Deterministic Wheeler Graphs

• Deterministic Wheeler graphs are edge-labeled directed graphs with a total vertex order that guarantees co-lexicographic interval properties for efficient pattern matching.
• They generalize the Burrows–Wheeler transform to graphs by aligning state intervals with FM-index search, thereby supporting compressed indexes and exact search.
• Their structure allows effective DFA minimization and canonical form construction, with applications in pangenome indexing and formal language analysis.

Searching arXiv for papers on deterministic Wheeler graphs and Wheeler DFAs. Deterministic Wheeler graphs are edge-labeled directed graphs whose vertices admit a total order compatible with edge labels in a Burrows–Wheeler–like manner. In automata-theoretic form, they are deterministic finite automata whose transition graph satisfies the Wheeler axioms, yielding a co-lexicographic organization of states and intervals that behave analogously to suffix-array intervals on strings. This structure places deterministic Wheeler graphs at the intersection of formal language theory, compressed indexing, and graph-based pattern matching, and underlies a body of work on exact search, matching statistics, minimization, pangenome indexing, and combinatorial problems such as $k$-mer counting (Alanko et al., 2020, Conte et al., 2023, Maso et al., 2 Jul 2026).

1. Definition and ordering structure

Let $\Sigma$ be a finite alphabet equipped with a total order. A Wheeler graph is a finite directed edge-labeled graph $G=(V,E,\lambda)$ for which there exists a total order $<$ on $V$ satisfying the Wheeler conditions: if two edges have labels $a\prec a'$, then their target vertices are ordered accordingly; if the labels are equal and the source of one edge precedes the source of the other, then the same monotonicity is inherited by the targets. In one standard formulation, if $e_1=(u_1\to v_1,\text{ label }a)$ and $e_2=(u_2\to v_2,\text{ label }a')$, then $a\prec a'\Rightarrow v_1<v_2$, and $a=a'$ with $\Sigma$0 (D'Agostino et al., 28 Jun 2026, Alanko et al., 2020).

A deterministic Wheeler graph is a Wheeler graph in which no vertex has two outgoing edges carrying the same label. Equivalently, the underlying automaton is a DFA with a partial transition function $\Sigma$1, a unique start state with no incoming edges, and a total order on states satisfying the same axioms. In state-labeled presentations, every non-start state has a unique incoming label $\Sigma$2, a property often called input-consistency and implied by the Wheeler axioms (Alanko et al., 2020, Becker et al., 8 Jul 2026).

Several equivalent viewpoints coexist. One may define Wheelerness directly on the graph; one may define it on a deterministic automaton; or one may characterize it via co-lexicographic ordering of strings reaching states. In the transducer-oriented treatment, a DFA $\Sigma$3 is Wheeler iff the partial order

$\Sigma$4

is in fact a total order (D'Agostino et al., 28 Jun 2026). In deterministic settings this order is particularly rigid: the order is unique whenever it exists, because the sets of strings reaching distinct states are disjoint (Cotumaccio, 8 Feb 2026). This uniqueness is one of the key structural simplifications distinguishing deterministic Wheeler graphs from the nondeterministic setting.

A central consequence is interval behavior. For every string $\Sigma$5, the set of states reachable by $\Sigma$6 forms a contiguous interval in Wheeler order; this is the path-coherence property attributed to Gagie et al. 2017 and used throughout the indexing literature (Maso et al., 2 Jul 2026). This interval property is the graph analogue of suffix-array intervals for strings and is the combinatorial basis for FM-index-style search.

2. Relation to the Burrows–Wheeler transform

Deterministic Wheeler graphs generalize the Burrows–Wheeler transform from a single text to labeled graphs. In the classical string case, an FM-index relies on the suffix array, the Burrows–Wheeler transform $\Sigma$7, a count array $\Sigma$8, and rank queries. Backward search maintains an interval $\Sigma$9 such that after processing a pattern suffix the interval corresponds exactly to the suffixes prefixed by that suffix. The recurrence is

$G=(V,E,\lambda)$0

from the initial interval $G=(V,E,\lambda)$1 (Maso et al., 2 Jul 2026).

On a Wheeler DFA, Gagie et al. and later Conte et al. observed that one can define graph analogues of the same ingredients. The ordered state set plays the role of the suffix array; the extremal strings entering each state induce a sequence $G=(V,E,\lambda)$2 of infimum and supremum strings; the $G=(V,E,\lambda)$3-array counts how many infimum/supremum blocks begin with letters smaller than a given symbol; and rank/select structures over edge labels support interval refinement (Maso et al., 2 Jul 2026). In the formulation used for deterministic Wheeler automata, the recurrence becomes

$G=(V,E,\lambda)$4

This is a left-to-right “forward-search” on the pattern, but formally it mirrors string backward search (Maso et al., 2 Jul 2026).

The importance of this correspondence is algorithmic rather than merely notational. It means that exact path queries in the graph can be answered by repeatedly updating a state interval, just as exact substring queries are answered by updating a suffix-array interval in an FM-index. The data structure thus inherits the two defining features that made BWT-based indexes effective on strings: compactness and direct search on the compressed representation (Maso et al., 2 Jul 2026, Alanko et al., 2020).

The I/O behavior, however, also carries over. The classical forward-search/backward-search strategy performs a rank query per pattern character, and in the I/O model this yields $G=(V,E,\lambda)$5 cache misses in the worst case for a pattern of length $G=(V,E,\lambda)$6. The recent cache-efficient literature on deterministic Wheeler pangenome graphs addresses precisely this inherited bottleneck (Maso et al., 2 Jul 2026).

3. Exact pattern matching and matching statistics

The interval structure of deterministic Wheeler graphs supports exact pattern matching in much the same style as the FM-index. Given an interval $G=(V,E,\lambda)$7 of states and a character $G=(V,E,\lambda)$8, one can compute the interval of all states reachable by an outgoing $G=(V,E,\lambda)$9-edge from some state in $<$0. In the framework described for Wheeler DFAs, this takes $<$1, or $<$2 for constant alphabet, and the interval structure “plays the role of SA-intervals for strings” (Conte et al., 2023).

A further extension is the introduction of an LCP array for Wheeler automata. For each state $<$3, let $<$4 be the set of backward-read infinite strings reaching it, and define $<$5 and $<$6 as the lexicographically minimum and maximum strings in $<$7. The Wheeler-automaton LCP array has length $<$8 and alternates comparisons $<$9 and $V$0. By Fine–Wilf, each finite LCP value is $V$1 or is infinite, and with a succinct RMQ structure one supports $V$2-time RMQ and $V$3 prev-smaller/next-smaller queries (Conte et al., 2023).

This LCP array enables a generalization of matching statistics from strings to deterministic Wheeler automata. For a pattern $V$4, the objective is to find, for each $V$5, the longest suffix $V$6 that labels a path in the automaton together with the corresponding interval of states. The algorithm proceeds exactly as in the string setting of Ohlebusch–Gog–Kügell, but replaces suffix-tree components with forward-search on states and uses the Wheeler-automaton LCP array to jump when extension by the next character fails. The resulting complexity is $V$7 time and $V$8 words of space, with the LCP/RMQ/PSV/NSV structures taking $V$9 bits (Conte et al., 2023).

The significance of this development is that it establishes a first clear step toward suffix-tree-like functionality on labeled graphs. The ingredients listed explicitly in the literature are the state order as an analogue of the suffix array, the Wheeler LCP array as an analogue of string LCP, and constant-time RMQ together with $a\prec a'$0 PSV/NSV (Conte et al., 2023). This suggests that a substantial part of compressed suffix-tree methodology may be transportable to the deterministic Wheeler setting, although full navigation primitives remain outside the present corpus.

4. Minimization, canonical forms, and language-theoretic characterization

Deterministic Wheeler graphs are not only indexable objects but also a language-theoretic subclass of DFAs. A language is Wheeler if it is recognized by some Wheeler graph, and the foundational characterization is a Wheeler version of Myhill–Nerode. For a language $a\prec a'$1, one refines the usual right congruence $a\prec a'$2 on prefixes into a convex, input-consistent relation $a\prec a'$3 by requiring that equivalent prefixes end in the same letter and that every prefix between them in co-lex order belongs to the same equivalence class. The Wheeler Myhill–Nerode theorem states that the following are equivalent: $a\prec a'$4 is recognized by some WNFA; $a\prec a'$5 has finite index; $a\prec a'$6 is a union of classes of some finite-index convex, input-consistent, right-invariant equivalence; and $a\prec a'$7 is recognized by a WDFA (Alanko et al., 2020).

This theorem gives both a recognition criterion for Wheeler languages and a route to construction. It also shows that determinization behaves unusually well: if $a\prec a'$8 is a WNFA with $a\prec a'$9 states, then its determinization $e_1=(u_1\to v_1,\text{ label }a)$0, whose states are intervals $e_1=(u_1\to v_1,\text{ label }a)$1, is a WDFA with at most $e_1=(u_1\to v_1,\text{ label }a)$2 states (Alanko et al., 2020). In the deterministic hierarchy, minimization is further sharpened by convexity constraints.

Two complementary minimization theories appear in the recent literature. The first is phrased in terms of convex refinements of Myhill–Nerode classes. For a Wheeler language $e_1=(u_1\to v_1,\text{ label }a)$3, the quotient automaton on the classes of $e_1=(u_1\to v_1,\text{ label }a)$4 is a deterministic Wheeler DFA recognizing $e_1=(u_1\to v_1,\text{ label }a)$5 and minimal among Wheeler DFAs for $e_1=(u_1\to v_1,\text{ label }a)$6 (D'Agostino et al., 28 Jun 2026). The second is phrased through Wheeler bisimulations. Standard bisimulation does not preserve the relevant convex order structure, so Wheeler bisimulations impose additional convexity-preservation conditions: images and preimages of convex sets must remain convex. The largest reflexive Wheeler autobisimulation induces a quotient $e_1=(u_1\to v_1,\text{ label }a)$7 that is again a Wheeler DFA, and any two minimal Wheeler DFAs for the same language are isomorphic (Cotumaccio, 8 Feb 2026). In the deterministic case, this retrieves the minimum Wheeler deterministic automaton of a given language (Cotumaccio, 8 Feb 2026).

Algorithmically, the deterministic case supports particularly strong results. Given a Wheeler DFA with $e_1=(u_1\to v_1,\text{ label }a)$8 states and $e_1=(u_1\to v_1,\text{ label }a)$9 edges, its quotient by the largest Wheeler autobisimulation can be constructed in $e_2=(u_2\to v_2,\text{ label }a')$0 time, assuming edge labels can be radix-sorted from an integer range $e_2=(u_2\to v_2,\text{ label }a')$1 (Cotumaccio, 8 Feb 2026). The algorithm marks equivalence-class boundaries via a bit-vector $e_2=(u_2\to v_2,\text{ label }a')$2, initializes boundaries using local tests on outgoing labels and finality, and then propagates distinctions along carefully selected predecessor positions. A matching $e_2=(u_2\to v_2,\text{ label }a')$3 lower bound holds in the pure comparison model (Cotumaccio, 8 Feb 2026).

A distinct but related problem is to compute the minimum Wheeler DFA from an ordinary minimum DFA recognizing a Wheeler language. For general DFAs, a recent algorithm builds the minimum equivalent Wheeler DFA in $e_2=(u_2\to v_2,\text{ label }a')$4 time, where $e_2=(u_2\to v_2,\text{ label }a')$5 is the number of transitions of the output automaton (Becker et al., 8 Jul 2026). The method starts from a spanning tree, co-lex sorts root-to-node paths to obtain an initial Wheeler subgraph, and then incrementally inserts the remaining transitions by redirecting, splitting, or cloning states while maintaining the Wheeler order and language equivalence. When the input language is not Wheeler, the process does not terminate with an empty pending set, thereby witnessing non-Wheelerness (Becker et al., 8 Jul 2026).

Closure properties differentiate Wheeler languages from regular languages in familiar ways. Deterministic Wheeler languages are closed under intersection, but not under arbitrary union or complement; they are not closed under general concatenation or Kleene star; and they are closed under certain inverse images preserving co-lex monotonicity, such as inverse images of deterministic Wheeler transductions in the transducer setting (Alanko et al., 2020, D'Agostino et al., 28 Jun 2026). The literature repeatedly emphasizes that the complexity of deciding whether an arbitrary regular language is equivalent to some Wheeler DFA remains open (D'Agostino et al., 28 Jun 2026).

5. Cache-efficient indexing and pangenome pattern matching

The most recent algorithmic development for deterministic Wheeler graphs addresses the high cache-miss rate of FM-style forward search on Wheeler DFAs. The Graph Suffix Array (Gr-SA) is designed specifically for deterministic Wheeler automata and mimics suffix-array pattern location by combining binary search with sequential scans (Maso et al., 2 Jul 2026).

The key primitive is the suffix-match oracle

$e_2=(u_2\to v_2,\text{ label }a')$6

which locates in co-lex order the maximal open interval of the implicit sequence $e_2=(u_2\to v_2,\text{ label }a')$7 whose entries are all suffixed by $e_2=(u_2\to v_2,\text{ label }a')$8. Since $e_2=(u_2\to v_2,\text{ label }a')$9 is represented implicitly by two pseudoforests, one for all $a\prec a'\Rightarrow v_1<v_2$0 and one for all $a\prec a'\Rightarrow v_1<v_2$1, a comparison between $a\prec a'\Rightarrow v_1<v_2$2 and one of these extremal strings is implemented by walking backward along a heavy-light decomposition of the corresponding pseudotree, costing

$a\prec a'\Rightarrow v_1<v_2$3

cache misses per walk. A binary search over $a\prec a'\Rightarrow v_1<v_2$4 adds another $a\prec a'\Rightarrow v_1<v_2$5 factor, so a call to $a\prec a'\Rightarrow v_1<v_2$6 costs

$a\prec a'\Rightarrow v_1<v_2$7

I/Os (Maso et al., 2 Jul 2026).

Pattern location in Gr-SA is organized in three phases for an input $a\prec a'\Rightarrow v_1<v_2$8. Phase 1 uses $a\prec a'\Rightarrow v_1<v_2$9 calls to $a=a'$0 to find the longest prefix $a=a'$1 such that $a=a'$2. Phase 2 checks whether $a=a'$3, identifies the unique candidate state $a=a'$4, and verifies via rank/select on the Wheeler-NFA representation whether some state in $a=a'$5 has an $a=a'$6-transition. Phase 3 decomposes the automaton into maximal unary paths and traverses the remaining suffix $a=a'$7 by sequentially scanning those paths in memory, costing $a=a'$8 plus $a=a'$9 random jumps between paths (Maso et al., 2 Jul 2026).

The resulting I/O bound is explicit. For a WDFA $\Sigma$00 with $\Sigma$01 states and a pattern $\Sigma$02 of length $\Sigma$03, there is a data structure of $\Sigma$04 RAM words supporting locate/count in

$\Sigma$05

I/Os, where $\Sigma$06 is the number of unary paths traversed in the sequential-scan phase. If one only needs to know whether $\Sigma$07 occurs at least twice, a single suffix-match call suffices, giving

$\Sigma$08

I/Os using $\Sigma$09 words of space (Maso et al., 2 Jul 2026).

The empirical evaluation was carried out on real-world Wheeler pangenome graphs derived from Finnish-subset human chromosomes 21 and 14. The WDFA for chromosome 21 has $\Sigma$10 states and $\Sigma$11 transitions; chromosome 14 has $\Sigma$12 states and $\Sigma$13 transitions. Both are extremely sparse, with average out-degree $\Sigma$14 (Maso et al., 2 Jul 2026).

Graph	Forward-search space	Gr-SA space
Chr 21	$\Sigma$15 MB	$\Sigma$16 MB
Chr 14	$\Sigma$17 MB	$\Sigma$18 MB

The space blow-up is approximately $\Sigma$19 on chromosome 21 and $\Sigma$20 on chromosome 14. The performance gain, however, is much larger. Over $\Sigma$21 random patterns of length $\Sigma$22 or $\Sigma$23 drawn from the graph, average time per character was $\Sigma$24 ns/char for forward search versus $\Sigma$25 ns/char for Gr-SA on chromosome 21 when $\Sigma$26, and $\Sigma$27 ns/char versus $\Sigma$28 ns/char when $\Sigma$29. On chromosome 14 the corresponding numbers were $\Sigma$30 ns/char versus $\Sigma$31 ns/char for $\Sigma$32, and $\Sigma$33 ns/char versus $\Sigma$34 ns/char for $\Sigma$35, reaching a reported speedup of $\Sigma$36 in the latter case. The implementation was also able to process a single character of the pattern in less than $\Sigma$37 ns (Maso et al., 2 Jul 2026).

These results do not change the asymptotic compactness frontier of FM-style WDFA indexes, but they show that a different memory-access regime can dominate in practice on sparse pangenome graphs. The paper reports that empirically $\Sigma$38, typically by two orders of magnitude on the tested graphs, so the sequential scans are dominant yet highly cache-friendly (Maso et al., 2 Jul 2026).

6. Applications, extensions, and open directions

A recurrent application domain is computational pangenomics. Variation graphs built from populations of haplotypes, including graphs formed from all recorded SNPs in dbSNP within a single chromosome, can be prefixed-sorted into a Wheeler order and thereby become WDFAs (Maso et al., 2 Jul 2026). This enables exact graph indexing, construction of minimum Wheeler DFAs from graph-derived DFAs, and direct application of FM-style or cache-efficient search. In experiments on 23 pangenome graphs from the Finnish subset of dbSNP variations for human chromosomes, a recent implementation for constructing minimum Wheeler DFAs from general DFAs achieved throughput of over $\Sigma$39 transitions per second on a standard workstation; on chromosomes where GCSA terminated, it produced identical WDFA sizes while additionally guaranteeing minimality (Becker et al., 8 Jul 2026).

Deterministic Wheeler graphs also support graph-combinatorial tasks that are hard in general graphs. Counting distinct $\Sigma$40-mers is $\Sigma$41P-hard even on connected deterministic DAGs, but in deterministic Wheeler graphs distinct $\Sigma$42-mers can be counted using $\Sigma$43 or $\Sigma$44 arithmetic operations (Alanko et al., 26 Sep 2025). The linear-in-$\Sigma$45 method uses dynamic programming over lengths together with the fact that in a deterministic Wheeler graph two consecutive in-neighbors of a vertex can share at most one $\Sigma$46-mer, detected via an LCS/RMQ structure on infimum and supremum strings. The same work shows how to represent the de Bruijn graph of the $\Sigma$47-mers in time $\Sigma$48 and space

$\Sigma$49

bits (Alanko et al., 26 Sep 2025).

Beyond pure automata, the Wheeler paradigm has been extended to sequential transducers. Sequential Wheeler transducers combine a Wheeler automaton with a monotonicity requirement on outputs; they are closed under composition, and Wheeler languages are closed under inverse images of Wheeler transductions (D'Agostino et al., 28 Jun 2026). For the automaton-only setting, this places deterministic Wheeler graphs inside a broader structural theory rather than treating them solely as indexes.

Several limitations and open problems remain explicit in the literature. Arbitrary union does not preserve Wheelerness, nor do complement, unrestricted concatenation, factors, suffixes, or inversion in general (Alanko et al., 2020). The Wheeler-realizability problem for regular languages remains open in the formulations summarized in the transducer paper, while for arbitrary NFAs the corresponding decision and construction problems are substantially harder, including PSPACE-hardness for building a minimum Wheeler automaton from a nondeterministic input (D'Agostino et al., 28 Jun 2026, Becker et al., 8 Jul 2026). The transducer-oriented literature also identifies open directions concerning logical characterizations, bounded co-lex width, and nondeterministic extensions (D'Agostino et al., 28 Jun 2026).

Taken together, these results situate deterministic Wheeler graphs as a robust subclass of labeled deterministic automata with three simultaneous properties: a canonical state order, strong interval regularity for pattern matching, and a nontrivial minimization theory. This suggests a unifying role for deterministic Wheeler graphs in compressed indexing and formal-language structure, particularly where graph-shaped references must support exact search at scale.