Optimal label alphabets and the hypergraph/simplicial extension of the Sleator-Tarjan-Thurston derivation-encoding scheme

Sleator, Tarjan, and Thurston (SIAM J. Discrete Math. 5(3):428-450, 1992) encode derivations of a graph grammar as sequences of vertex labels over a small alphabet, obtaining the counting bound $|R(G,\Gamma,m)| \le A^{n+rm}$, where $A$ is the alphabet size, and thereby tight $\Theta(n\log n)$ lower bounds for transforming associative-commutative expressions and numbered plane triangulations. The paper leaves the encoding scheme itself unfinished in two ways it explicitly flags: the leader-follower technique reduces the alphabet from $|\Gamma|+b+1$ to $|\Gamma|+d+1$ by choosing leaders and directed spanning trees on the left sides of productions, and the authors state it 'may be possible to further reduce the alphabet size by making use of the flexibility' in that choice - a choice they never optimize; and the scheme is asserted to hold 'for more general combinatorial structures such as hypergraphs and simplicial complexes' without any formal construction, decodability lemma, or counting theorem for those structures. This problem asks for (A) the alphabet-optimization problem - minimizing the number $d$ of distinct direction labels over leader/spanning-tree choices, and more generally the minimal alphabet of any valid decodable encoding of the derivations of a given grammar, together with the tightness of the resulting counting bounds - and (B) the formal generalization of the encoding construction and its counting theorem to hypergraphs and simplicial complexes. The known applications - flip distance, tree metrics, growth rates - do not settle either question.

Background

A graph grammar is a finite set of productions $\{L_i \to R_i\}$: the left side $L_i$ is a connected graph whose vertices have degree at most $b$ and whose edge ends carry distinct edge-end labels from $\{1,\dots,b\}$ (these labels act as a local rotation system, recording how a replacement is glued into the surrounding graph); the right side $R_i$ is a similar graph with half-edges; and a fixed one-to-one correspondence matches the half-edges of $L_i$ to those of $R_i$. A production applies to a graph $G$ containing a subgraph isomorphic to $L_i$ (including edge-end labels) by replacing that occurrence with a copy of $R_i$ glued along the correspondence; one application is an action, and a derivation is a sequence of actions transforming a starting graph $G$ into a final graph $G'$.

Sleator, Tarjan, and Thurston observed that such derivations are redundant: independent actions commute, so what matters is which substructures get rewritten, not the interleaving order. They exploit this by giving every vertex of the derivation a label from a finite alphabet - label $0$ for vertices never destroyed, positive labels identifying the production and position that destroys the vertex - so that a derivation is encoded by the sequence of labels of its vertices, of length at most $n+rm$ ($n$ the size of $G$, $r$ the largest right side), and a decoder reconstructs the final graph by repeatedly applying any ready action, meaning one whose left-side pattern is matched by the current labels; the paper's Lemma 2.2 proves that label matches occur exactly at ready actions. Since each graph reachable in $m$ actions has such an encoding over an alphabet of size $A$, Theorem 2.3 gives $|R(G,\Gamma,m)| \le A^{n+rm}$: the number of graphs reachable in at most $m$ actions from $G$ is exponentially bounded with base equal to the alphabet size. This counting bound converts directly into distance lower bounds: if a starting graph has more reachable companions than the bound allows, two of them must be far apart. With this, the paper proved tight $\Theta(n\log n)$ worst-case bounds for transforming one fully parenthesized expression over a binary associative, commutative operation into an equivalent one by applications of the associative and commutative laws (closing a problem of Culik and Wood, who had the matching upper bound), and for transforming one $n$-vertex numbered plane triangulation into another (a plane triangulation being a maximal plane graph, all of whose faces are triangles, and 'numbered' meaning the vertices carry distinct labels, so triangulations are counted individually rather than up to isomorphism) by diagonal flips (a flip replaces the edge shared by two triangular faces by the other diagonal of the quadrilateral they bound), improving the previously known quadratic upper bound to $O(n\log n)$; the paper is a successor of the authors' 1988 study of rotation distance between triangulations via hyperbolic geometry.

The alphabet size $A$ is therefore not an implementation detail: it is the base of the exponential counting bound and the per-symbol cost of the compact encodings themselves. The paper reduces $A$ by refinements. Zero-elimination drops the label $0$ whenever every reachable graph admits a labeling with no ready action. The leader-follower technique designates one vertex of each left side the leader and the rest followers, chooses a spanning tree of each left side (possible because left sides are connected), and labels a follower $v$ by $|\Gamma|+\mathrm{DIR}(v)$, where $\mathrm{DIR}(v)$ is the edge-end label at $v$ of the first tree edge on the path from $v$ to the leader; following $\mathrm{DIR}$ labels from any follower leads to its leader, which is what the decoder's matching procedure checks. This reduces the alphabet from $|\Gamma|+b+1$ to $|\Gamma|+d+1$, where $d$ is the number of distinct direction labels actually used. In the applications the authors hand-tune these choices (the associative-commutative grammar goes from $6$ to $4$ labels; the flip grammar ends at $3$ labels plus a restriction to $8$ of the $9$ possible labelings of a created pair, giving the reachable-triangulation bound $3^{2n-4}\cdot 8^m$), but they never optimize the choice, writing: 'It may be possible to further reduce the alphabet size by making use of the flexibility that exists in choosing which spanning tree to use on each left side.'

Two further gaps are flagged in the same paper. The construction is stated to hold 'for more general combinatorial structures such as hypergraphs and simplicial complexes' - a hypergraph being a generalization of a graph in which an edge may contain any number of vertices, and a simplicial complex being a downward-closed family of such subsets (its faces), the natural home of higher-dimensional analogs of diagonal flips such as bistellar (Pachner) moves, local replacements in which, on a set of $k+2$ vertices of a triangulated $k$-dimensional complex, the $k$-dimensional simplices present are exchanged for the complementary ones on the same vertex set (for $k=2$ this is exactly the diagonal flip) - but no formal grammar model, encoding, decodability lemma, or counting theorem is given for those structures. And the authors 'envision several other applications' - compact representation of structures close to a given one, in the spirit of persistent data structures (data structures that keep old versions readable while they are being modified), and enumeration of grammar-generated graphs with far fewer redundant isomorphic copies - without algorithms or complexity analyses. A hypergraph or simplicial formalization must in particular find the analog of the directed spanning tree on a left side that is not a graph, and reprove the ready-action, label-match correspondence for sub-hypergraph matching.

Problem Statement

Part A (alphabet optimality). Fix a graph grammar $\Gamma$ with degree bound $b$ whose left sides are connected. The leader-follower encoding is valid for every choice of a leader and a spanning tree on each left side, and its alphabet size is $|\Gamma|+d+1$, where $d$ is the number of distinct edge-end labels occurring as first-edge directions toward leaders ($d \le b$); zero-elimination removes one further symbol when its hypothesis holds. Determine how small the alphabet can be made: (i) the optimization problem: given $\Gamma$, choose leaders and spanning trees on the left sides so as to minimize $d$ - give an algorithm (exact, or with a proven approximation guarantee) and settle the computational complexity of this optimization; (ii) the foundational problem: determine the minimum alphabet size of any valid decodable encoding of the derivations of $\Gamma$ - that is, over all vertex-label assignments of the paper's form: each vertex labelled from data about the production and position that destroys it (label $0$ if it survives), encoding length at most $n+rm$, order-independent, and injective on the reachable graphs - not restricted to the leader-follower form, and decide whether the counting bounds $|R(G,\Gamma,m)| \le A^{n+rm}$ so obtained are tight, i.e., whether they match information-theoretic lower bounds on $|R(G,\Gamma,m)|$ for the paper's associative-commutative grammar and diagonal-flip grammar, or for a wider grammar class delimited in the answer, so that no further alphabet reduction of this method is possible.

Part B (general structures). Construct the formal counterpart of the encoding scheme for the broader combinatorial structures the paper asserts but does not develop: a grammar model for hypergraphs and simplicial complexes (productions replacing connected sub-hypergraphs or subcomplexes, with appropriate notions of edge-end labels and correspondence), the ENCODE/DECODE construction, a proof of the ready-action, label-match correspondence (the analog of the paper's Lemma 2.2, including whatever replaces the directed spanning tree on a left side that is not a graph), and the counting theorem bounding the number of reachable structures; identify the applicability limits and the adaptations the generalization requires.

Relationship between the parts: A and B concern the same encoding construction - its single governing parameter and its scope. Together they answer how tight and how general the Sleator-Tarjan-Thurston derivation-encoding machinery is: every unit of alphabet reduction in Part A directly lowers the base of the exponential counting bound and hence sharpens every lower bound derivable by the method (as the paper's own hand-tuning demonstrates), while Part B determines which rewriting systems the method covers at all. The applied directions the paper additionally envisions (compact persistent multi-version representations, and enumeration of grammar-generated graphs with fewer redundant isomorphic copies) are context rather than deliverables: the source states no complexity targets for them, so they are outside the completion boundary of this problem.

References

1. Daniel D. Sleator, Robert E. Tarjan, and William P. Thurston, Short Encodings of Evolving Structures, SIAM Journal on Discrete Mathematics 5(3):428-450, 1992. DOI: 10.1137/0405034, https://doi.org/10.1137/0405034 (author's copy: https://www.cs.cmu.edu/~sleator/papers/evolving-structures.pdf)
2. Karel Culik II and Derick Wood, A note on some tree similarity measures, Information Processing Letters 15(1):39-42, 1982. DOI: 10.1016/0020-0190(82)90083-7, https://doi.org/10.1016/0020-0190(82)90083-7
3. Emo Welzl, Encoding graphs by derivations and implications for the theory of graph grammars, in Automata, Languages and Programming (ICALP 1984), Lecture Notes in Computer Science 172, Springer, 1984, pp. 503-513. DOI: 10.1007/3-540-13345-3_47, https://doi.org/10.1007/3-540-13345-3_47
4. James R. Driscoll, Neil Sarnak, Daniel D. Sleator, and Robert E. Tarjan, Making data structures persistent, Journal of Computer and System Sciences 38(1):86-124, 1989. DOI: 10.1016/0022-0000(89)90034-2, https://doi.org/10.1016/0022-0000(89)90034-2
5. Daniel D. Sleator, Robert E. Tarjan, and William P. Thurston, Rotation distance, triangulations, and hyperbolic geometry, Journal of the American Mathematical Society 1(3):647-681, 1988. DOI: 10.1090/S0894-0347-1988-0928904-4, https://doi.org/10.1090/S0894-0347-1988-0928904-4
6. Ming Li and Louxin Zhang, Better Approximation of Diagonal-Flip Transformation and Rotation Transformation, in Computing and Combinatorics (COCOON 1998), Lecture Notes in Computer Science 1449, Springer, 1998, pp. 85-94. DOI: 10.1007/3-540-68535-9_12, https://doi.org/10.1007/3-540-68535-9_12
7. Ming Li and Louxin Zhang, Twist–Rotation Transformations of Binary Trees and Arithmetic Expressions, Journal of Algorithms 32(2):155-166, 1999. DOI: 10.1006/jagm.1999.1010, https://doi.org/10.1006/jagm.1999.1010
8. Paul Balister, Béla Bollobás, and Stefanie Gerke, Sequences with Changing Dependencies, SIAM Journal on Discrete Mathematics 22(3):1149-1154, 2008. DOI: 10.1137/060663611, https://doi.org/10.1137/060663611
9. Prosenjit Bose and Sander Verdonschot, A History of Flips in Combinatorial Triangulations, in Computational Geometry (XIV Spanish Meeting on Computational Geometry, EGC 2011), Lecture Notes in Computer Science 7579, Springer, 2012, pp. 29-44. DOI: 10.1007/978-3-642-34191-5_3, https://doi.org/10.1007/978-3-642-34191-5_3
10. Alex Gavryushkin, Chris Whidden, and Frederick A. Matsen IV, The combinatorics of discrete time-trees: theory and open problems, Journal of Mathematical Biology 76(5):1101-1121, 2018. DOI: 10.1007/s00285-017-1167-9, https://doi.org/10.1007/s00285-017-1167-9 (published online 29 July 2017; print issue date April 2018)
11. Anna Lubiw and Vinayak Pathak, Flip distance between two triangulations of a point set is NP-complete, Computational Geometry 49:17-23, 2015. DOI: 10.1016/j.comgeo.2014.11.001, https://doi.org/10.1016/j.comgeo.2014.11.001
12. Iyad Kanj, Eric Sedgwick, and Ge Xia, Computing the Flip Distance Between Triangulations, Discrete & Computational Geometry 58(2):313-344, 2017. DOI: 10.1007/s00454-017-9867-x, https://doi.org/10.1007/s00454-017-9867-x
13. Kasra Rafi and Jing Tao, Uniform growth rate, Proceedings of the American Mathematical Society 144(4):1415-1427, 2015. DOI: 10.1090/PROC/12816, https://doi.org/10.1090/PROC/12816