More Analysis of Double Hashing for Balanced Allocations

Abstract

Abstract. With double hashing, for a key x, one generates two hash values f (x) and g(x), and then uses combinations (f (x) + ig(x)) mod n for i = 0, 1, 2, . . . to generate multiple hash values in the range [0, n − 1] from the initial two. For balanced allocations, keys are hashed into a hash table where each bucket can hold multiple keys, and each key is placed in the least loaded of d choices. It has been shown previously that asymptotically the performance of double hashing and fully random hashing is the same in the balanced allocation paradigm using fluid limit methods. Here we extend a coupling argument used by Lueker and Molodowitch to show that double hashing and ideal uniform hashing are asymptotically equivalent in the setting of open address hash tables to the balanced allocation setting, providing further insight into this phenomenon. We also discuss the potential for and bottlenecks limiting the use this approach for other multiple choice hashing schemes.

Arithmetic Progression Hypergraphs: Examining the Second Moment Method

“…, a + (d − 1)b modulo n. That is, the d choices are constrained to form an arithmetic progression; as such, only two random numbers modulo n are chosen to determine the choices, instead of d random numbers. It has been shown that, even with this more limited randomness, not only does the maximum load remain log log n/ log d + O(1) with high probability, but that the asymptotic fraction of bins of each constant load is the same as when the d choices are all perfectly random [21,22]. Similar results have been found for other data structures that use multiple hash functions, including Bloom filters [17] and cuckoo hashing [18,23].…”

Linear Probing Revisited: Tombstones Mark the Death of Primary Clustering

“…At one extreme is uniform probing, where each probe is to a random location, thus sacrificing locality in the probe sequence [119]. There has been intensive work in analyzing variations on uniform probing, including in the presence of deletions [28,68,70,79,93,94,123,137].…”

“…Double hashing [21,60,75,85,86,93,143] is a classic alternative to uniform probing, in which a primary hash function determines the first probe and a secondary hash function determines jump size between indices in the probe sequence. Double hashing has been shown to have short probe sequences similar to that of uniform hashing, but like uniform probing, these short probe sequences come with a corresponding loss in locality.…”