Minimal locked trees

Brad Ballinger, David Charlton, Erik D. Demaine, Martin L. Demaine, John Iacono, Ching Hao Liu, Sheung Hung Poon

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Abstract

    Locked tree linkages have been known to exist in the plane since 1998, but it is still open whether they have a polynomial-time characterization. This paper examines the properties needed for planar trees to lock, with a focus on finding the smallest locked trees according to different measures of complexity, and suggests some new avenues of research for the problem of algorithmic characterization. First we present a locked linear tree with only eight edges. In contrast, the smallest previous locked tree has 15 edges. We further show minimality by proving that every locked linear tree has at least eight edges. We also show that a six-edge tree can interlock with a four-edge chain, which is the first locking result for individually unlocked trees. Next we present several new examples of locked trees with varying minimality results. Finally, we provide counterexamples to two conjectures of [12], [13] by showing the existence of two new types of locked tree: a locked orthogonal tree (all edges horizontal and vertical) and a locked equilateral tree (all edges unit length).

    Original languageEnglish (US)
    Title of host publicationAlgorithms and Data Structures - 11th International Symposium, WADS 2009, Proceedings
    Pages61-73
    Number of pages13
    Volume5664 LNCS
    DOIs
    StatePublished - 2009
    Event11th International Symposium on Algorithms and Data Structures, WADS 2009 - Banff, AB, Canada
    Duration: Aug 21 2009Aug 23 2009

    Publication series

    NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
    Volume5664 LNCS
    ISSN (Print)03029743
    ISSN (Electronic)16113349

    Other

    Other11th International Symposium on Algorithms and Data Structures, WADS 2009
    CountryCanada
    CityBanff, AB
    Period8/21/098/23/09

    Fingerprint

    Polynomials
    Minimality
    Equilateral
    Locking
    Linkage
    Counterexample
    Polynomial time
    Horizontal
    Vertical
    Unit

    ASJC Scopus subject areas

    • Computer Science(all)
    • Theoretical Computer Science

    Cite this

    Ballinger, B., Charlton, D., Demaine, E. D., Demaine, M. L., Iacono, J., Liu, C. H., & Poon, S. H. (2009). Minimal locked trees. In Algorithms and Data Structures - 11th International Symposium, WADS 2009, Proceedings (Vol. 5664 LNCS, pp. 61-73). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5664 LNCS). https://doi.org/10.1007/978-3-642-03367-4_6

    Minimal locked trees. / Ballinger, Brad; Charlton, David; Demaine, Erik D.; Demaine, Martin L.; Iacono, John; Liu, Ching Hao; Poon, Sheung Hung.

    Algorithms and Data Structures - 11th International Symposium, WADS 2009, Proceedings. Vol. 5664 LNCS 2009. p. 61-73 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5664 LNCS).

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Ballinger, B, Charlton, D, Demaine, ED, Demaine, ML, Iacono, J, Liu, CH & Poon, SH 2009, Minimal locked trees. in Algorithms and Data Structures - 11th International Symposium, WADS 2009, Proceedings. vol. 5664 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 5664 LNCS, pp. 61-73, 11th International Symposium on Algorithms and Data Structures, WADS 2009, Banff, AB, Canada, 8/21/09. https://doi.org/10.1007/978-3-642-03367-4_6
    Ballinger B, Charlton D, Demaine ED, Demaine ML, Iacono J, Liu CH et al. Minimal locked trees. In Algorithms and Data Structures - 11th International Symposium, WADS 2009, Proceedings. Vol. 5664 LNCS. 2009. p. 61-73. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-642-03367-4_6
    Ballinger, Brad ; Charlton, David ; Demaine, Erik D. ; Demaine, Martin L. ; Iacono, John ; Liu, Ching Hao ; Poon, Sheung Hung. / Minimal locked trees. Algorithms and Data Structures - 11th International Symposium, WADS 2009, Proceedings. Vol. 5664 LNCS 2009. pp. 61-73 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
    @inproceedings{1f3137cf4cd64eb3a505ed725298f5f1,
    title = "Minimal locked trees",
    abstract = "Locked tree linkages have been known to exist in the plane since 1998, but it is still open whether they have a polynomial-time characterization. This paper examines the properties needed for planar trees to lock, with a focus on finding the smallest locked trees according to different measures of complexity, and suggests some new avenues of research for the problem of algorithmic characterization. First we present a locked linear tree with only eight edges. In contrast, the smallest previous locked tree has 15 edges. We further show minimality by proving that every locked linear tree has at least eight edges. We also show that a six-edge tree can interlock with a four-edge chain, which is the first locking result for individually unlocked trees. Next we present several new examples of locked trees with varying minimality results. Finally, we provide counterexamples to two conjectures of [12], [13] by showing the existence of two new types of locked tree: a locked orthogonal tree (all edges horizontal and vertical) and a locked equilateral tree (all edges unit length).",
    author = "Brad Ballinger and David Charlton and Demaine, {Erik D.} and Demaine, {Martin L.} and John Iacono and Liu, {Ching Hao} and Poon, {Sheung Hung}",
    year = "2009",
    doi = "10.1007/978-3-642-03367-4_6",
    language = "English (US)",
    isbn = "3642033660",
    volume = "5664 LNCS",
    series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
    pages = "61--73",
    booktitle = "Algorithms and Data Structures - 11th International Symposium, WADS 2009, Proceedings",

    }

    TY - GEN

    T1 - Minimal locked trees

    AU - Ballinger, Brad

    AU - Charlton, David

    AU - Demaine, Erik D.

    AU - Demaine, Martin L.

    AU - Iacono, John

    AU - Liu, Ching Hao

    AU - Poon, Sheung Hung

    PY - 2009

    Y1 - 2009

    N2 - Locked tree linkages have been known to exist in the plane since 1998, but it is still open whether they have a polynomial-time characterization. This paper examines the properties needed for planar trees to lock, with a focus on finding the smallest locked trees according to different measures of complexity, and suggests some new avenues of research for the problem of algorithmic characterization. First we present a locked linear tree with only eight edges. In contrast, the smallest previous locked tree has 15 edges. We further show minimality by proving that every locked linear tree has at least eight edges. We also show that a six-edge tree can interlock with a four-edge chain, which is the first locking result for individually unlocked trees. Next we present several new examples of locked trees with varying minimality results. Finally, we provide counterexamples to two conjectures of [12], [13] by showing the existence of two new types of locked tree: a locked orthogonal tree (all edges horizontal and vertical) and a locked equilateral tree (all edges unit length).

    AB - Locked tree linkages have been known to exist in the plane since 1998, but it is still open whether they have a polynomial-time characterization. This paper examines the properties needed for planar trees to lock, with a focus on finding the smallest locked trees according to different measures of complexity, and suggests some new avenues of research for the problem of algorithmic characterization. First we present a locked linear tree with only eight edges. In contrast, the smallest previous locked tree has 15 edges. We further show minimality by proving that every locked linear tree has at least eight edges. We also show that a six-edge tree can interlock with a four-edge chain, which is the first locking result for individually unlocked trees. Next we present several new examples of locked trees with varying minimality results. Finally, we provide counterexamples to two conjectures of [12], [13] by showing the existence of two new types of locked tree: a locked orthogonal tree (all edges horizontal and vertical) and a locked equilateral tree (all edges unit length).

    UR - http://www.scopus.com/inward/record.url?scp=69949184033&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=69949184033&partnerID=8YFLogxK

    U2 - 10.1007/978-3-642-03367-4_6

    DO - 10.1007/978-3-642-03367-4_6

    M3 - Conference contribution

    SN - 3642033660

    SN - 9783642033667

    VL - 5664 LNCS

    T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

    SP - 61

    EP - 73

    BT - Algorithms and Data Structures - 11th International Symposium, WADS 2009, Proceedings

    ER -