1 / 28

Complexity of Graph Self-Assembly: Accretive and Self-Destructible Systems

Explore the complexity of self-assembly in accretive and self-destructible systems, analyzing Hamiltonian paths, sorting, and the AGAP-DGAP hierarchy.

Download Presentation

Complexity of Graph Self-Assembly: Accretive and Self-Destructible Systems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. 1 Complexity of Graph Self-Assembly in Accretive Systems and Self-Destructible Systems Peng Yin Joint with John H. Reif and Sudheer Sahu Department of Computer Science, Duke University DNA11, June 7th, 2005

  2. Self-Assembly: Small objects autonomously associate into larger complex Scientific importance: Ubiquitous phenomena in nature Crystal salt Eukaryotic cell Engineering significance: Powerful nano-scale & meso-scale construction methods Algorithmic DNA lattice (Rothemund et al 04) Autonomous DNA walker (Yin et al 04) 2 Motivation: Self-Assembly

  3. 3 Motivation: Complexity Theoretical Study of Self-Assembly How complex?

  4. Hamiltonian Path ? How many H Paths? …… Sorting PSPACE 11, 3, 10, 25, 6 Who wins? Playing GO PSPACE-Complete 3 < 6 < 10 < 11 < 25 #P Counting Hamiltonian Path #P- Complete NP Hamiltonian Path NP-Complete P Sorting 4 Complexity 101 Complexity Hierarchy

  5. Complexity Hierarchy Self-Assembly Model, Problems …… ? PSPACE PSPACE-Complete Formalize #P #P-Complete NP NP-Complete P 5 Motivation: Complexity Theoretical Study of Self-Assembly

  6. AGAP-PAGAP-#AGAP-DGAP Accretive Graph Assembly Problem • AGAP is NP-complete • Planar AGAP is NP-complete • #AGAP/Stochastic AGAP is #P-complete Self-Destructible Graph Assembly Problem • DGAP is PSPCAE-complete 6 Roadmap Complexity Hierarchy …… PSPACE PSPACE-Complete #P #P-Complete NP NP-Complete P

  7. AGAP-PAGAP-#AGAP-DGAP 7 Roadmap Complexity Hierarchy Accretive Graph Assembly Problem …… • AGAP is NP-complete • Planar AGAP is NP-complete • #AGAP/Stochastic AGAP is #P-complete PSPACE PSPACE-Complete Self-Destructible Graph Assembly Problem #P • DGAP is PSPCAE-complete #P-Complete NP NP-Complete P

  8. AGAP-PAGAP-#AGAP-DGAP Seed vertex Temperature Graph Weight function Temperature: τ = 2 Seed vertex 8 Accretive Graph Assembly System Sequentially constructible?

  9. AGAP-PAGAP-#AGAP-DGAP 9 Example: An assembly ordering Assembly Ordering Temperature =2 Seed vertex

  10. AGAP-PAGAP-#AGAP-DGAP 10 Example Temperature = 2 Stuck!

  11. AGAP-PAGAP-#AGAP-DGAP 11 Accretive Graph Assembly Problem Seed vertex Temperature Graph Weight function Temperature: τ = 2 Accretive Graph Assembly Problem: Given an accretive graph assembly system, determine whether there exists an assembly ordering to sequentially assemble the given target graph. Seed vertex

  12. AGAP-PAGAP-#AGAP-DGAP Hamiltonian Path ? 12 Roadmap Complexity Hierarchy Accretive Graph Assembly Problem …… • AGAP is NP-complete • Planar AGAP is NP-complete • #AGAP/Stochastic AGAP is #P-complete PSPACE PSPACE-Complete Self-Destructible Graph Assembly Problem #P • DGAP is PSPCAE-complete #P-Complete NP NP-Complete P

  13. AGAP-PAGAP-#AGAP-DGAP • Restricted 3SAT: each variable appears ≤ 3, literal ≤ 2 Top v. Literal v. Bottom v. 13 AGAP is NP-complete • AGAP is in NP • AGAP is NP-hard, using reduction from 3SAT

  14. AGAP-PAGAP-#AGAP-DGAP Seed vertex 2 2 2 2 2 2 2 2 2 2 2 -1 -1 -1 -1 2 2 2 2 2 2 -1 -1 2 2 2 14 AGAP is NP-complete • AGAP is in NP • AGAP is NP-hard, using reduction from 3SAT • Restricted 3SAT: each variable appears < 3, literal < 2 Top v. Literal v. Bottom v. Temperature = 2

  15. AGAP-PAGAP-#AGAP-DGAP 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 15 AGAP is NP-complete Proposition: φ is satisfiable⇔ exists an assembly ordering Seed vertex -1 -1 -1 -1 -1 -1 Temperature = 2

  16. AGAP-PAGAP-#AGAP-DGAP 2 2 Stage 1 2 2 2 2 2 2 2 2 2 Stage 2 2 2 2 Stage 4 2 2 2 2 2 2 Stage 3 16 AGAP is NP-complete φ is satisfiable ⇒ exists an assembly ordering T T T FF T T F T Seed vertex -1 -1 -1 -1 -1 -1 Temperature = 2

  17. AGAP-PAGAP-#AGAP-DGAP 2 2 2 2 2 2 2 2 2 2 2 2 F F F -1 2 2 2 2 2 2 2 2 2 17 AGAP is NP-complete φ is satisfiable⇐ exists an assembly ordering Exists at least one TRUE literal in each clause; proof by contradiction Total support ≤-1+2=1< 2 = temperature! Seed vertex -1 -1 -1 -1 -1 -1 Temperature = 2

  18. AGAP-PAGAP-#AGAP-DGAP 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 18 AGAP is NP-complete • Theorem: AGAP is NP-complete Seed vertex -1 -1 -1 -1 -1 -1 Temperature = 2

  19. AGAP-PAGAP-#AGAP-DGAP Hamiltonian Path ? 19 Roadmap Complexity Hierarchy Accretive Graph Assembly Problem …… • AGAP is NP-complete • Planar AGAP is NP-complete • #AGAP/Stochastic AGAP is #P-complete PSPACE PSPACE-Complete Self-Destructible Graph Assembly Problem #P • DGAP is PSPCAE-complete #P-Complete NP NP-Complete P

  20. AGAP-PAGAP-#AGAP-DGAP Seed vertex 20 Planar-AGAP • Planar AGAP is NP-complete;reduction from Planar-3SAT Reduction gadget Planar-3SAT

  21. AGAP-PAGAP-#AGAP-DGAP How many H Paths? 21 Roadmap Complexity Hierarchy Accretive Graph Assembly Problem …… • AGAP is NP-complete • Planar AGAP is NP-complete • #AGAP/Stochastic AGAP is • #P-complete PSPACE PSPACE-Complete Self-Destructible Graph Assembly Problem #P • DGAP is PSPCAE-complete #P-Complete NP NP-Complete P

  22. AGAP-PAGAP-#AGAP-DGAP 22 #AGAP is #P-complete • Parsimonious reduction from PERMANENT, i.e., counting number of perfect matchings in a bipartite graph Reduction gadget PERMANENT

  23. AGAP-PAGAP-#AGAP-DGAP temperature = 2 b 2 -2 c a 6 23 Roadmap Complexity Hierarchy Accretive Graph Assembly Problem …… • AGAP is NP-complete • Planar AGAP is NP-complete • #AGAP/Stochastic AGAP is #P-complete PSPACE PSPACE-Complete Self-Destructible Graph Assembly Problem #P • DGAP is PSPCAE-complete #P-Complete NP NP-Complete P

  24. AGAP-PAGAP-#AGAP-DGAP Nature: e.g. programmed cell death Programmed cell death (NASA) • Engineering: • e.g. remove scaffolds Tower Scaffold 24 Self-Destructible System

  25. AGAP-PAGAP-#AGAP-DGAP Association rule Slot Graph Seed Vertex set Weight func. Slot Graph Vertex set Seed Association rule: M⊆ S X V Weight func: V(sa) X V(sb) → Z, (sa, sb) ∈E 25 Self-Destructible Graph Assembly System Temperature Self-Destructible Graph Assembly Problem: Given a self-destructible graph assembly system, determine whether there exists a sequence of assembly operations to sequentially assemble a target graph.

  26. AGAP-PAGAP-#AGAP-DGAP Playing GO 26 Roadmap Complexity Hierarchy Accretive Graph Assembly Problem …… • AGAP is NP-complete • Planar AGAP is NP-complete • #AGAP/Stochastic AGAP is #P-complete PSPACE PSPACE-Complete Self-Destructible Graph Assembly Problem #P • DGAP is PSPCAE-complete #P-Complete NP NP-Complete P

  27. AGAP-PAGAP-#AGAP-DGAP • Classical tiling • TM simulation • (Rothemund & Winfree 00) • Our cyclic • gadget • Integration 27 DGAP is PSPACE complete • DGAP is PSPACE-complete • Reduction from IN-PLACE ACCEPTANCE • Proof Scheme

  28. AGAP-PAGAP-#AGAP-DGAP Features • Genaral graph • Repulsion • Self-destructible Related work • Self-assembly of DNA graphs (Jonoska et al 99) Future • “Towards a mathematical theory of self-assembly” • (Adleman99) 28 Conclusion Summary Complexity Hierarchy …… Accretive Graph Assembly Problem • AGAP is NP-complete • Planar AGAP is NP-complete PSPACE • #AGAP/Stochastic AGAP is #P-complete PSPACE-Complete Self-Destructible Graph Assembly Problem • DGAP is PSPCAE-complete #P #P-Complete • Self-assembly using graph grammars (Klavins et al 04) NP • Tiling scheme(Wang61, Rothemund & Winfree00) NP-Complete P

More Related