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Documents authored by Williams, Aaron

Document
DOI: 10.4230/LIPIcs.ISAAC.2022.53
Pop & Push: Ordered Tree Iteration in 𝒪(1)-Time

Authors: Paul Lapey and Aaron Williams

Published in: LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

Abstract
The number of ordered trees (also known as plane trees) with n nodes is the (n-1)st Catalan number C_{n-1}. An ordered tree can be stored directly using nodes and pointers, or represented indirectly by a Dyck word. This paper presents a loopless algorithm for generating ordered trees with n nodes using pointer-based representations. In other words, we spend 𝒪(C_{n-1})-time to generate all of the trees, and moreover, the delay between consecutive trees is worst-case 𝒪(1)-time. To achieve this run-time, each tree must differ from the previous by a constant amount. In other words, the algorithm must create a type of Gray code order. Our algorithm operates on the children of a node like a stack, by popping the first child off of one node’s stack and pushing the result onto another node’s stack. We refer to this pop-push operation as a pull, and consecutive trees in our order differ by one or two pulls. There is a simple two-case successor rule that determines the pulls to apply directly from the current tree. When converted to Dyck words, our rule corresponds to a left-shift, and these shift generate a cool-lex variant of lexicographic order. Our results represent the first pull Gray code for ordered trees, and the first fully published loopless algorithm for ordered trees using pointer representations. More importantly, our algorithm is incredibly simple: A full implementation in C, including initialization and output, uses only three loops and three if-else blocks. Our work also establishes a simultaneous Gray code for Dyck words, ordered trees, and also binary trees, using cool-lex order.
Cite as

Paul Lapey and Aaron Williams. Pop & Push: Ordered Tree Iteration in 𝒪(1)-Time. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 53:1-53:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{lapey_et_al:LIPIcs.ISAAC.2022.53,
  author =	{Lapey, Paul and Williams, Aaron},
  title =	{{Pop \& Push: Ordered Tree Iteration in 𝒪(1)-Time}},
  booktitle =	{33rd International Symposium on Algorithms and Computation (ISAAC 2022)},
  pages =	{53:1--53:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-258-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{248},
  editor =	{Bae, Sang Won and Park, Heejin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.53},
  URN =		{urn:nbn:de:0030-drops-173380},
  doi =		{10.4230/LIPIcs.ISAAC.2022.53},
  annote =	{Keywords: combinatorial generation, Gray code, simultaneous Gray code, ordered trees, plane trees, Dyck words, binary trees, Catalan objects, loopless algorithm, cool-lex order}
}
Document
DOI: 10.4230/LIPIcs.FUN.2022.6
Rolling Polyhedra on Tessellations

Authors: Akira Baes, Erik D. Demaine, Martin L. Demaine, Elizabeth Hartung, Stefan Langerman, Joseph O'Rourke, Ryuhei Uehara, Yushi Uno, and Aaron Williams

Published in: LIPIcs, Volume 226, 11th International Conference on Fun with Algorithms (FUN 2022)

Abstract
We study the space reachable by rolling a 3D convex polyhedron on a 2D periodic tessellation in the xy-plane, where at every step a face of the polyhedron must coincide exactly with a tile of the tessellation it rests upon, and the polyhedron rotates around one of the incident edges of that face until the neighboring face hits the xy plane. If the whole plane can be reached by a sequence of such rolls, we call the polyhedron a plane roller for the given tessellation. We further classify polyhedra that reach a constant fraction of the plane, an infinite area but vanishing fraction of the plane, or a bounded area as hollow-plane rollers, band rollers, and bounded rollers respectively. We present a polynomial-time algorithm to determine the set of tiles in a given periodic tessellation reachable by a given polyhedron from a given starting position, which in particular determines the roller type of the polyhedron and tessellation. Using this algorithm, we compute the reachability for every regular-faced convex polyhedron on every regular-tiled (≤ 4)-uniform tessellation.
Cite as

Akira Baes, Erik D. Demaine, Martin L. Demaine, Elizabeth Hartung, Stefan Langerman, Joseph O'Rourke, Ryuhei Uehara, Yushi Uno, and Aaron Williams. Rolling Polyhedra on Tessellations. In 11th International Conference on Fun with Algorithms (FUN 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 226, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{baes_et_al:LIPIcs.FUN.2022.6,
  author =	{Baes, Akira and Demaine, Erik D. and Demaine, Martin L. and Hartung, Elizabeth and Langerman, Stefan and O'Rourke, Joseph and Uehara, Ryuhei and Uno, Yushi and Williams, Aaron},
  title =	{{Rolling Polyhedra on Tessellations}},
  booktitle =	{11th International Conference on Fun with Algorithms (FUN 2022)},
  pages =	{6:1--6:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-232-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{226},
  editor =	{Fraigniaud, Pierre and Uno, Yushi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2022.6},
  URN =		{urn:nbn:de:0030-drops-159761},
  doi =		{10.4230/LIPIcs.FUN.2022.6},
  annote =	{Keywords: polyhedra, tilings}
}
Document
DOI: 10.4230/LIPIcs.FUN.2022.22
All Your bases Are Belong to Us: Listing All Bases of a Matroid by Greedy Exchanges

Authors: Arturo Merino, Torsten Mütze, and Aaron Williams

Published in: LIPIcs, Volume 226, 11th International Conference on Fun with Algorithms (FUN 2022)

Abstract
You provide us with a matroid and an initial base. We say that a subset of the bases "belongs to us" if we can visit each one via a sequence of base exchanges starting from the initial base. It is well-known that "All your base are belong to us". We refine this classic result by showing that it can be done by a simple greedy algorithm. For example, the spanning trees of a graph can be generated by edge exchanges using the following greedy rule: Minimize the larger label of an edge that enters or exits the current spanning tree and which creates a spanning tree that is new (i.e., hasn't been visited already). Amazingly, this works for any graph, for any labeling of its edges, for any initial spanning tree, and regardless of how you choose the edge with the smaller label in each exchange. Furthermore, by maintaining a small amount of information, we can generate each successive spanning tree without storing the previous trees. In general, for any matroid, we can greedily compute a listing of all its bases matroid such that consecutive bases differ by a base exchange. Our base exchange Gray codes apply a prefix-exchange on a prefix-minor of the matroid, and we can generate these orders using "history-free" iterative algorithms. More specifically, we store O(m) bits of data, and use O(m) time per base assuming O(1) time independence and coindependence oracles. Our work generalizes and extends a number of previous results. For example, the bases of the uniform matroid are combinations, and they belong to us using homogeneous transpositions via an Eades-McKay style order. Similarly, the spanning trees of fan graphs belong to us via face pivot Gray codes, which extends recent results of Cameron, Grubb, and Sawada [Pivot Gray Codes for the Spanning Trees of a Graph ft. the Fan, COCOON 2021].
Cite as

Arturo Merino, Torsten Mütze, and Aaron Williams. All Your bases Are Belong to Us: Listing All Bases of a Matroid by Greedy Exchanges. In 11th International Conference on Fun with Algorithms (FUN 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 226, pp. 22:1-22:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{merino_et_al:LIPIcs.FUN.2022.22,
  author =	{Merino, Arturo and M\"{u}tze, Torsten and Williams, Aaron},
  title =	{{All Your bases Are Belong to Us: Listing All Bases of a Matroid by Greedy Exchanges}},
  booktitle =	{11th International Conference on Fun with Algorithms (FUN 2022)},
  pages =	{22:1--22:28},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-232-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{226},
  editor =	{Fraigniaud, Pierre and Uno, Yushi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2022.22},
  URN =		{urn:nbn:de:0030-drops-159928},
  doi =		{10.4230/LIPIcs.FUN.2022.22},
  annote =	{Keywords: Matroids, base exchange, Gray codes, combinatorial generation, greedy algorithms, spanning trees}
}
Document
DOI: 10.4230/LIPIcs.FUN.2016.13
Super Mario Bros. is Harder/Easier Than We Thought

Authors: Erik D. Demaine, Giovanni Viglietta, and Aaron Williams

Published in: LIPIcs, Volume 49, 8th International Conference on Fun with Algorithms (FUN 2016)

Abstract
Mario is back! In this sequel, we prove that solving a generalized level of Super Mario Bros. is PSPACE-complete, strengthening the previous NP-hardness result (FUN 2014). Both our PSPACE-hardness and the previous NP-hardness use levels of arbitrary dimensions and require either arbitrarily large screens or a game engine that remembers the state of off-screen sprites. We also analyze the complexity of the less general case where the screen size is constant, the number of on-screen sprites is constant, and the game engine forgets the state of everything substantially off-screen, as in most, if not all, Super Mario Bros. video games. In this case we prove that the game is solvable in polynomial time, assuming levels are explicitly encoded; on the other hand, if levels can be represented using run-length encoding, then the problem is weakly NP-hard (even if levels have only constant height, as in the video games). All of our hardness proofs are also resilient to known glitches in Super Mario Bros., unlike the previous NP-hardness proof.
Cite as

Erik D. Demaine, Giovanni Viglietta, and Aaron Williams. Super Mario Bros. is Harder/Easier Than We Thought. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{demaine_et_al:LIPIcs.FUN.2016.13,
  author =	{Demaine, Erik D. and Viglietta, Giovanni and Williams, Aaron},
  title =	{{Super Mario Bros. is Harder/Easier Than We Thought}},
  booktitle =	{8th International Conference on Fun with Algorithms (FUN 2016)},
  pages =	{13:1--13:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-005-7},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{49},
  editor =	{Demaine, Erik D. and Grandoni, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2016.13},
  URN =		{urn:nbn:de:0030-drops-58802},
  doi =		{10.4230/LIPIcs.FUN.2016.13},
  annote =	{Keywords: video games, computational complexity, PSPACE}
}
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