Bounded In-degree Case With Bounded Cost

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Bounded In-degree Case With Bounded Cost

Since an exponential lower bound was proven in a general case, we consider a special but practical case, in which the maximal in-degree is bounded by a constant D. First, we consider the case D=2.

Proposition $\Omega(n^2)$ experiments are necessary for identification even if the maximum in-degree is 2 and all nodes are AND nodes, where we assume that the maximum cost is bounded by a fixed constant C.

$\begin{proof}First, consider the case of $C=2$ . Assume that $\neg x \wedge \neg... ...ssible ones). \\ For cases of $C > 3 $ , similar arguments work. \end{proof}$
If C is not bounded, the above proposition does not hold. It is possible to identify the above pair (x,y) by $O(\log(n))$ experiments of maximum cost n, using a strategy based on binary search. Although this strategy might be generalized for other cases, we do not investigate it because experiments with high cost are not realistic. (The cells simply die if they are heavily mutated.) Next, we consider the upper bound.

Proposition O(n⁴) experiments with maximum cost 4 are sufficient for identification if the maximum in-degree is 2.

$\begin{proof}We assume (w.l.o.g) that all nodes are of in-degree 2 since identi... ...re examined, in total $0(n^{4})$\space experiments are sufficient. \end{proof}$
The above property holds even for an unstable graph because c is consistent under any experiment on $\{a,b,x,y\}$ if $f_c \equiv g(a,b)$ .
$\begin{theorem} $O(n^{2D})$\space experiments with maximal cost $2D$\space are ... ... worst case if cost of each experiment is bounded by a constant. \end{theorem}$

Next: More Efficient Strategies for Up: Identification of Gene Regulatory Previous: Upper and Lower Bounds

Peer Itsik
2001-03-04