algorithmic structure when a miss may wait. In our per-replacement maximum-delay model, loading a pending page costs one unit of movement plus the age of its oldest outstanding request and clears the whole page-specific episode. Equivalently, the instantaneous holding rate is the number of pending pages, rather than the number of pending requests. The classical competitive hierarchy survives this change. For cache size $k$, we give a deterministic $(5k+3)$-competitive threshold-LRU algorithm and a randomized $5H_k$-competitive algorithm against an oblivious adversary; classical lower-bound instances give matching $\Omega(k)$ and $\Omega(H_k)$ orders. The randomized algorithm uses cache-independent temporal windows to create an ordinary-paging sequence fixed before any random choices; a shadow paging algorithm is then projected onto nonproactive physical replacements. The offline picture is less classical. We give an exact $O(nk)$ dynamic program with one hole, an exact configuration dynamic program for a fixed number of holes, and a deterministic nonproactive polynomial-time $5$-approximation without fixing that number. Yet farthest-next-use victim selection can be suboptimal in the physical delayed problem already with three pages.

cs.DS↗

arXiv2026

Online Service with Per-Batch Maximum Delay

We study online service with one maximum-waiting-time charge per service batch. The persistent server endpoint prevents a phase-by-phase comparison with the offline optimum: an offline schedule may merge many online phases, share movement globally, and finish at unrelated endpoints. Our main contribution is a metric-independent \emph{group--trajectory certificate framework} that restores such a comparison. For ordered request groups in disjoint time windows, a certificate value is bounded both by the window length and by the metric Steiner cost of the group. After normalizing the offline schedule into consecutive arrival blocks, strictly interior groups are charged to offline delay, while boundary groups induce connectors of congestion at most two along the offline trajectory. One color class therefore has certificate sum at most $2\OPT$; a parity decomposition yields $\sum_h C_h\le4\OPT$. Consequently, any phase rule whose cost is at most $\alpha C_h$ is $4\alpha$-competitive. For visible service, this theorem yields deterministic ratios $10$ on a line, $12$ on a weighted tree, and $20$ on an arbitrary finite metric; the last algorithm is polynomial and uses a phase-local terminal-MST envelope, while an exact metric-Steiner oracle gives ratio $12$. Structurally, elective and automatic schedules can have different event structures but equal offline optimal values. The common value is computable exactly in polynomial time on lines and explicitly represented weighted trees, whereas exact optimization on arbitrary finite metrics is NP-hard. Finally, we use spatial blindness---announced requests whose locations are revealed only when visited---as a stress test: dyadic exploration preserves a constant ratio on a known finite line, while a single hidden request on a star forces a loss linear in its degree.

cs.DS↗

arXiv2026

Online Multi-Level Aggregation with Per-Batch Maximum Delay

We study online multi-level aggregation on finite rooted trees with a per-batch maximum-delay objective. A service pays for a rooted subtree and for the maximum waiting time among the requests cleared by that service. We show that the offline optimum admits a consecutive-arrival-block normal form and can be computed by a polynomial-time dynamic program. The same dynamic program defines the deadlines of a family of online algorithms, which we call DP-Envelope. Its deterministic endpoint is $2$-competitive. Sampling one global parameter with density $e^\theta/(e-1)$ leads to an $e/(e-1)$-competitive randomized algorithm against an oblivious adversary. The deterministic guarantee matches the known fixed-node lower bound, and we prove a matching randomized lower bound. Thus, both guarantees are optimal on every nondegenerate rooted tree. We first develop the line metric as a warm-up, where the algorithm and its nested block partitions have a direct geometric interpretation. Finally, we show that the upper bounds extend to every realizable static service system with a normalized, nondecreasing, submodular joint service cost.

cs.DS↗

arXiv2026

Learning-Augmented and Randomized Algorithms for Line Aggregation with Delays

This paper studies learning-augmented and randomized online aggregation with delays on a line metric. We consider advice given as online suggested service lengths, and evaluate the algorithms in terms of robustness and consistency. For each $\lambda \in (0,1]$, we first propose a deterministic learning-augmented \textsc{Balance} algorithm that is $(4/\lambda+1/\lambda^2)$-robust and $(4+\lambda)$-consistent. We also propose a randomized algorithm for the problem in the classical adversarial model, which is $(e+1)$-competitive against an oblivious adversary, improving over the deterministic $5$-competitive \textsc{Balance} benchmark~\cite{bienkowski2013chain}. Notably, this competitive ratio is even lower than the lower bound of $4$ for deterministic online algorithms. Moreover, we establish a lower bound of $e$ on the competitive ratio of randomized online algorithms, improving the previous lower bound of $e/(e-1)$. Besides, we combine the two ideas and obtain a randomized learning-augmented algorithm that is $(e/\lambda+1/\lambda^2)$-robust and $(e+\lambda)$-consistent. Finally, we conduct numerical experiments to complement our theoretical analysis and evaluate the empirical performance of our algorithms.

cs.LG↗

arXiv2026

Learning-Augmented Algorithms for Online Vertex Cover

This paper studies learning-augmented online weighted vertex cover with local advice and a tradeoff parameter $\lambda \in (0,1)$. We consider two graph settings: bipartite graphs and general graphs. In both settings, the online algorithm must maintain a feasible vertex cover under irrevocable decisions. We show that these problems admit the same robustness--consistency tradeoffs as learning-augmented ski rental. For the bipartite graph model, we give a randomized algorithm that is $\frac{1}{1-e^{-\lambda}}$-robust and $\frac{\lambda}{1-e^{-\lambda}}$-consistent. For the general graph model, we give a deterministic algorithm that is $(1+\frac{1}{\lambda})$-robust and $(1+\lambda)$-consistent. We prove that the tradeoffs above are optimal in both settings. We also validate the proposed algorithms through experiments on synthetic and real-world datasets.

cs.CC↗

arXiv2025

Online matching with delays and stochastic arrival times

This paper presents a new research direction for the Min-cost Perfect Matching with Delays (MPMD) - a problem introduced by Emek et al. (STOC'16). In the original version of this problem, we are given an $n$-point metric space, where requests arrive in an online fashion. The goal is to minimise the matching cost for an even number of requests. However, contrary to traditional online matching problems, a request does not have to be paired immediately at the time of its arrival. Instead, the decision of whether to match a request can be postponed for time $t$ at a delay cost of $t$. For this reason, the goal of the MPMD is to minimise the overall sum of distance and delay costs. Interestingly, for adversarially generated requests, no online algorithm can achieve a competitive ratio better than $O(\log n/\log \log n)$ (Ashlagi et al., APPROX/RANDOM'17). Here, we consider a stochastic version of the MPMD problem where the input requests follow a Poisson arrival process. For such a problem, we show that the above lower bound can be improved by presenting two deterministic online algorithms, which, in expectation, are constant-competitive. The first one is a simple greedy algorithm that matches any two requests once the sum of their delay costs exceeds their connection cost, i.e., the distance between them. The second algorithm builds on the tools used to analyse the first one in order to obtain even better performance guarantees. This result is rather surprising as the greedy approach for the adversarial model achieves a competitive ratio of $Ω(m^{\log \frac{3}{2}+\varepsilon})$, where $m$ denotes the number of requests served (Azar et al., TOCS'20). Finally, we prove that it is possible to obtain similar results for the general case when the delay cost follows an arbitrary positive and non-decreasing function, as well as for the MPMD variant with penalties to clear pending requests.

cs.DS↗

arXiv2024

Online Multi-level Aggregation with Delays and Stochastic Arrivals

This paper presents a new research direction for online Multi-Level Aggregation (MLA) with delays. In this problem, we are given an edge-weighted rooted tree $T$, and we have to serve a sequence of requests arriving at its vertices in an online manner. Each request $r$ is characterized by two parameters: its arrival time $t(r)$ and location $l(r)$ (a vertex). Once a request $r$ arrives, we can either serve it immediately or postpone this action until any time $t > t(r)$. We can serve several pending requests at the same time, and the service cost of a service corresponds to the weight of the subtree that contains all the requests served and the root of $T$. Postponing the service of a request $r$ to time $t > t(r)$ generates an additional delay cost of $t - t(r)$. The goal is to serve all requests in an online manner such that the total cost (i.e., the total sum of service and delay costs) is minimized. The current best algorithm for this problem achieves a competitive ratio of $O(d^2)$ (Azar and Touitou, FOCS'19), where $d$ denotes the depth of the tree. Here, we consider a stochastic version of MLA where the requests follow a Poisson arrival process. We present a deterministic online algorithm which achieves a constant ratio of expectations, meaning that the ratio between the expected costs of the solution generated by our algorithm and the optimal offline solution is bounded by a constant. Our algorithm is obtained by carefully combining two strategies. In the first one, we plan periodic oblivious visits to the subset of frequent vertices, whereas in the second one, we greedily serve the pending requests in the remaining vertices. This problem is complex enough to demonstrate a very rare phenomenon that ``single-minded" or ``sample-average" strategies are not enough in stochastic optimization.

cs.DS↗

arXiv2024

Modeling Online Paging in Multi-Core Systems

Web requests are growing exponentially since the 90s due to the rapid development of the Internet. This process was further accelerated by the introduction of cloud services. It has been observed statistically that memory or web requests generally follow power-law distribution, Breslau et al. INFOCOM'99. That is, the $i^{\text{th}}$ most popular web page is requested with a probability proportional to $1 / i^α$ ($α> 0$ is a constant). Furthermore, this study, which was performed more than 20 years ago, indicated Zipf-like behavior, i.e., that $α\le 1$. Surprisingly, the memory access traces coming from petabyte-size modern cloud systems not only show that $α$ can be bigger than one but also illustrate a shifted power-law distribution -- called Pareto type II or Lomax. These previously not reported phenomenon calls for statistical explanation. Our first contribution is a new statistical {\it multi-core power-law} model indicating that double-power law can be attributed to the presence of multiple cores running many virtual machines in parallel on such systems. We verify experimentally the applicability of this model using the Kolmogorov-Smirnov test (K-S test). The second contribution of this paper is a theoretical analysis indicating why LRU and LFU-based algorithms perform well in practice on data satisfying power-law or multi-core assumptions. We provide an explanation by studying the online paging problem in the stochastic input model, i.e., the input is a random sequence with each request independently drawn from a page set according to a distribution $π$. We derive formulas (as a function of the page probabilities in $π$) to upper bound their ratio-of-expectations, which help in establishing O(1) performance ratio given the random sequence following power-law and multi-core power-law distributions.

cs.DS↗

arXiv2020

The Min-Cost Matching with Concave Delays Problem

We consider the problem of online min-cost perfect matching with concave delays. We begin with the single location variant. Specifically, requests arrive in an online fashion at a single location. The algorithm must then choose between matching a pair of requests or delaying them to be matched later on. The cost is defined by a concave function on the delay. Given linear or even convex delay functions, matching any two available requests is trivially optimal. However, this does not extend to concave delays. We solve this by providing an $O(1)$-competitive algorithm that is defined through a series of delay counters. Thereafter we consider the problem given an underlying $n$-points metric. The cost of a matching is then defined as the connection cost (as defined by the metric) plus the delay cost. Given linear delays, this problem was introduced by Emek et al. and dubbed the Min-cost perfect matching with linear delays (MPMD) problem. Liu et al. considered convex delays and subsequently asked whether there exists a solution with small competitive ratio given concave delays. We show this to be true by extending our single location algorithm and proving $O(\log n)$ competitiveness. Finally, we turn our focus to the bichromatic case, wherein requests have polarities and only opposite polarities may be matched. We show how to alter our former algorithms to again achieve $O(1)$ and $O(\log n)$ competitiveness for the single location and for the metric case.

cs.DS↗