DecreaseKeys are Expensive for External Memory Priority Queues

One of the biggest open problems in external memory data structures is the priority queue problem with DecreaseKey operations. If only Insert and ExtractMin operations need to be supported, one can design a comparison-based priority queue performing $O((N\mathrm{/}B){\lg }_{M\mathrm{/}B}N)$ I/Os over a sequence of $N$ operations, where $B$ is the disk block size in number of words and $M$ is the main memory size in number of words. This matches the lower bound for comparison-based sorting and is hence optimal for comparison-based priority queues. However, if we also need to support DecreaseKeys, the performance of the best known priority queue is only $O((N\mathrm{/}B){\lg }_{2}N)$ I/Os. The big open question is whether a degradation in performance really is necessary. We answer this question affirmatively by proving a lower bound of $\Omega((N/B) \lg_{\lg N} B)$I\mathrm{/}Osforprocessingasequenceof$N$ intermixed Insert, ExtraxtMin and DecreaseKey operations. Our lower bound is proved in the cell probe model and thus holds also for non-comparison-based priority queues.