Derandomization Procedures

• Derandomization procedures are algorithmic frameworks that convert stochastic steps into deterministic processes while maintaining accuracy and optimal performance.
• They employ techniques such as conditional expectations, union bounds, and pseudorandom generators, with applications in quantum algorithms, parallel computation, and combinatorial optimization.
• These methods offer robust alternatives to randomness, balancing enhanced reliability with computational overhead and trade-offs in uniformity and efficiency.

Derandomization procedures refer to algorithmic or structural frameworks devised to systematically eliminate or limit the role of randomness in computational processes, replacing stochastic steps with deterministic constructs while matching or surpassing the performance of their randomized analogs. Across theoretical computer science—spanning quantum algorithms, distributed computing, parallel computation, circuit complexity, combinatorial optimization, and machine learning—derandomization is fundamental both as a practical tool and as a route to deeper understanding of the power of randomness.

1. Formal Frameworks of Derandomization

Derandomization, in its strict sense, involves transforming a randomized protocol into a deterministic procedure that preserves essential guarantees—accuracy, efficiency, or optimality—of the original. The underlying methods depend on context:

• Conditional Expectations: Fix random choices sequentially, minimizing expected failure at each step (e.g., derandomizing randomized measurements in quantum estimators (Huang et al., 2021)).
• Enumeration-Based Methods: Exploit finiteness of possible random seeds or inputs, often by union bounds to guarantee the existence of a "good" deterministic choice (e.g., derandomizing distributed LOCAL algorithms via "lying about n" (Dahal et al., 2023, Ghaffari et al., 2019)).
• Hardness-vs-Randomness and Pseudorandom Generators (PRGs): Use computational hardness to build PRGs, replacing uniform randomness with pseudorandom sequences indistinguishable for polynomial-size circuits (see negative results in (Lin, 2023)).
• LP-based Derandomization: Explicitly maintain and update small distributions over algorithmic states, usually by extracting extreme points of linear programs (submodular maximization (Buchbinder et al., 2015)).
• Coupling and Marginal Sampling in MCMC: Enumerate a bounded set of random choices for coupling arguments in Markov chains, producing deterministic approximate samplers for partition functions (Markov chain derandomization via "coupling towards the past" (Feng et al., 2022)).
• Sketching and Sparsification: Compress input space to avoid exhaustive amplification over random bits, yielding non-uniform deterministic algorithms with nearly no slowdown (Grossman et al., 2015).

These frameworks serve both as existential proofs of derandomizability, and as design blueprints for efficient algorithms.

2. Algorithmic Methodologies

Central derandomization techniques include:

• Method of Conditional Expectations: Fix each random variable to the value minimizing the conditional expectation of the failure or error event. For estimation of Pauli observables, this yields a deterministic measurement array with performance at least as good as the randomized counterpart (Huang et al., 2021). Each measurement configuration is fixed by evaluating a closed-form confidence bound, maintaining monotonicity of performance guarantees.
• Enumeration via Union Bounds: For algorithms with input-dependent randomness, derive an overall error probability small enough so that a union bound over the finite set of inputs or sketches ensures a deterministic strategy exists that works for all inputs (Dahal et al., 2023, Ghaffari et al., 2019, Grossman et al., 2015).
• Local Rounding and Iterative Coloring: In parallel graph algorithms, fractional solutions are deterministically rounded to integral solutions using a carefully designed iterative process. Potential functions (linear or quadratic) track progress to ensure deterministic shrinkage and objective maintenance, with conflict graphs colored in defective fashion to manage dependencies (Ghaffari et al., 22 Apr 2025).
• Explicit Distribution Maintenance in Value-Oracle Models: Randomized greedy-like algorithms for submodular maximization rely on implicit probability distributions across partial solutions. Derandomization keeps such distributions explicit and small by LP-based splitting, recursively maintaining support size growth that is polynomial and extracting optimal or near-optimal deterministic solutions (Buchbinder et al., 2015).
• Marginal Sampling via Coupling Towards the Past (CTTP): For Markov chain Monte Carlo, derandomization proceeds by locally reconstructing the chain backwards via recursive enumeration of random choices, as the number of required random bits is logarithmic in system size. This provides deterministic approximate samplers for hypergraph independent sets and colorings (Feng et al., 2022).
• Online Randomness Extraction: For barely-random online algorithms in the random order model, deterministic bit extraction mechanisms are devised using input arrival permutations, with worst-case bias precisely quantified (e.g., worst-case bias $2-\sqrt{2}$). Such extraction enables simulation of 1-bit randomized competitive algorithms as deterministic ROM algorithms (Borodin et al., 20 Oct 2025).

3. Performance Guarantees and Complexity Tradeoffs

Derandomization procedures are typically accompanied by precise, often optimal, sample complexity, runtime, or approximation guarantees:

• Quantum Estimation: The method derandomizes classical-shadow measurement by ensuring that deterministic protocols match the sample complexity of randomized ones ($M=O(3^w\log L/\varepsilon^2)$ for $w$-local Pauli observables) and strictly outperform in regimes with high operator weight (Huang et al., 2021).
• Distributed Algorithms: Any randomized LOCAL algorithm solving a component-wise verifiable problem in $T_R(n)$ rounds can be derandomized to a deterministic protocol in $T_R(2^{n^2})$ rounds, regardless of individual random bit bounds (Dahal et al., 2023). In specific settings, polylogarithmic time randomized algorithms with exponentially small error can be derandomized to polylogarithmic time deterministic algorithms (Ghaffari et al., 2019).
• Parallel Work Efficiency: Recent advances move deterministic parallel algorithms for maximal independent set, matching, and hitting-set closer to work-efficiency thresholds, achieving $m\cdot \text{poly}(\log\log n)$ total work and $\text{poly}(\log n)$ depth (Ghaffari et al., 22 Apr 2025).
• Submodular Maximization: Deterministic unconstrained maximization achieves a $1/2$-approximation (optimal) in $O(n^2)$ value-oracle queries, matching the best randomized ratio. Cardinality-constrained deterministic algorithms obtain $1/e$-approximation with $O(k^2n)$ queries (Buchbinder et al., 2015).
• Limits of PRG/HSG-Based Methods: Under unconditional separations such as $\mathsf{P} \subsetneq \mathsf{BPP}$, any uniform derandomization method relying on $O(\log n)$ random bits or seed length will fail to simulate general randomized algorithms efficiently. This is formalized in the impossibility results for PRGs and hitting-set generators in (Lin, 2023).

4. Derandomization in Applied Domains

Applications extend from foundational theory to practical algorithmics:

• Quantum Algorithms: Derandomization yields deterministic quantum algorithms for triangle finding in edge-weighted graphs, matching the best bounded-error complexities ($O(n^{9/7})$ queries) by exploiting nested quantum walks, exact amplitude amplification, and dimensional reduction to constant subspaces (Li et al., 2023). The approach generalizes to other quantum search problems.
• Submodular Maximization and Value-Oracles: The LP-based explicit distribution maintenance procedure bridges the gap to randomized performance for non-monotone unconstrained and cardinality-constrained maximization, even in value-oracle settings where classical conditional expectation fails (Buchbinder et al., 2015).
• Group Testing and Coding: Extractors, condensers, and lossless expanders are employed to construct error-correcting and threshold group testing schemes with nearly optimal measurement complexity, achieving robustness against adversarial noise and tight identification of sparse supports (Cheraghchi, 2010, Cheraghchi, 2011).
• Markov Chain Monte Carlo: CTTP-based derandomization supplies deterministic FPTAS for partition functions in statistical models satisfying uniform marginal lower bounds, enabling efficient counting for hypergraph independent sets and proper colorings in regimes matching randomized mixing times (Feng et al., 2022).
• Online Algorithms and ROM: Deterministic extraction-based simulation in the random order model provides competitive ratios close to the best 1-bit randomized algorithms (multiplicative loss factor $2.41$), covering knapsack, weighted interval selection, string guessing, and throughput scheduling (Borodin et al., 20 Oct 2025).

5. Structural and Theoretical Implications

• Circuit Complexity and Lower Bounds: Derandomization is intimately connected to non-trivial circuit lower bounds via hardness-vs-randomness tradeoffs, e.g., quantified derandomization for $TC^0$ circuits of superlinear wire complexity yields consequences for $\mathsf{NEXP}$ noncontainment in $TC^0$ (Tell, 2017). Hitting-set based techniques for promise AM protocols can yield downward separations and subsume subexponential derandomization with advice (Stull, 2017).
• Limits, Barriers, and Optimality: Tight lower bounds for randomness reduction, error boosting, and advised derandomization indicate that any substantial improvement would imply breakthroughs in deterministic algorithms and complexity theory—for instance, efficient deterministic network decomposition in polylog time for distributed graph problems (Ghaffari et al., 2019).
• Game-Theoretic Probability: Derandomization in non-measure-theoretic settings is characterized via reality strategies that force events previously known to hold almost surely under randomness—mixing deterministic play against hypothetical adversaries in perfect-information games such as the unbounded forecasting game (Miyabe et al., 2014).

6. Practical Tradeoffs and Open Problems

Derandomization procedures often entail increased complexity (e.g., enumeration overhead, amplified running time by a factor related to input sketch size, or elaborate LP solves). Several tradeoffs persist:

• Non-uniformity vs Efficiency: Most frameworks yield non-uniform deterministic algorithms, with efficiency closely tied to the size of input sketches or enumeration space (Grossman et al., 2015).
• Limits of Explicitness and Uniformity: Explicit, uniformly computable objects—condensers, extractors, seeds—are essential for practical derandomization but are still limited in several regimes (e.g., small alphabet linear extractors for coding, optimal lossless condensers for group testing).
• Probabilistic vs Adversarial Models: The extent to which derandomization matches adversarially randomized algorithms is unresolved in certain ROM models and for algorithms requiring $k>1$ bits; optimal unbiased extraction remains open (Borodin et al., 20 Oct 2025).
• Extensions to MCMC and Statistical Models: Generalizing CTTP and related coupling-based techniques to zero-marginal regimes or non-Gibbs sampling is an active research area (Feng et al., 2022).

Open directions include unifying uniform derandomization methods in complexity theory, extending derandomization to sparse instances, more work-efficient parallel algorithms, and improving extraction bias in online randomized-to-deterministic reductions.