Optimal Logarithmic-Time Scheduling

• Logarithmic-time scheduling is a framework for executing scheduling operations in O(log N) time using efficient data structures and algorithms, applicable across dynamic systems and distributed networks.
• It leverages methods such as balanced trees, circulant graphs, and Lyapunov drift analysis to optimize performance in dynamic interval scheduling, wireless queueing, and parallel communication collectives.
• These techniques offer scalability and reduced compute times, enabling provably optimal or near-optimal performance in large-scale machine learning and real-time system applications.

Logarithmic-time scheduling refers to algorithmic frameworks and data structures that achieve scheduling decisions, updates, or coordination in computational systems with asymptotic time complexity $O(\log N)$ in the key system parameter %%%%1%%%%. This property is realized across a range of domains, notably parallel communication collectives, online interval scheduling, queue-based wireless control, and large-scale machine learning optimizers, where logarithmic-time operations are crucial for scalable performance. Recent research demonstrates that by leveraging hierarchical, circulant, or dynamically balanced data structures, and by exploiting the information-theoretic and graph-theoretic structure of the problem, a suite of scheduling problems can be solved in provably optimal or near-optimal logarithmic time per operation.

1. Data Structures and Algorithms for Dynamic Scheduling

Logarithmic-time scheduling was first rigorously realized in the domain of dynamic interval scheduling, where the set $I$ of intervals evolves online and must support efficient queries for optimal compatible subsets. Using advanced data structures—balanced binary search trees (BSTs), splay-trees, interval-trees, and link–cut dynamic trees—Gavruskin et al. constructed algorithms supporting $\mathsf{insert}$, $\mathsf{remove}$, and $\mathsf{query}$ in amortized $O(\log n)$ or $O(d\log^2 n)$ time, where $n$ is the number of intervals and $d$ the maximum degree of overlap (Gavryushkin et al., 2014).

For sets where intervals are monotonic (no containment), both queries and updates admit $O(\log n)$ amortized cost. The key structures are the Compatibility Forest (CF) and the Linearised Tree (LT), which maintain directed forests or binary trees reflecting the optimal greedy selection induced by the right-compatible successors. The LT structure achieves strictly local updates, yielding the optimal per-operation runtime. The CF employs heavy–light decomposition to control expose costs when general overlaps are present, with update costs scaling as $O(d\log^2 n)$. Empirical validation shows tight correspondence between the theoretical bound and observed performance, especially for large $n$ under high query or update pressure.

2. Logarithmic-Time Schedule Computation for Collective Communication

In distributed and parallel systems, optimal communication primitives require schedule computation for broadcast, reduction, and all-gather operations. For a fully-connected $p$-processor network, the minimal round complexity for broadcasting $n$ blocks is $(n-1)+\lceil\log_2 p\rceil$ under the one-ported, bidirectional communication model. The central advance in this area is the construction of per-processor send/receive schedules and circulant communication graphs in $O(\log p)$ time and space (Träff, 2024).

The method centers on the skip array and directed circulant graphs: each processor precomputes arrays of length $q=\lceil\log_2 p\rceil$ for send and receive operations. The schedule ensures that at each round, a processor receives and forwards blocks according to simple index calculations, without metadata or communication overhead. Algorithms—CirculantSkips, baseBlock, recvSchedule, sendSchedule—are all $O(\log p)$ and operate independently per processor. Correctness is established by showing that each phase propagates $q$ distinct blocks network-wide, and that the total number of rounds exactly matches the information-theoretic lower bound.

The table summarizes key complexities:

Phase	Time ($O(\cdot)$)	Space ($O(\cdot)$)	Rounds
Precompute R, S	$\log p$	$\log p$	–
Broadcast	–	–	$n-1 + \lceil\log_2 p\rceil$
Allgather/Allreduce	–	–	$n-1 + \lceil\log_2 p\rceil$
Reduce	–	–	$n-1 + \lceil\log_2 p\rceil$

This approach generalizes immediately to all-broadcast, reduction, and all-reduction collectives, with the same per-processor preprocessing bound and minimal communication rounds. Schedules are symmetric and require no global coordination, enabling scalable implementations for MPI collectives such as MPI_Bcast, MPI_Allgatherv, MPI_Reduce, and MPI_Reduce_scatter (Träff, 2024).

3. Logarithmic-Time Scheduling in Wireless Queueing and Control

In stochastic queueing systems with time-varying service and arrival processes, logarithmic backlog and scheduling times are key for power-efficient operation under delay constraints. Theoretical lower bounds established by Neely et al. assert that any policy attaining average power within $\epsilon$ of the optimum must incur average queue $E[Q] = \Omega(\log(1/\epsilon))$ (Neely, 2014). Further, prior art achieved convergence times $T(\epsilon)$ only as fast as $O(1/\epsilon^2)$.

The drift-plus-penalty framework achieves these logarithmic-time backlogs using the following control:

• At slot $t$, with queue $Q(t)$ and channel state $\omega(t)$, set $p(t) = 1$ if $Q(t)\cdot\omega(t) \geq V$; $p(t)=0$ otherwise, transmitting $\mu(t) = p(t)\omega(t)$.
• Select $V = \Theta(\log(1/\epsilon))$.

The main results are:

• Steady-state queue size: $\limsup_t E[Q(t)] = O(\log(1/\epsilon))$, matching the lower bound.
• Convergence time to achieve $\epsilon$-approximation: $T(\epsilon)=O((\log(1/\epsilon))/\epsilon)$; this matches the necessary lower bound up to a logarithmic factor.

This scheduling decision requires only the current queue and observable channel state, with no knowledge of arrival rates or channel distribution. The method relies on Lyapunov drift analysis and interval partitioning of the queue state, bounding time fractions in “good” and “bad” intervals via exponential moment arguments. A notable implementation variant, using LIFO service order, can dramatically reduce observed packet delay without compromising the total queue bound (Neely, 2014).

4. Logarithmic-Time Scheduling in Large-Scale Machine Learning

Recent advances in optimizers for LLMs reveal the importance of logarithmic-time scheduling in momentum and weight decay hyperparameters. The underlying phenomenon is linked to the power-law growth of block entropy in language data (Hilberg’s law), which suggests that the "useful" training signal in language modeling increases sublinearly with the number of tokens, motivating a scheduled increase in optimizer memory horizon (Ferbach et al., 5 Feb 2026).

Formally, for iterations $t=1,2,...$,

• Momentum schedule: $\beta_1(t) = 1 - \delta/(\delta + t)$
• Second moment schedule: $\beta_2(t) = 1 - \delta/(\delta + t)$
• Weight-decay: $\lambda(t) = \omega/t$

The memory horizon then scales as $O(\log t)$, aligning with the power-law decay of additional information in the data. Naive application of log-time momentum schedules, however, leads to instability due to the accumulation of gradient noise. The ADANA optimizer uses explicit damping via $\alpha(t) = \tilde\alpha (1+t)^{1-\kappa}$, with $\kappa = 1/(2\rho)$ where $\rho$ is the power-law random features exponent, to guarantee stability and acceleration.

Extensive empirical analysis demonstrates:

• Up to 40% compute savings versus standard AdamW, especially for transformer models with 45M to 2.6B parameters.
• Compute benefits and scaling law gains persist and even improve at large model sizes, distinguishing ADANA from other optimizers whose gains vanish at $>1$B parameters.
• Variants such as DANAMK4 and DANASTAR integrate SNR clipping and time-effective rescaling to ensure robustness, even with sparse-gradient updates characteristic of embeddings and MoE architectures, preserving the efficacy of logarithmic-time scheduling (Ferbach et al., 5 Feb 2026).

5. Proofs, Optimality, and Theoretical Underpinnings

The fundamental theoretical claims underpinning logarithmic-time scheduling are rooted in lower bounds (e.g., $\Omega(\log(1/\epsilon))$ backlog for scheduling, $(n-1) + \lceil\log_2 p\rceil$ rounds for broadcast), and in the construction of algorithms matching these bounds to within logarithmic factors (Neely, 2014, Träff, 2024). The proofs employ heavy–light decompositions for dynamic trees, circulant graph traversals for broadcasts, and Lyapunov drift inequalities for queueing.

For dynamic interval scheduling, heavy–light decomposition ensures that the expose operation, which may otherwise traverse $O(n)$ edges, is contained within $O(\log n)$ light arcs, amortizing query and update costs (Gavryushkin et al., 2014). For communication collectives, circulant graphs with skip arrays admit a unique decomposition of processor indices into skip-paths, enabling deterministic $O(\log p)$ schedule computation for each processor.

The power-law motivated optimizer schedules rely on information-theoretic arguments: block entropy scaling as $T^\beta$ and the resultant decay in per-token information justifies log-time momentum; stability is ensured via sublinear damping derived from PLRF theory (Ferbach et al., 5 Feb 2026).

6. Extensions, Open Questions, and Practical Considerations

Several open questions and extensions are highlighted:

• Multi-queue and multi-user settings: The generalization of drift-plus-penalty scheduling to more complex queue/interference structures is established, but whether logarithmic convergence scaling is preserved remains open (Neely, 2014).
• Optimality of schedule computation: For distributed collectives, it is unresolved whether further reductions in computational overhead for schedule generation (below $O(\log p)$) are possible without sacrificing universality or locality (Träff, 2024).
• Logarithmic-time optimizers for sparse, nonstationary, or adversarial regimes: Robustification via SNR clipping or time-effective normalization is effective, but theoretical convergence characterizations in highly non-i.i.d. contexts require further analysis (Ferbach et al., 5 Feb 2026).
• Real-world implementations: For all areas above, the transition from theoretical optimality to practical performance is sensitive to hardware, communication protocol design, and system-level engineering.

A plausible implication is that logarithmic-time scheduling, while established as optimal or near-optimal in canonical frameworks, prompts further research on robustness, compositionality, and adaptation in heterogeneous, large-scale environments.