Efficient search approaching the reach-imposed bumping floor

Determine whether an efficient search procedure for PHastR can approach the lower bound on the number of bumped keys imposed by the keys’ reachable slot ranges, thereby reducing bumping by approximately a factor of ten and saving about 0.05 bits per key.

Background

PHastR places each key within a reach of W slots and experimentally bumps about ten times as many keys as the structural constraints on reaches require. The paper derives a lower bound of approximately 0.1/W on the fraction of bumped keys and estimates that the best possible placement bumps about 1/(2W) of the keys. The unresolved issue is whether a more powerful search can efficiently approach this floor rather than the substantially higher bumping rate achieved by PHastR’s current seed-search procedure.

References

There are several open problems and further research directions. On the practical side, the small final levels could be replaced by a more compact structure. More interestingly, PHastR bumps about ten times as many keys as the reaches force: it is open whether a search can approach this floor efficiently (which would save about $0.05$ bits per key), and whether bump-free placements, which would use just $S/\lambda$ bits per key, can be found efficiently within subsets of exactly known size, in spite of the long-range constraints that made simple backtracking fail in our prototype.

— PHastR: Faster Minimal Perfect Hashing with Rotated Layouts  (2610.17833 - Vigna, 13 Oct 2026) in Section Conclusions