Tight Bounds for Sorting Under Partial Information

Sorting is one of the fundamental algorithmic problems in theoretical computer science. It has a natural generalization, introduced by Fredman in 1976, called sorting under partial information. The input consists of: –a ground set $X$ of size $n$, –a partial oracle $O_{F}$ (where partial oracle queries for any $(x_{i},x_{j})$ output whether $x_{i}\prec _{P}x_{j}$, for some partial order $P$), –a linear oracle $O_{L}$ (where linear oracle queries for any $(x_{i},x_{j})$ output whether $x_{i} < _{L}x_{j}$ and the order $L$ extends $P$) The goal is to recover the linear order $L$ on $X$ using the fewest number of linear oracle queries. In this problem, we measure algorithmic complexity through three metrics: the number of linear oracle queries to $O_{L}$, the number of partial oracle queries to $O_{P}$, and the time spent (the number of algorithmic instructions required to identify for which pairs $(x_{i},x_{j})$ a partial or linear oracle query is performed). Let $e(P)$ denote the number of linear extensions of $P$. Any algorithm requires worst-case $\log_{2}e(P)$ linear oracle queries to recover the linear order on $X$. In 1984, Kahn and Saks presented the first algorithm that uses $\Theta(\log e(P))$ linear oracle queries (using $O(n^{2})$ partial oracle queries and exponential time). Since then, both the general problem and restricted variants have been consistently studied. The state-of-the-art for the general problem is by Cardinal, Fiorini, Joret, Jungers and Munro who at STOC'10 manage to separate the linear and partial oracle queries into a preprocessing and query phase. They can preprocess $P$ using $O(n^{2})$ partial oracle queries and $O(n^{2.5})$ time. Then, given $O_{L}$, they uncover the linear order on $X$ in $\Theta(\log e(P)$ linear oracle queries and $O(n+\log e(P))$ time - which is worst-case optimal in the number of linear oracle queries but not in the time spent. We present the first algorithm that uses a subquadratic number of partial oracle queries. For any constant $c\geq 1$, our algorithm can preprocess $O_{F}$ using O $O(n^{1+\frac{1}{c}})$ partial oracle queries and time. Given ${OL}$, we uncover the linear order on $X$ using $\Theta(c\log e(P))$ linear oracle queries and time, which is worst-case optimal. We show a matching lower bound for the prepossessing also, as we show that there exist positive constants $((y,\beta)$ where for any constant $c\geq 3$, any algorithm that uses at most $\alpha\cdot n^{1+\frac{1}{c}}$ partial oracle queries must use worst-case at least $\beta\cdot c\log e(P)$ linear oracle queries. Thus, we solve the problem of sorting under partial information through an algorithm that is asymptotically tight across all three metrics.