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# [theory-seminar] Theory Lunch 08/05: William Kuszmaul (MIT)

David Wajc wajc at stanford.edu
Mon Aug 2 08:00:14 PDT 2021

Hi all,

This week's theory lunch will take place Thursday at noon (PDT), at this
zoom room
<https://stanford.zoom.us/j/98932206471?pwd=YXdubytLVGNTbXhGeXFxNmJaVnhrUT09>.
As
usual, we'll start with some socializing, followed by a talk starting at
12:30pm. Pras will be MCing. Thanks Pras!
Bill will tell us about: *Linear Probing Revisited: Tombstones Mark the
Death of Primary Clustering*

*Abstract:* First introduced in 1954, linear probing is one of the oldest
data structures in computer science, and due to its unrivaled data
locality, it continues to be one of the fastest hash tables in practice. It
is widely believed and taught, however, that linear probing should never be
used at high load factors; this is because primary-clustering effects cause
insertions at load factor $1 - 1/x$ to take expected time $\Theta(x^2)$
(rather than the ideal $\Theta(x)$). The dangers of primary clustering,
first discovered by Knuth in 1963, have been taught to generations of
computer scientists, and have influenced the design of some of many widely
used hash tables.

We show that primary clustering is not the foregone conclusion that it is
reputed to be. We demonstrate that small design decisions in how deletions
are implemented have dramatic effects on the asymptotic performance of
insertions, so that, even if a hash table operates continuously at a load
factor $1 - \Theta(1/x)$, the expected amortized cost per operation is
$\tilde{O}(x)$. This is because tombstones created by deletions actually
cause an anti-clustering effect that combats primary clustering.

We also present a new variant of linear probing (which we call graveyard
hashing) that completely eliminates primary clustering on any sequence of
operations: if, when an operation is performed, the current load factor is
$1 - 1/x$ for some $x$, then the expected cost of the operation is $O(x)$.
One corollary is that, in the external-memory model with a data blocks of
size $B$, graveyard hashing offers the following remarkable guarantee: at
any load factor $1-1/x$ satisfying $x=o(B)$, graveyard hashing achieves
$1+o(1)$ expected block transfers per operation. Past external-memory hash
tables have only been able to offer a $1+o(1)$ guarantee when the block
size $B$ is at least $\Omega(x^2)$.

Based on joint work with Michael A. Bender, Bradley C. Kuszmaul

Cheers, David
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