Cache-Oblivious Range Reporting with Optimal Queries Requires Superlinear Space

Abstract

We consider a number of range reporting problems in two and three dimensions and prove lower bounds on the amount of space used by any cacheoblivious data structure for these problems that achieves the optimal query bound of O(log B N + K/B) block transfers, where K is the size of the query output.The problems we study are three-sided range reporting, 3-d dominance reporting, and 3-d halfspace range reporting. We prove that, in order to achieve the above query bound or even a bound of f (log B N, K/B), for any monotonically increasing function f (·, ·), the data structure has to use Ω(N(log log N) ε ) space. This lower bound holds also for the expected size of any Las-Vegas-type data structure that achieves an expected query bound of at most f (log B N, K/B) block transfers. The exponent ε depends on the function f (·, ·) and on the range of permissible block sizes.Our result has a number of interesting consequences. The first one is a new type of separation between the I/O model and the cache-oblivious model, as deterministic I/O-efficient data structures with the optimal query bound in the worst case and using linear or O(N log * N) space are known for the above problems. The second conse- Discrete Comput Geom (2011) 45: 824-850 825 quence is the non-existence of linear-space cache-oblivious persistent B-trees with optimal 1-d range reporting queries.

Classic and new data structure problems in external memory

“…In most cases, the cache-oblivious bounds match their cache-aware versions, and it has always be an interesting problem to see for what problems do we have a separation between the cache-oblivious model and the cache-aware model. Until today there have been only three separation results [2,14,19]; our lower bound adds to that list, furthering our understanding of cache-obliviousness.…”

“…1 − 1/2 Ω(B) (by the parameters chosen and the assumption that N > Ω(M B 2c log U )), for all the bad functions in H, we have X 2 3 λ f k. Consequently, since the bad index area can only accommodate B • λ f /ρ keys in the fast zone, at least 2 3 λ f k − Bλ f /ρ cannot be in the fast zone. The memory zone can accept at most M keys, so the number of keys in the overflow zone is at least…”

“…Meanwhile, the F 1 based summaries usually have size s ε = Θ(1/ε), so (2) and (3) canbe satisfied, too. In Section 5.4 we will formally prove the α-exponentially decomposable property for all the F 1 based summaries mentioned in Section 5.1.2.…”