1. The Complexity and Viability of DNA Computations
Martyn Amos, Alan Gibbons and Paul E. Dunne
Proc. Bio-computing and Emergent Computation (BCEC97)

2. © 2002 SNU CSE Biointelligence Lab
Abstract
Complexity issues are paramount in the search for so-called “killer applications”.
Strong model of computation which provides better estimates of the resources required by DNA algorithms.
Compare the complexities of published algorithm within this new model and the weaker, extant model
© 2002 SNU CSE Biointelligence Lab

3. © 2002 SNU CSE Biointelligence Lab
Introduction
Computational paradigm employed by Adleman requires exponentially sized initial solution of DNA
NP-complete problems require exponential sequential running time. A DNA computation, in seeking to reduce this to sub-exponential parallel running time, will certainly require an exponential volume of DNA.
Complexity class NC
Polylogarithmic running time: O(log(n)k) for some k
© 2002 SNU CSE Biointelligence Lab

4. © 2002 SNU CSE Biointelligence Lab
Introduction
The weak model of DNA computation and the strong model.
Compare time complexities of extant algorithms within both the weak and strong models.
Discuss the complexity of one extant Turing-complete model of DNA computation in the context of the strong model.
Review current search for the killer application.
© 2002 SNU CSE Biointelligence Lab

5. © 2002 SNU CSE Biointelligence Lab
Weak and strong models
The weak model
Remove(U, {Si})
Union({Ui}, U)
Copy(U, {Ui})
Select(U)
Pour(U, U´)
The strong model
Remove, union, copy operation takes O(i) time.
© 2002 SNU CSE Biointelligence Lab

6. Complexity comparisons in the weak and strong models
Problem: Permutations
Solution
Input: all string of the form p1i1p2i2…pnin.
Algorithm
for j = 1 to n do
begin
copy(U, {U1, U2, …, Un})
for i = 1, 2, …, n and all k > j
remove(Ui, {pj¬I, pki})
union({U1, U2, …, Un}, U)
end
PnU
Complexity: O(n2) parallel-time
© 2002 SNU CSE Biointelligence Lab

7. Fully-algorithmic DNA computations
Simulation of Boolean circuits within a model of DNA computation, Ogihara and Ray.
- real-time simulation of the class NC in time proportional to the depth of the circuit.
Time complexity should be proportional to the size of the circuit.
polynomial running time in the strong model.
At each level, it requires sequential pour operation, so in the general case it’s time complexity is O(C(S), not O(D(S)).
© 2002 SNU CSE Biointelligence Lab

8. © 2002 SNU CSE Biointelligence Lab
Conclusions
Complexity considerations are important to the identification of “killer applications” for DNA computation.
Strong model of DNA computation allows realistic assessment of the time complexities of algorithms.
If we were to establish polylogarithmic time computations using only a polynomial volume of DNA, the vast potential for parallelisation would yield feasible solutions to very much larger problem sizes than could be achived using existing, silicon-based parallel machines.
© 2002 SNU CSE Biointelligence Lab