amortized analysis

This paper addresses the forward error analysis of sum-product algorithms under stochastic rounding (SR). We propose a probabilistic error bounding method grounded in martingale theory. Our key contributions are threefold: (1) We introduce the first automated martingale construction framework tailored to multilinear computational structures—encompassing addition, subtraction, multiplication, and intermediate result reuse; (2) We extend SR error analysis to algorithms with structural reuse, notably Karatsuba polynomial multiplication—previously unaddressed in SR literature; (3) Leveraging the Azuma–Hoeffding inequality, we derive a tight probabilistic error bound of $O(sqrt{n},u)$, markedly improving upon the classical worst-case bound $O(n,u)$. Our framework uniformly recovers known error guarantees for pairwise summation and Horner’s method, and—crucially—provides the first rigorous SR error guarantee for Karatsuba multiplication.

Probabilistic programs pose significant challenges for expected-cost analysis due to the intricate coupling of stochasticity, recursion, and evaluation strategies, which complicates semantic modeling. Method: We introduce CBPV-cert—the first verifiable denotational semantics framework that simultaneously accounts for probabilistic behavior and expected resource consumption. Grounded in the Call-by-Push-Value (CBPV) metalanguage, it provides a unified operational semantics and two denotational interpretations: one for accumulated cost and one for expected cost. We formally define an *effect simulation* property to relate these semantics and prove, in Coq-style mechanized reasoning, that the expected-cost semantics is canonical (i.e., minimal) and operationally adequate (i.e., sound and complete). Contribution/Results: Evaluated on randomized algorithms and stochastic processes, CBPV-cert demonstrates expressive power and yields the first sound, verifiable theoretical foundation for expected-resource analysis of probabilistic programs.

This study addresses the online admission control problem for two customer classes in a loss queue with reusable resources. By decomposing total cost into variability cost and uncertainty cost, and leveraging stochastic queueing models, deterministic relaxation upper bounds, and the full-information offline optimum, the authors conduct an asymptotic analysis in large-scale systems ($N \to \infty$). They establish that the $\Theta(\log N)$ growth of total cost arises entirely from uncertainty cost, while variability cost remains $O(1)$. Furthermore, they demonstrate that a lookahead window of merely $\Omega(\log N / N)$ suffices to substantially reduce operational costs. These findings reveal that in overloaded loss networks, even a vanishingly small—yet scale-dependent—degree of foresight can eliminate most of the excess cost, offering theoretical justification for highly efficient online resource allocation strategies.

Evaluating learning-augmented online algorithms under uncertainty remains challenging, as conventional metrics focus narrowly on worst-case prediction errors, neglecting both prediction accuracy and risk sensitivity. Method: We propose a dual-track evaluation framework grounded in decision theory, jointly incorporating distance-based prediction error quantification (deterministic aspect) and risk-sensitive modeling (stochastic aspect). By embedding decision-theoretic loss functions into online algorithm analysis, we integrate prediction error modeling with risk-controllable optimization, designing novel learning-augmented algorithms for contract scheduling and 1-max search. Contribution/Results: Our approach achieves provable robustness to prediction errors, performance guarantees with tight bounds, and explicit risk controllability. It is the first to unify prediction accuracy, worst-case robustness, and risk preference within a single theoretical framework—establishing a systematic evaluation paradigm and design principle for learning-augmented online algorithms.

This study addresses the computational bottlenecks in scientific computing arising from the infeasibility of exact algorithms for large-scale problems. Through a systematic evaluation of approximation methods across 118 core algorithmic problems—integrating complexity analysis, taxonomies of approximation algorithms, and historical context—the work presents the first large-scale empirical evidence demonstrating that only approximately 20% of these problems derive substantial benefit from approximation. Notably, one-quarter of exponential-time-hard problems admit polynomial-time approximation schemes, and the adoption of approximation strategies increases the proportion of linear-time solvable problems by 23%. By quantifying the trade-offs between accuracy and efficiency, this research offers theoretical insights to guide the design of AI-driven and high-performance algorithms.

This work addresses the challenge of analyzing the cost of randomized programs involving nonlinear operations such as max/min, which traditional moment propagation techniques struggle to handle, while full distribution propagation remains computationally infeasible. The authors propose a compositional cost analysis method that solves local recurrence equations bottom-up using hierarchical cost expressions and summarizes subproblems via surrogate distributions composed of an exact short-time prefix and a parameterized tail. This approach enables efficient computation of cost moments—including mean and higher-order moments—with provable error bounds. To the best of our knowledge, it is the first technique to support scalable, error-bounded moment analysis for randomized programs with nonlinear constructs. The framework is implemented in the tool DICKENS and validated on case studies including quantum repeater waiting times, RFID collision resolution, and fork-join completion times, demonstrating both accuracy and efficiency.

This study addresses the problem of dynamic fair allocation of indivisible goods (or tasks) in an online setting, where items arrive sequentially and must be allocated immediately upon arrival. Under a broad range of models—including normalized and non-normalized utilities as well as identical and general additive utility functions—the work designs constructive online algorithms within the competitive analysis framework, targeting multiple fairness criteria such as EF1 and PROP1. For most settings considered, the paper not only presents algorithms achieving optimal competitive ratios but also establishes matching theoretical upper bounds, thereby substantially expanding the theoretical foundations of online fair division.

Computing fixed points of non-monotonic operators—such as those arising from negation-as-failure or hypothetical updates—is notoriously challenging, as traditional monotonic methods do not apply and existing approximation techniques are either imprecise or computationally expensive. This work introduces, for the first time, controlled incompleteness into approximate fixed-point computation, integrating abstract interpretation, Approximation Fixpoint Theory (AFT), lattice theory, and partitioning-based optimization to devise a practical algorithm. The proposed method guarantees termination, polynomial-time complexity, and soundness over finite lattices while substantially improving approximation precision. Empirical evaluations demonstrate its effectiveness: deployed as an accelerating preprocessor in Answer Set Programming and applied to speculative program analysis, it significantly reduces rollback frequency, thereby validating both its efficiency and practical utility.

Floating-point round-off errors are ubiquitous in numerically intensive programs arising in fields such as scientific computing and optimization. As floating-point errors potentially lead to unexpected and catastrophic program failures, one must derive guaranteed round-off thresholds to ensure the correctness of these programs. However, deterministic round-off thresholds tend to be too conservative to be usable in practice, since they often involve large round-off errors that occur with small probability. Probabilistic thresholds relax deterministic ones by specifying that the probability of the round-off error exceeding a threshold is below a given confidence. In this work, we propose a novel approach to probabilistic round-off analysis, by applying concentration inequalities over the Taylor expansion from FPTaylor (TOPLAS 2018). A major obstacle in applying concentration inequalities is that the Taylor expansion involves absolute value operators that make the calculation of the expected values of the first order partial differential terms difficult. Our first step to overcome this obstacle is a sound over-approximation that removes the absolute value operators in polynomial expressions. Then, we show how to handle fractional expressions by a transformation into polynomial case. Finally, we show how to improve our approach with range partitioning. Our approach is scalable since the key computational part is the calculation of expected values of polynomial expressions with independent variables, for which the linear and independence properties of expectation boost the computation. Experimental results show that our approach is orders of magnitude more time efficient, while producing thresholds with comparable precision against the state of the art.