Group Testing Problem

• Main Authors: Su-Tzu Juan, 阮夙姿
• Other Authors: Gerard J. Chang
• Format: Others
• Language: en_US
• Published: 2000
• Online Access: http://ndltd.ncl.edu.tw/handle/51083617530691999042

博士 === 國立交通大學 === 應用數學系 === 88 === This thesis studies group testing problems. The idea of group testing originated from the blood testing in 1942 by Dorfman. Li was the first who studied the combinatorial group testing as follows. Consider a population $V$ of $n$ items consisting of an unknown subset $D\subseteq V$ of $d$ defectives. The problem is to identify the set $D$ by a sequence of group tests. Each test is on a subset $X$ of $V$ with two possible outcomes: a {\it negative} outcome indicates that $X\cap D=\emptyset$, and a {\it positive} outcome indicates that $X\cap D\neq\emptyset$. The goal is to minimize the number $M[d,n]$ of tests under the worst scenario. We may generalize the problem as follows. Consider a population $V$ of $n$ items and a sample space $S \subseteq 2^V$. The problem is to identify an unknown $D \in S$ by a sequence of group tests. Each test is on a subset $X$ of $V$ which partitions $S$ into $S_1=\{D \in S:D \cap X = \emptyset\}$ and $S_2= \{D \in S: D \cap X \neq \emptyset \}$. The goal is to identify the unknown $D$ using a minimum number $M[S]$ of tests under the worst scenario. Chapter 2 considers the case when $G=(V,S)$ is a graph. We also use $M[G]$ for $M[S]$. Damaschke proved that $\lceil\log_2 e(G)\rceil \leq M[G] \leq \lceil\log_2 e(G)\rceil +1$ for any graph $G$, where $e(G)$ is the number of edges of $G$. Chapter 2 gives an improved bound for general graphs $G$. Namely, if $G$ with $2^{k-1}< e(G) \leq 2^{k-1} + 2^{k-9}+ 2^{k-15} + 2^{\frac{k+1}{2}} + 2^{\frac{k-5}{2}} + 2^{\frac{k-7}{2}} + 2^{\frac{k-9}{2}}$ and $k\geq 15$, then $M[G]= \lceil \log_2e(G) \rceil$. While there are infinitely many complete graphs $G$ with $M[G]=\lceil \log_2e(G) \rceil +1$, it was conjectured by Chang and Hwang that $M[G]=\lceil \log_2 e(G) \rceil$ for all bipartite graphs $G$. Chapter 2 also verifies the conjecture for bipartite graphs $G$ with $e(G)\leq 2^5$ or $2^{k-1}<e(G) \leq2^{k-1}+2^{k-3}+2^{k-4}+2^{k-5}+2^{k-6}+2^{k-7}+ 27\cdot 2^{\frac{k-8}{2}}-1$ for $k\ge 6$. Chapter 3 considers group testing on hypergraphs $H=(V,S)$. We also use $M[H]$ for $M[S]$. Triesch proved for any hypergraph $H=(V,E)$ of rank $r$, it is the case that $ M[H] \leq \log_2 e(H) +r-1,$ where $e(H)$ means the number of edges of $H$. This chapter proves that for any hypertree $H=(V,E)$, we have $ M[H] = \lceil \log_2 e(H) \rceil.$ Chapter 4 considers another kind of group testing called the consecutive positive problem. In a set $V_n$ of $n$ items, with linear order $\prec$, each item has an associated {\it state} positive or negative. The set $V_n$ has the {\it d-consecutive positive property} if the set of positive items is a consecutive set (under the order $\prec$), and contains at most $d$ items. Chapter 4 studies the problem of finding these consecutive positive items from $V_n$. Colbourn proved that this can be accomplished with $O(\log_2 d + \log_2 n)$ times for at most $d$ consecutive positives in a linearly ordered $n$-set of items. Colbourn didn''t give the exact value of times needed. Chapter 4 gives the exact values for $d=1,2,3$, and gives an upper bound $\lceil \log_2 (nd)\rceil+6$ for $d \geq 4$.