Weak Heaps and Friends: Recent Developments

A weak heap is a variant of a binary heap where, for each node, the heap ordering is enforced only for one of its two children. In 1993, Dutton showed that this data structure yields a simple worst-case-efficient sorting algorithm. In this paper we review the refinements proposed to the basic data structure that improve the efficiency even further. Ultimately,

minimum

and

insert

operations are supported in

O

(1) worst-case time and

extract-min

operation in

$O(\lg n)$

worst-case time involving at most

$\lg n + O(1)$

element comparisons. In addition, we look at several applications of weak heaps. This encompasses the creation of a sorting index and the use of a weak heap as a tournament tree leading to a sorting algorithm that is close to optimal in terms of the number of element comparisons performed. By supporting

insert

operation in

O

(1) amortized time, the weak-heap data structure becomes a valuable tool in adaptive sorting leading to an algorithm that is constant-factor optimal with respect to several measures of disorder. Also, a weak heap can be used as an intermediate step in an efficient construction of binary heaps. For graph search and network optimization, a weak-heap variant, which allows some of the nodes to violate the weak-heap ordering, is known to be provably better than a Fibonacci heap.