Tight Bounds for Sorting Under Partial Information

Sorting has a natural generalization where the input consists of: (1) a ground set $X$ of size $n$, (2) a partial oracle $O_P$ specifying some fixed partial order $P$ on $X$ and (3) a linear oracle $O_L$ specifying a linear order $L$ that 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: oracle queries to $O_L$, oracle queries to $O_P$, and the time spent. Any algorithm requires worst-case ${\log }_{2}e(P)$ linear oracle queries to recover the linear order on $X$. Kahn and Saks presented the first algorithm that uses $\mathrm{\Theta }(\log e(P))$ linear oracle queries (using $O(n^2)$ partial oracle queries and exponential time). 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 $\mathrm{\Theta }(\log e(P))$ 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. For $c\ge 1$, our algorithm can preprocess $O_P$ using $O(n^{1+\frac{1}{c}})$ queries and time. Given $O_L$, we uncover $L$ using $\mathrm{\Theta }(c\log e(P))$ queries and time. We show a matching lower bound, as there exist positive constants $(\alpha ,\beta )$ where for any constant $c\ge 1$, any algorithm that uses at most $\alpha \cdot n^{1+\frac{1}{c}}$ preprocessing 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.