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Hamiltonian Cycles in Triangular Grids. Valentin Polishchuk joint work with Estie Arkin and Joseph Mitchell. Applied Math and Statistics Stony Brook University. Grids. Grids, grids. Grids, grids, grids…. Square Grid Graph. Subset S of Z 2 vertices: S
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Hamiltonian Cycles in Triangular Grids Valentin Polishchuk joint work with Estie Arkin and Joseph Mitchell Applied Math and Statistics Stony Brook University
Square Grid Graph • Subset S of Z2 • vertices: S • edge (i,j) if |Si –Sj | = 1 • Solid grid • no “holes” • all bounded faces – unit squares
Hamiltonicity of Square Grids • NP-compete in general [Itai, Papadimitriou, and Szwarcfiter ’82] • even if max deg = 3 [Papadimitriou and Vazirani ’84, Buro ‘03] • Solid grids • polynomial [Umans and Lenhart ’96] • short covering tour [Arkin, Fekete, Mitchell ‘92] • length (6/5)N • linear time
Tilings • Square grid • unit squares
Tilings • Square grid • unit squares • Triangular grid • unit equilateral triangles
Triangular Grid Graph Subset S • vertices: S • edge (i,j) if |Si –Sj | = 1
Solid Triangular Grid No “holes” all bounded faces – unit equilateral triangles
Previous Work • Other triangulations [Dillencourt ’92, Arkin, Held, Mitchell, Skiena ’96, Cimikowski ’90, ’93, Dogrusoz and Krishnamoorthy ’95, Flatland ‘04] • Long cycles through faces • stripification [Bushan, Diaz-Gutierrez, Eppstein, Gopi’ 04,’06] • possible subdivision • relaxed notion of Hamiltonicity [Demaine, Eppstein, Erickson, Hart, O'Rourke ‘01] Hamiltonian cycle through vertices?
Our Contribution • NP-compete in general • even if max deg = 4 • Solid triangular grids • polynomial • cut-free – Hamiltonian • not the Star of David • linear-time to find a cycle • linear-time solution to TSP
The Reduction • Same idea as for square grids [Itai, Papadimitriou, and Szwarcfiter ‘82, Papadimitriou and Vazirani ‘84] • Hamiltonian Cycle • undirected planar bipartite graphs • max deg 3 G0 Embed 0o, 60o, 120o segments
Traversing Tentacles Black node-tentacle connection • a cut Return path Cross path returns to white node connects white and black nodes
HC in G HC in G0 Any node gadget adjacent to 2 cross paths Return path Cross path • Edges of G0 in HC Cross paths • Edges of G0 not in HC Return paths from white nodes
Max degree 4 Grids • No degree-6 vertices • Degree-5 vertices Modified gadgets
Assumption No cut vertex • removal disconnects G WLOG • o.w. – no cycle
Boundary • deg < 6 vertices • Internal vertices • deg-6 vertices • No cut vertex • cycle B around the boundary • visits every bd vertex once
Idea • B – cycle around the boundary • Local modifications • attach to B internal vertices • cost: 1 per vertex
Priority: L , V , Z • L • V • Z
Wedges • Sharp • 60o turn • Wide • 120o turn
The Main Lemma Until B passes through ALL internal vertices • either L, V, or Z may be applied small print: unless G is the Star of David
Internal vertex v not in B • A neighbor u is in B • Crossed edges • not in B • o.w. – apply L
How is u visited? WLOG, 1 is in B
s L cannot be applied s is in B How is s visited?
Sharp Wedge Z s V s
Wide Wedge L cannot be applied t is in B
s Deja Vu Rhombus • edge of B • vertex not in B • vertex in B Unless • t is a wide wedge • modification! • welcome new vertex to B
Another Wide Wedge Yet Another vertex • Yet Another rhombus Yet Another wide wedge
And so on… Star of David!
Summary • Triangular grids • solid • Hamiltonian Cycle Problem • NP-compete even if max deg = 4 • Solid triangular grids • cut-free – Hamiltonian • not the Star of David • linear-time to find a cycle • linear-time solution to TSP ”Topological” proof
Extensions • Grids with holes • no “local cut” • Manifolds (meshes) • some assumptions
Open • Max degree 3 • Hexagonal grids Thank you!