69 research outputs found
Tight Thresholds for Cuckoo Hashing via XORSAT
We settle the question of tight thresholds for offline cuckoo hashing. The
problem can be stated as follows: we have n keys to be hashed into m buckets
each capable of holding a single key. Each key has k >= 3 (distinct) associated
buckets chosen uniformly at random and independently of the choices of other
keys. A hash table can be constructed successfully if each key can be placed
into one of its buckets. We seek thresholds alpha_k such that, as n goes to
infinity, if n/m <= alpha for some alpha < alpha_k then a hash table can be
constructed successfully with high probability, and if n/m >= alpha for some
alpha > alpha_k a hash table cannot be constructed successfully with high
probability. Here we are considering the offline version of the problem, where
all keys and hash values are given, so the problem is equivalent to previous
models of multiple-choice hashing. We find the thresholds for all values of k >
2 by showing that they are in fact the same as the previously known thresholds
for the random k-XORSAT problem. We then extend these results to the setting
where keys can have differing number of choices, and provide evidence in the
form of an algorithm for a conjecture extending this result to cuckoo hash
tables that store multiple keys in a bucket.Comment: Revision 3 contains missing details of proofs, as appendix
The Multiple-orientability Thresholds for Random Hypergraphs
A -uniform hypergraph is called -orientable, if there
is an assignment of each edge to one of its vertices such
that no vertex is assigned more than edges. Let be a
hypergraph, drawn uniformly at random from the set of all -uniform
hypergraphs with vertices and edges. In this paper we establish the
threshold for the -orientability of for all and
, i.e., we determine a critical quantity such that
with probability the graph has an -orientation if
.
Our result has various applications including sharp load thresholds for
cuckoo hashing, load balancing with guaranteed maximum load, and massive
parallel access to hard disk arrays.Comment: An extended abstract appeared in the proceedings of SODA 201
On the insertion time of random walk cuckoo hashing
Cuckoo Hashing is a hashing scheme invented by Pagh and Rodler. It uses
distinct hash functions to insert items into the hash table. It has
been an open question for some time as to the expected time for Random Walk
Insertion to add items. We show that if the number of hash functions
is sufficiently large, then the expected insertion time is per item.Comment: 9 page
On the Insertion Time of Cuckoo Hashing
Cuckoo hashing is an efficient technique for creating large hash tables with
high space utilization and guaranteed constant access times. There, each item
can be placed in a location given by any one out of k different hash functions.
In this paper we investigate further the random walk heuristic for inserting in
an online fashion new items into the hash table. Provided that k > 2 and that
the number of items in the table is below (but arbitrarily close) to the
theoretically achievable load threshold, we show a polylogarithmic bound for
the maximum insertion time that holds with high probability.Comment: 27 pages, final version accepted by the SIAM Journal on Computin
Load thresholds for cuckoo hashing with overlapping blocks
Dietzfelbinger and Weidling [DW07] proposed a natural variation of cuckoo
hashing where each of objects is assigned intervals of size
in a linear (or cyclic) hash table of size and both start points are chosen
independently and uniformly at random. Each object must be placed into a table
cell within its intervals, but each cell can only hold one object. Experiments
suggested that this scheme outperforms the variant with blocks in which
intervals are aligned at multiples of . In particular, the load threshold
is higher, i.e. the load that can be achieved with high probability. For
instance, Lehman and Panigrahy [LP09] empirically observed the threshold for
to be around as compared to roughly using blocks.
They managed to pin down the asymptotics of the thresholds for large ,
but the precise values resisted rigorous analysis.
We establish a method to determine these load thresholds for all , and, in fact, for general . For instance, for we
get . The key tool we employ is an insightful and general
theorem due to Leconte, Lelarge, and Massouli\'e [LLM13], which adapts methods
from statistical physics to the world of hypergraph orientability. In effect,
the orientability thresholds for our graph families are determined by belief
propagation equations for certain graph limits. As a side note we provide
experimental evidence suggesting that placements can be constructed in linear
time with loads close to the threshold using an adapted version of an algorithm
by Khosla [Kho13]
Random hypergraphs for hashing-based data structures
This thesis concerns dictionaries and related data structures that rely on providing several random possibilities for storing each key. Imagine information on a set S of m = |S| keys should be stored in n memory locations, indexed by [n] = {1,âŠ,n}. Each object x [ELEMENT OF] S is assigned a small set e(x) [SUBSET OF OR EQUAL TO] [n] of locations by a random hash function, independent of other objects. Information on x must then be stored in the locations from e(x) only. It is possible that too many objects compete for the same locations, in particular if the load c = m/n is high. Successfully storing all information may then be impossible. For most distributions of e(x), however, success or failure can be predicted very reliably, since the success probability is close to 1 for loads c less than a certain load threshold c^* and close to 0 for loads greater than this load threshold. We mainly consider two types of data structures: âą A cuckoo hash table is a dictionary data structure where each key x [ELEMENT OF] S is stored together with an associated value f(x) in one of the memory locations with an index from e(x). The distribution of e(x) is controlled by the hashing scheme. We analyse three known hashing schemes, and determine their exact load thresholds. The schemes are unaligned blocks, double hashing and a scheme for dynamically growing key sets. âą A retrieval data structure also stores a value f(x) for each x [ELEMENT OF] S. This time, the values stored in the memory locations from e(x) must satisfy a linear equation that characterises the value f(x). The resulting data structure is extremely compact, but unusual. It cannot answer questions of the form âis y [ELEMENT OF] S?â. Given a key y it returns a value z. If y [ELEMENT OF] S, then z = f(y) is guaranteed, otherwise z may be an arbitrary value. We consider two new hashing schemes, where the elements of e(x) are contained in one or two contiguous blocks. This yields good access times on a word RAM and high cache efficiency. An important question is whether these types of data structures can be constructed in linear time. The success probability of a natural linear time greedy algorithm exhibits, once again, threshold behaviour with respect to the load c. We identify a hashing scheme that leads to a particularly high threshold value in this regard. In the mathematical model, the memory locations [n] correspond to vertices, and the sets e(x) for x [ELEMENT OF] S correspond to hyperedges. Three properties of the resulting hypergraphs turn out to be important: peelability, solvability and orientability. Therefore, large parts of this thesis examine how hyperedge distribution and load affects the probabilities with which these properties hold and derive corresponding thresholds. Translated back into the world of data structures, we achieve low access times, high memory efficiency and low construction times. We complement and support the theoretical results by experiments.Diese Arbeit behandelt WörterbĂŒcher und verwandte Datenstrukturen, die darauf aufbauen, mehrere zufĂ€llige Möglichkeiten zur Speicherung jedes SchlĂŒssels vorzusehen. Man stelle sich vor, Information ĂŒber eine Menge S von m = |S| SchlĂŒsseln soll in n SpeicherplĂ€tzen abgelegt werden, die durch [n] = {1,âŠ,n} indiziert sind. Jeder SchlĂŒssel x [ELEMENT OF] S bekommt eine kleine Menge e(x) [SUBSET OF OR EQUAL TO] [n] von SpeicherplĂ€tzen durch eine zufĂ€llige Hashfunktion unabhĂ€ngig von anderen SchlĂŒsseln zugewiesen. Die Information ĂŒber x darf nun ausschlieĂlich in den PlĂ€tzen aus e(x) untergebracht werden. Es kann hierbei passieren, dass zu viele SchlĂŒssel um dieselben SpeicherplĂ€tze konkurrieren, insbesondere bei hoher Auslastung c = m/n. Eine erfolgreiche Speicherung der Gesamtinformation ist dann eventuell unmöglich. FĂŒr die meisten Verteilungen von e(x) lĂ€sst sich Erfolg oder Misserfolg allerdings sehr zuverlĂ€ssig vorhersagen, da fĂŒr Auslastung c unterhalb eines gewissen Auslastungsschwellwertes c* die Erfolgswahrscheinlichkeit nahezu 1 ist und fĂŒr c jenseits dieses Auslastungsschwellwertes nahezu 0 ist. HauptsĂ€chlich werden wir zwei Arten von Datenstrukturen betrachten: âą Eine Kuckucks-Hashtabelle ist eine Wörterbuchdatenstruktur, bei der jeder SchlĂŒssel x [ELEMENT OF] S zusammen mit einem assoziierten Wert f(x) in einem der SpeicherplĂ€tze mit Index aus e(x) gespeichert wird. Die Verteilung von e(x) wird hierbei vom Hashing-Schema festgelegt. Wir analysieren drei bekannte Hashing-Schemata und bestimmen erstmals deren exakte Auslastungsschwellwerte im obigen Sinne. Die Schemata sind unausgerichtete Blöcke, Doppel-Hashing sowie ein Schema fĂŒr dynamisch wachsenden SchlĂŒsselmengen. âą Auch eine Retrieval-Datenstruktur speichert einen Wert f(x) fĂŒr alle x [ELEMENT OF] S. Diesmal sollen die Werte in den SpeicherplĂ€tzen aus e(x) eine lineare Gleichung erfĂŒllen, die den Wert f(x) charakterisiert. Die entstehende Datenstruktur ist extrem platzsparend, aber ungewöhnlich: Sie ist ungeeignet um Fragen der Form âist y [ELEMENT OF] S?â zu beantworten. Bei Anfrage eines SchlĂŒssels y wird ein Ergebnis z zurĂŒckgegeben. Falls y [ELEMENT OF] S ist, so ist z = f(y) garantiert, andernfalls darf z ein beliebiger Wert sein. Wir betrachten zwei neue Hashing-Schemata, bei denen die Elemente von e(x) in einem oder in zwei zusammenhĂ€ngenden Blöcken liegen. So werden gute Zugriffszeiten auf Word-RAMs und eine hohe Cache-Effizienz erzielt. Eine wichtige Frage ist, ob Datenstrukturen obiger Art in Linearzeit konstruiert werden können. Die Erfolgswahrscheinlichkeit eines naheliegenden Greedy-Algorithmus weist abermals ein Schwellwertverhalten in Bezug auf die Auslastung c auf. Wir identifizieren ein Hashing-Schema, das diesbezĂŒglich einen besonders hohen Schwellwert mit sich bringt. In der mathematischen Modellierung werden die Speicherpositionen [n] als Knoten und die Mengen e(x) fĂŒr x [ELEMENT OF] S als Hyperkanten aufgefasst. Drei Eigenschaften der entstehenden Hypergraphen stellen sich dann als zentral heraus: SchĂ€lbarkeit, Lösbarkeit und Orientierbarkeit. Weite Teile dieser Arbeit beschĂ€ftigen sich daher mit den Wahrscheinlichkeiten fĂŒr das Vorliegen dieser Eigenschaften abhĂ€ngig von Hashing Schema und Auslastung, sowie mit entsprechenden Schwellwerten. Eine RĂŒckĂŒbersetzung der Ergebnisse liefert dann Datenstrukturen mit geringen Anfragezeiten, hoher Speichereffizienz und geringen Konstruktionszeiten. Die theoretischen Ăberlegungen werden dabei durch experimentelle Ergebnisse ergĂ€nzt und gestĂŒtzt
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