Efficient Algorithms for Sorting in Trees

TL;DR: This work considers the problem of sorting in trees, a particular case of partial orders, and parametrize the complexity with respect to d, the maximum degree of an element in the tree, as d is usually much smaller than w in trees.

Abstract: . Sorting is a foundational problem in computer science that is typically employed on se-quences or total orders. More recently, a more general form of sorting on partially ordered sets (or posets), where some pairs of elements are incomparable, has been studied. General poset sorting algorithms have a lower-bound query complexity of Ω ( wn + n log n ) , where w is the width of the poset. We consider the problem of sorting in trees, a particular case of partial orders, and parametrize the complexity with respect to d , the maximum degree of an element in the tree, as d is usually much smaller than w in trees. For example, in complete binary trees, d = Θ (1) , w = Θ ( n ) . We present a randomized algorithm for sorting a tree poset in worst-case expected O ( dn log n ) query and time complexity. This improves the previous upper bound of O ( dn log 2 n ) . Our algorithm is the first to be optimal for bounded-degree trees. We also provide a new lower bound of Ω ( dn + n log n ) for the worst-case query complexity of sorting a tree poset. Finally, we present the first deterministic algorithm for sorting tree posets that has lower total complexity than existing algorithms for sorting general partial orders.