We introduce the hollow heap , a very simple data structure with the same amortized efficiency as the classical Fibonacci heap. All heap operations except delete and delete - min take O (1) time, worst case as well as amortized; delete and delete - min take O (log n ) amortized time on a heap of n items. Hollow heaps are the simplest structure to achieve these bounds. Hollow heaps combine two novel ideas: the use of lazy deletion and re-insertion to do decrease - key operations and the use of a dag (directed acyclic graph) instead of a tree or set of trees to represent a heap. Lazy deletion produces hollow nodes (nodes without items), giving the data structure its name.