Download presentation

Presentation is loading. Please wait.

Published byAllie Tickner Modified over 3 years ago

1
Ashish Goel, ashishg@stanford.edu 1 A simple analysis Suppose complementary DNA strands of length U always stick, and those of length L never stick (eg: U=20, L=10) Suppose you have n randomly chosen DNA strands of length U Probability that there will be some undesired interaction of length at least L · U 2 n 2 4 -L Basis of Adlemans construction and also DNA tiles

2
Ashish Goel, ashishg@stanford.edu 2 Tile Systems and Program Size A tile system that uniquely assembles into a shape is analogous to a program for computing the shape Program size: The number of tile types used Goal: Find smallest program size for interesting classes of shapes [Adleman 99][Rothemund and Winfree 00] Can often derive lower bounds on program size Need n distinct tile types to assemble a line of length n (pumping lemma) Need Ω io (log n/log log n) distinct tile types to construct squares of side n (Kolmogorov complexity) Also need a notion of assembly time or running time

3
Ashish Goel, ashishg@stanford.edu 3 Tile Systems and Assembly Time d Seed A BC = 2

4
Ashish Goel, ashishg@stanford.edu 4 Tile Systems and Assembly Time d Seed A B C A B A B A: 50%, B:30%, C: 20% 0.5 0.3 0.2 0.3 0.5 Assembly time = Average time to go from seed to terminal state

5
Ashish Goel, ashishg@stanford.edu 5 Example: Assembling Lines Require n tiles to assemble line of length n Eg. suppose tiles B and F are the same Could pump the tiles BCDE to get infinite line Therefore program size is n For fastest assembly, all tiles must have the same concentration (1/n) Expected assembly time is n 2 Can assemble thicker rectangles (2 £ n, log n £ n, n £ n etc.) faster and with less tile types! BCDE FGHIA BCDEBCDEBCDE

6
Ashish Goel, ashishg@stanford.edu 6 Self-assembling Squares Would like to assemble n x n squares as fast as possible and with as few tile types Kolmogorov lower bound of Ω io (log n/log log n) on the program size (i.e. number of different tiles) Assembly time has to be Ω(n) in accretion model Motivation: Simple canonical problem, leads to useful general techniques Also interesting in its own right (e.g. assembling computer memories)

7
Ashish Goel, ashishg@stanford.edu 7 Rothemund-Winfree construction Basic Idea: Easy to extend a self-assembled rectangle into a self-assembled square =2 Inefficient to start with a 1 £ n rectangle –Need (n) tile types, (n 2 ) time for lines –Solution: Use counters to make rectangles B C A x z y x z z w w w y

8
Ashish Goel, ashishg@stanford.edu 8 Assembly Time Analysis Technique Seed A B C 2 BC A S 12 1 Equivalent Acyclic Subgraph Terminal assembly BC A S 1 2 2 22 1 1 1 Graph G: a vertex for each tile in final assembly, directed edges between adjacent vertices

9
Ashish Goel, ashishg@stanford.edu 9 Assembly Time Analysis Technique Let L be the length of the longest path in an equivalent acyclic subgraph of G Often, much smaller than longest path in Markov Chain Let C m be the smallest concentration C i Theorem: The assembly time of the given tile system is O(L/C m ) with high probability Find the right equivalent acyclic subgraph No Markov chains anymore, just combinatorial analysis Analysis technique is tight: there always exists an EAG which gives a tight bound if C i s are identical

10
Ashish Goel, ashishg@stanford.edu 10 Proof for identical C i s t i,j : exponential random variable with mean 1/C L = length of longest path in equivalent acyclic graph Number of paths · 3 L For any path P in graph, let t P = i,j 2 P t i,j Assembly-time · sd max P t P E[t P ] · L/C; also sharp concentration (Chernoff bounds) ) Expected assembly time = O(L/C) t i,j = time for tile to attach at position (i,j) after it becomes attachable

11
Ashish Goel, ashishg@stanford.edu 11 Squares via counters Self-assemble a log n £ n rectangle by simulating a log n bit counter Start with a seed row that has log n tiles, labeled 0/1 Define an increment operation which adds one to the row Crucial step: assembly must stop when all the tiles in a row are labeled 1 Requires (log n) tile-types to make the seed and (1) other tiles Can be improved to the Kolmogorov optimum of (log n/log log n) Then, use triangulation on one side of the counter to covert the counter into a square

12
Ashish Goel, ashishg@stanford.edu 12 Counter tiles

13
Ashish Goel, ashishg@stanford.edu 13 Counting example From Cheng, Moisset, Goel; Optimal self-assembly of counters at temperature 2 (basic ideas developed over many papers)

14
Ashish Goel, ashishg@stanford.edu 14 Equivalent Acyclic Subgraph for Counters 1111110 ……….. 0000000 1111111 ……….. 0000000 00000000 11111111 01111111 Only time a directed edge is needed from left column 00000000 L(n) = 2 L(n/2) + (log n) = (n) T(n) = L(n)/C m = (n)

15
Ashish Goel, ashishg@stanford.edu 15 Constructing squares and Counters Can extend this basic idea to Assemble n £ n squares in time O(n), using O(log n/log log n) tiles (provably optimum) Count optimally in binary using the same assembly time and program size General design and analysis techniques A library of subroutines (counting, base-conversion, triangulating the line) If this is so efficient, why didnt nature learn to count? Possible conjecture: Not evolvable, not robust Open problems: general analysis techniques for assembly time in reversible models?

Similar presentations

OK

Ashish Goel, 1 The Source of Errors: Thermodynamics Rate of correct growth ¼ exp(-G A ) Probability of incorrect growth ¼ exp(-G A.

Ashish Goel, 1 The Source of Errors: Thermodynamics Rate of correct growth ¼ exp(-G A ) Probability of incorrect growth ¼ exp(-G A.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Pdf to ppt online convert Ppt on different forms of power sharing in india Ppt on project management process Ppt on aravind eye care system How to run ppt on ipad 2 Slides for ppt on robotics Ppt on end of life care Ppt on job evaluation and job rotation Ppt on artificial intelligence techniques Ppt on any topic of science