Download presentation

Presentation is loading. Please wait.

Published byMaximilian Eastland Modified over 4 years ago

1
Alternative Tile Assembly Models and Complexity Results Tianqi Song

2
All screenshot pictures are from original papers.

3
Outline Multiple Temperature model and complexity results. Staged assembly model and complexity results. Flexible glue model, time-dependent glue model and complexity results.

4
Multiple Temperature Model A tile system S= under multiple temperature model[Aggarwal et al. SODA 2004]. {τ i } i=1 is temperature sequence. Temperature complexity: the number of items in the temperature sequence. k k

5
Assembly Process One pot reaction Multiple phases: each phase has its own temperature. Assemble and disassemble

6
Assembly process Assemble and disassemble under temperature τ 1 for long enough time. Assemble and disassemble under temperature τ k for long enough time. Phase 1 Phase k The terminal product of phase k is the terminal product of this k-temperature system.

7
Basic Tool Bit-Flipping[Kao et al. SODA 2006].

8
Binary Strings with O(1) Tiles Tile set[Kao et al. SODA 2006]:

9
Binary Strings with O(1) Tiles Example: 1010010

10
Binary Strings with O(1) Tiles

11
Building n×n Square in O(1) Tiles

12
Staged Assembly Model A tile system S= under staged assembly model[Demaine et al. Nat Comput 2008]. Assumption: merge, split, extract M r,b : r-stage b-bin mix graph M. {T i,j }: T i,j is the tile set of bin i,j. {τ i,j }: τ i,j is the temperature of bin i,j. Stage complexity, bin complexity

13
Example of Staged Assembly System

14
Assembly of 1×n lines Special case: there is a planar temperature-1 staged assembly system that uniquely produce a(full connected) 1×2 k line using 3 glues, 6 tiles, 6 bins, and O(k) stages [Demaine et al. Nat Comput 2008]. Planar system:

15
Assembly of 1×n lines Strategy: divide and conquer a b a cc b b ba a a c b

16
Assembly of 1×n lines General case: standard trick, build 1×2 i line for the ith bit of binary encoding of n if the ith bit is 1. Theorem: there is a planar temperature-1 staged assembly system that uniquely produce a(full connected) 1×n line using 3 glues, 6 tiles, 7 bins, and O(logn) stages [Demaine et al. Nat Comput 2008].

17
Assembly of n×n Squares Theorem: there is a planar temperature-1 staged assembly of a full connected n×n square using 9 glues, O(1) tiles, O(1) bins, and O(logn) stages[Demaine et al. Nat Comput 2008].

18
Jigsaw Technique

19
Decompose Algorithm

22
Flexible Glue Model Remove: g(x, y)=0 for xy[Aggarwal et al. SODA 2004]. Assembly of N×N square: the tile complexity of self-assembling N×N squares is O(logN) under the flexible glue model.

23
Assembly of k×N Rectangles under Standard Model Theorem: the tile complexity of self- assembling a k×N rectangle is O(N^(1/k)+k) for standard model[Aggarwal et al. SODA 2004]. Trick: a k-digit base-m counter, where m=ceil(N^(1/k)). What if N is not a power of m? By initializing.

24
Assembly of k×N Rectangles under Standard Model

25
C tiles: counting for the 0 th bit of the counter. H tiles: counting for the other bits of the counter. R,P tiles: transfer carry and flip bits back to 0.

26
Assembly of N×N Square under Flexible Glue Model Theorem: the tile complexity of self- assembling N×N squares is O(logN) under the flexible glue model[Aggarwal et al. SODA 2004]. Thought 1: the complexity of assembling N×N square is almost the complexity of assembling a (logN)-digit counter. Thought 2: how to assemble a (logN)-digit counter by O(logN) tile types.

27
Assembly of N×N Square under Flexible Glue Model Thought 3: the complexity of assembling a counter is almost the complexity of assembling the first row of the counter. Thought 4: how to assemble the first row of a (logN)-digit counter by O(logN) tile types. Target: Assemble the first row of a n-digit counter by O( n ) tile types.

28
Assembly of N×N Square under Flexible Glue Model Tile set:

29
Assembly of N×N Square under Flexible Glue Model Example:

30
Time-dependent Glue Model The glue strength between glue x, y is a function of time t: f x,y (t)[Sahu et al. DNA 2005].

31
Time-dependent Glue Model Minimum interaction time. Complexity result: the tile complexity of assembling a k×N rectangle is O(logN/loglogN) under time-dependent glue model, where k

Similar presentations

OK

CSE 8351 Computer Arithmetic Fall 2005 Instructors: Peter-Michael Seidel.

CSE 8351 Computer Arithmetic Fall 2005 Instructors: Peter-Michael Seidel.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To ensure the functioning of the site, we use **cookies**. We share information about your activities on the site with our partners and Google partners: social networks and companies engaged in advertising and web analytics. For more information, see the Privacy Policy and Google Privacy & Terms.
Your consent to our cookies if you continue to use this website.

Ads by Google

Ppt on alternative sources of energy vs fossil fuels Ppt on object-oriented programming concepts in c++ Ppt on earth hour day Ppt on 7 wonders of the world 2011 Ppt on product management process Ppt on pin diode limiters Ppt on power line communication modem Ppt on beer lambert law problems Ppt on topic iso 9000 Ppt on power system harmonics