Greedy Sequencing Rule Overview

• The greedy sequencing rule is a method that builds sequences by selecting the minimal candidate that avoids forbidden configurations, playing a key role in combinatorics, number theory, and approximation theory.
• It yields sharp bounds and reveals surprising connections across fields such as fractal geometry, Banach space bases, and algorithm design, with practical applications in change-making and metagenomic assembly.
• This rule underpins classical constructs like nonaveraging and Stanley sequences and extends to modern problems including decentralized blockchain ordering and low-discrepancy sequence generation.

The greedy sequencing rule is a unifying principle underlying several important areas in combinatorics, analysis, number theory, approximation theory, and computational applications. It is typically characterized by constructing sequences or orderings step-by-step, each time making the "greedy" or locally optimal choice according to a defined exclusion or minimization criterion. Despite the local nature of its construction, the greedy sequencing rule frequently yields rich mathematical structures, sharp bounds, and surprising connections to fractal, algebraic, or analytic behavior. It has major roles in nonaveraging sequences, extremal set theory, coin-making algorithms, Banach space bases, arithmetic-progression-avoiding sets, block design in discrepancy theory, metagenomic assembly, and decentralized consensus mechanisms.

1. Foundational Definitions and General Frameworks

The essence of the greedy sequencing rule is to iteratively build an ordered sequence $(a_k)$ such that each new element $a_{k+1}$ is the minimal or optimal candidate (with respect to some constraint), given previous choices. A canonical prototype is greedy nonaveraging integer sequences: fix $m\geq3$, set $a_0=0$, and for $k\geq0$ define

$$a_{k+1} = \min \Bigl\{n>a_k : \nexists \text{ distinct } x_1,\dots,x_m \in \{a_0,\dots,a_k,n\} \text{ with } x_1+\dots+x_{m-1} = (m-1)x_m \Bigr\}.$$

This rule prohibits introducing a forbidden configuration at each extension step. Many instances reduce to hereditary forbidden pattern avoidance, greedy minimization of a cost function, or thresholding algorithms with respect to some basis or metric (Tseng, 2011, Berasategui et al., 2023, Kiss et al., 2017, Berasategui et al., 2022).

Variants exist across settings:

• Nonaveraging sequences (e.g., Szekeres/Layman sequences)
• Block-wise selection in Banach spaces (sequential greedy-type bases)
• AP-avoiding (Stanley) sequences and their generalizations
• Greedy partitions of the integers into distinct combinatorial structures
• Low-discrepancy sequences in numerical integration
• Orderings in decentralized transaction execution (blockchain MEV)

2. Classical and Generalized Nonaveraging Sequences

Szekeres and Layman provided closed forms for $m=3,4$ in the greedy nonaveraging sequence:

• For $m=3$, $S_3$ consists of all nonnegative integers whose base-3 expansions omit the digit "2":

$S_3=\{n: \text{in base 3, all digits %%%%8%%%% or %%%%9%%%%}\}$

• For $a_{k+1}$0, structure depends on blocks in base 4 and digit restrictions.

The general solution for $a_{k+1}$1 involves a digit-expansion characterization with block length $a_{k+1}$2 and residue set $a_{k+1}$3: $a_{k+1}$4 with explicit formulas for $a_{k+1}$5 and $a_{k+1}$6, depending on parity of $a_{k+1}$7. Weighted average avoidance and further generalizations admit analogous digit-system (digital set) criteria (Tseng, 2011). These constructions have direct implications for extremal set theory, additive combinatorics, and the structure of Thue–Morse and related words.

3. Greedy Sequencing in Approximation Theory and Banach Spaces

In Banach and quasi-Banach spaces, greedy sequencing rules give rise to the Thresholding Greedy Algorithm (TGA) and its sequential or restricted variants. Given a basis $a_{k+1}$8, the greedy sum of order $a_{k+1}$9 is the projection onto the indices of the $m\geq3$0 largest coefficients in absolute value: $m\geq3$1 Sequential greedy-type rules restrict attention to a given (possibly gapped) sequence of orders $m\geq3$2, leading to properties such as $m\geq3$3-quasi-greedy, $m\geq3$4-democratic, and their extensions. If $m\geq3$5 has bounded quotient or additive gaps, these properties collapse to the classical theory; unbounded gaps induce strictly intermediate phenomena (Berasategui et al., 2023, Berasategui et al., 2022).

The $m\geq3$6-almost greedy property interpolates between classical and blockwise behaviors, with equivalence to classical almost-greediness in the case of bounded gaps. Growth conditions on the index sequence dictate whether or not new types of bases can exist.

4. Greedy Algorithms in Combinatorial and Number-Theoretic Contexts

A major instance is the Stanley sequence (and its generalizations): given a finite set $m\geq3$7 containing no $m\geq3$8-term arithmetic progression (AP), generate the infinite sequence by repeatedly adding the minimal integer which, when included, keeps the set $m\geq3$9-AP-free: $a_0=0$0 More generally, the greedy partitioning of $a_0=0$1 into disjoint $a_0=0$2-free sets underlies a fractal and base-expansion theory, as in the base 3/2–Stanley connection (Khovanova et al., 2020). For $a_0=0$3, this sequence (OEIS A005836) grows like $a_0=0$4, with the exact constants and finer asymptotics remaining an open topic (Kiss et al., 2017). Greedy partition algorithms exhibit deep links to digital expansions, fractal grid structures, and sequence growth theory.

5. Greedy Rules in Applied Computational Contexts

The greedy sequencing paradigm pervades applied algorithms:

• Change-making problems: Classify those sequences of coin denominations for which the greedy algorithm always gives the minimal number of coins for any amount; such sequences are "totally greedy" if every prefix subset is greedy. For positive-homogeneous and alternating-sign second-order linear recurrences with specific bounds, every prefix is greedy (Pérez-Rosés, 2024).
• Shotgun genome sequencing: The greedy assembly rule, merging reads by greatest overlap, reconstructs source sequences with high probability if and only if the read length $a_0=0$5 exceeds explicit entropic thresholds in terms of the number and length of genomes. This admits sharp phase transitions depending on $a_0=0$6, $a_0=0$7, $a_0=0$8 and ensures algorithmic reliability in metagenomic assembly at explicit information-theoretic thresholds (Herring, 2022).

Sequence Family	Greedy Rule Applied	Structure/Property
Szekeres/Layman (Sₘ)	m-nonaveraging	Digit-based structure, closed form (Tseng, 2011)
Stanley	AP$a_0=0$9-free	Lex minimal completion avoiding forbidden APs (Kiss et al., 2017)
Banach bases	Thresholding greedy sum	Best $k\geq0$0-term error control, democracy/quasi-greedy equivalences (Berasategui et al., 2022)
Change-making (totally greedy)	Coin substitution sequences	Greedy sufficiency for all prefixes, recurrence links (Pérez-Rosés, 2024)
Shotgun sequencing	Read-merging via overlap	Phase transition in identifiability via greedy assembly (Herring, 2022)

6. Greedy Sequencing in Discrepancy Theory and Blockchain Design

Recent work demonstrates the effectiveness of greedy sequencing in high-precision uniform distribution:

• Low-discrepancy sequences: By choosing each point in $k\geq0$1 to greedily minimize $k\geq0$2-star discrepancy, the rule achieves lower scaled discrepancy than classical Kronecker or van der Corput constructions, self-correcting even from adversarial initializations (Clément, 2024).
• DeFi/Blockchain sequencing: In decentralized finance, the greedy sequencing rule (GSR) constrains allowable transaction orderings to prevent asymptotic price drift and front-running, guaranteeing that price moves "back and forth" around a fixed benchmark. Algorithmic analysis demonstrates that under zero-fee regimes, miners’ optimal strategies under GSR are computable in polynomial time and always give users at least their stand-alone surplus; for constant fees, optimization becomes NP-hard (Li et al., 2023). These results guide the design of sequencing rules in permissionless systems to align miner incentives and user welfare.

7. Growth Rates, Sharp Thresholds, and Theoretical Implications

For classical greedy-ruled sequences, growth rates are determined via digital structure and combinatorial exclusion:

• Nonaveraging sequences: Counting 0/1-digit expansions yields exponents $k\geq0$3 for $k\geq0$4 and generalizations.
• Stanley-type sequences: Densities are known to interpolate between $k\geq0$5 and $k\geq0$6 for order $k\geq0$7.
• Greedy shotgun assembly: Succeeds with high probability for $k\geq0$8, fails for $k\geq0$9, with information-theoretic sharpness (Herring, 2022).

In Banach space approximation and democracy-type properties, boundedness of selection sequences (gaps) is the dividing line between reduction to the classical theory and the emergence of new, strictly intermediate greedy bases (Berasategui et al., 2023, Berasategui et al., 2022).

These sharp behaviors illuminate the deep structure provided by greedy sequencing mechanisms, controlling the emergence of forbidden configurations, the efficiency of approximation, and the phase behaviors of assembly and partition processes.

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References: For detailed proofs, formal definitions, and additional context on specific greedy sequencing rules, see (Tseng, 2011, Kiss et al., 2017, Pérez-Rosés, 2024, Berasategui et al., 2023, Berasategui et al., 2022, Herring, 2022, Li et al., 2023, Khovanova et al., 2020, Clément, 2024).