Deterministic Near-Linear Time Algorithms

Updated 24 October 2025

Deterministic near-linear time algorithms are methods achieving O(n·polylog(n)) worst-case time using only deterministic procedures, critical in graph theory, computational geometry, and stringology.
They combine advanced techniques like recursive decomposition, deterministic sparsification, and efficient data structures to replicate randomized algorithms' performance without probabilistic elements.
These algorithms offer robust, reproducible performance in applications such as minimum cuts, geometric matching, and submodular maximization, with practical benefits in areas requiring predictable outcomes.

A deterministic near-linear time algorithm is one that, for input of size $n$ , achieves asymptotic running time $O(n \cdot \operatorname{polylog}(n))$ using only deterministic procedures (i.e., no randomization) throughout its computation. Such algorithms have been sought across diverse fields, including graph theory, computational geometry, differential privacy, combinatorial optimization, online learning, and stringology, often as replacements for earlier randomized approaches. Problems for which deterministic near-linear time algorithms now exist range from global min-cut in weighted graphs, plane Steiner spanners, geometric matching/transportation, submodular maximization over matroids, and edit distance approximation, to edge coloring and longest common subsequence approximation.

1. Key Technical Principles

The design of deterministic near-linear time algorithms relies on efficiently combining problem-specific structural results, sparing data structures, and de-randomization techniques to match the (previously randomized) state-of-the-art in worst-case guarantee and speed. Many methods implement advanced partition or clustering procedures (e.g., expander or tree-based clustering), recursive frameworks, greedy search strategies, or derandomized versions of classical sampling or embedding tools.

A typical schematic, exemplified by recent deterministic minimum-cut algorithms in weighted graphs (Henzinger et al., 11 Jan 2024, Li, 2021), is as follows:

Recursive Decomposition: The input is partitioned into clusters or minor subproblems which locally preserve global constraints such as minimum cut values.
Structural Theorems and Certificates: These clusters satisfy guarantees that enable the preservation of all (near-)optimal substructures (such as minimum cuts, spanners, or matches).
Deterministic Sparsification/Reductions: Advanced sparsification, such as Benczúr–Karger graph sparsification derandomized via pessimistic estimators or explicit expander decompositions (with deterministic union-bounding over all cuts), yields sparse representations that allow faster further processing.
Efficient Data Structures: Specialized implementations, such as augmented segment trees, heavy path decompositions, or dynamic trees, are engineered to ensure near-linear performance on bottleneck subprocedures.
Bridging Approximate and Exact: Most methods first achieve an approximate solution with a strong certificate and then recursively refine or boost this to an exact or $(1+\epsilon)$ -approximate solution.

2. Deterministic Near-Linear Time Algorithms in Graph Theory

A substantial body of work has been devoted to graph algorithms:

Minimum Cut in Weighted/Unweighted Graphs

Recent algorithms (Henzinger et al., 11 Jan 2024, Li, 2021) achieve $\tilde{O}(m)$ or $m^{1+o(1)}$ deterministic running time for minimum cut in weighted graphs, solving a longstanding open problem posed by Karger. The key ingredients are deterministic clustering (partitioning vertices into strongly interconnected clusters), flow-based uncrossing (unit-flow push–relabel algorithm with capped heights), and deterministic cut-sparsifiers constructed via pessimistic estimators and expander decomposition. The new clustering approach ensures that all near-minimum cuts can be efficiently represented and preserved in contracted or sparsified graphs.

Edge Connectivity and Cactus Representations

In simple graphs, deterministic near-linear algorithms contract clusters identified via low-conductance cuts using deterministic PageRank routines (Kawarabayashi et al., 2014). This approach facilitates construction of cactus representations of all min-cuts in near-linear time.

Balanced Sparse Cuts and Vertex Connectivity

The subquadratic deterministic cut framework for balanced sparse cuts and $k$ -vertex connectivity (Gao et al., 2019) interlocks PageRank matrix inversion, Madry’s $J$ -tree decompositions, and recursive expander-based sparsification to achieve $O(m^{1.5 + o(1)})$ time bounds, matching natural fine-grained lower-bound barriers.

Edge Coloring

A deterministic $(1+\varepsilon)\Delta$ -edge-coloring algorithm (Elkin et al., 19 Jan 2024) uses recursive degree splitting and merges local colorings, achieving $O((m \log n)/\varepsilon)$ time. On graphs of arboricity $\alpha$ , an advanced orientation and alternating-direction path decomposition allow for $(\Delta+\varepsilon \alpha)$ -coloring in $O((m \log n)/\varepsilon^7)$ time.

3. Computational Geometry and Geometric Optimization

Plane Steiner Spanners and TSP

The plane Steiner spanner construction for well-spaced planar point sets (Borradaile et al., 2012) leverages deterministic placement of portals on Delaunay triangulations and recursive “wedged” spanner connections inside triangles. This yields $O(n \sqrt{\log \log n})$ deterministic time for $(1+\varepsilon)$ -spanners and near-linear deterministic TSP approximation schemes in the Euclidean plane, with spanner weight $O((\beta/\alpha) \cdot \mathrm{MST})$ for bounded minimal Delaunay angles $\alpha$ .

Geometric Bipartite Matching and Transportation

For geometric transportation and bipartite matching in $\mathbb{R}^d$ , deterministic warped quadtree constructions yield sparse spanner graphs preserving Euclidean distances up to $(1+O(\varepsilon))$ (Fox et al., 2022, Agarwal et al., 2022). Preconditioning with an oblivious greedy framework (Sherman-type solver) and explicit deterministic derandomization of cell boundaries allow a near-linear-time $(1+\varepsilon)$ -approximation for minimum-cost transportation ( $O(n\varepsilon^{-(d+2)}\log^5 n \log \log n)$ ) and geometric matching ( $n(\varepsilon^{-1} \log n)^{O(d)}$ ).

Deterministic Separators and Plane Partitioning

In higher dimensions, separator hyperplanes can be constructed deterministically in $O(nf(n))$ time ( $f(n)$ polylogarithmic) to nearly halve the centers of disjoint unit balls, intersecting only $o(n)$ balls (Hoffmann et al., 2014), critical for geometric divide-and-conquer and imprecise point space decomposition.

4. Stringology and Edit Distance Approximation

Deterministic near-linear time algorithms now exist for several classical string problems previously resistant to derandomization. Notably:
- Edit Distance: A multi-scale recursion based on interval matching, coloring with soft potentials, and recursive alignment distance (using “distance oracles” for substrings) achieves a constant-factor approximation to edit distance in $O(n^{1+o(1)})$ time (Andoni et al., 2020). Core components include the construction of cluster graphs on intervals, distortion-resilient potential assignments, and fast approximation data structures.
- Sparse Pattern Matching: The “Constellation” problem (determining if $P + s \subseteq Q$ for sets $P, Q \subseteq [N]$ ) is solved deterministically in $O(n \cdot \operatorname{polylog}(N))$ time (Fischer, 2023) by reducing to structured Partial Convolution and applying the Baur–Strassen theorem for efficient differentiation in arithmetic circuits, bypassing FFT-based randomization.

5. Submodular Maximization and Online Learning

Submodular Maximization: The TwinGreedyFast framework (Han et al., 2020) deterministically constructs two solution sets in parallel, using greedy marginal gains and thresholding to obtain a $\frac{1}{4} - \epsilon$ approximation with $O((n/\epsilon)\log(r/\epsilon))$ time for non-monotone submodular maximization under a matroid constraint.
Online Learning: In dynamic or adversarial-loss settings, mixing logarithmic-regret static solvers in an adaptive manner (via scheduled hyper-experts and weighted aggregation) yields deterministic near-linear time algorithms achieving near-logarithmic regret per switch (Gokcesu et al., 2021). Complexity per time is kept sub-polynomial by controlling hyper-expert schedules, a key advance over earlier quadratic-time frameworks.

6. Broader Algorithmic Themes and Impact

Deterministic near-linear time algorithms are typically built upon:

De-randomization of core algorithmic primitives (e.g., sampling, hashing, recursion branches)
Concentration and union-bound tools (pessimistic estimators, explicit search over directions or colorings)
Exploitation of hidden structural regularities (canonical partitions, representative sets)
Hierarchical decomposition (recursive partitioning, dynamic graph contractions, or geometric hierarchical coverings)

Their impact includes:

Enabling predictable and reproducible high-performance algorithms, critical for applications where failed randomization is unacceptable
Making subroutine performance worst-case robust (including timing attacks in privacy, networking, or online settings)
Serving as foundations or subroutines for broader frameworks (dynamic graphs, distributed algorithms, privacy-preserving analytics)

This class of algorithms now provides state-of-the-art methods in deterministic optimization, computational geometry, graph partitioning, privacy, and beyond, with several ongoing advances expanding their reach.