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Paper text:
hash table is a common data structure used in large set of data storage and retrieval .
it has an o(1 ) lookup time on average , but the worst case lookup time can be as bad as o(@xmath1)(@xmath1 is the size of the hash table )
. such a time variation is essentially caused by possibly many collisions during keys hashing . in this paper
, we present a set of hash table schemes called cascade hash tables which consist of several levels(@xmath2 ) of hash tables with different size .
after constant probes , if an item ca nt find a free  cell"(slot ) in the first level table , it will try to find a cell in the second level , or subsequent lower levels . with this simple strategy
, these hash tables will have descendent load factors , therefore descendent collision probabilities .
so finally the probability that the item can not find an empty cell in any hash table is slight .
this enables the whole hash table to reach a high load factor with constant probes in random generated test set , before a _
crisis_(the situation that when an item comes , we can not find a free slot in limited probes in any hash table ) happens .
a hash table is a data structure that associates keys with values .
the primary operation it supports efficiently is a _
lookup _ : given a key ( e.g. a person s name ) , find the corresponding value ( e.g. that person s telephone number ) .
it works by transforming the key using a _ hash function _ into a hash  a number that the hash table uses to locate the desired value .
a hash function is a many - to - one mapping , which maps keys in a large domain to hashes in a relative small range . so collisions among keys which are mapped to the same hash are inevitable .
differences among hash table schemes lie in hash function and collision resolution strategy .
generally a string type key is hashed into an integer by a hash function , then mapped into an index not bigger than the table size(a common method is to compute the hash value modulo the table size .
there are various hash functions on strings , such as crc , _
lookup2 _ and md5 . as to integer type keys ,
they are directly mapped into indices .      if two keys hash to the same index , the corresponding records can not be stored in the same location .
so , if it s already occupied , we must find another location to store the new record , and do it so that we can find it when we look it up later on .
the most popular collision resolution techniques are _ chaining _ and _ open addressing_.    in the chained hash table technique , each slot in the array references a linked list of inserted records that collide to the same slot .
insertion requires finding the correct slot , and appending to either end of the list in that slot ; deletion requires searching the list and removal .
this technique is intuitive and the performance degrades gracefully when the load factor increases .
but if the record size is small , the overhead of the linked list is significant .
additionally , traversing a linked list has poor cache performance .
some popular hash table implementations , such as stl , use this technique .
open addressing hash tables store the colliding records directly within the array .
a hash collision is resolved by probing through alternate locations in the array(the probe sequence ) until either the target record is found , or an unused array slot is found , which indicates that there is no such key in the table .
well known probe sequences include : linear probing , in which the interval between probes is fixed  often at 1 ; quadratic probing , in which the interval between probes increases linearly ( hence , the indices are described by a quadratic function ) ; double hashing , in which the interval between probes is fixed for each record but is computed by another hash function .
any of these methods may probe indefinite number of locations , even as many as @xmath1 in the worst case !
we call these methods _ unlimited_. a method is _ limited _ , if the number of probes can not exceed some limit .
in the implementation of @xmath3-level cascade hash table , we have @xmath3 hash tables , and use limited double hashing in every level of table . in turn , the hash table size is half of its preceding hash table(the proportion @xmath4 is chosen empirically ) .
we limit the total number of probes to @xmath5 .
thus in every level , the probe number is @xmath6 . here
@xmath3 is a factor of @xmath5 , so @xmath7 . if an item ca nt find a free cell in level @xmath8 in @xmath9 probes , it will probe in level 2 , and if still with bad luck , it turns to search lower levels .
if a crisis happens , the hash table will be enlarged and rehashed .
the lookup procedure is similar to the insertion procedure .
it also takes not more than @xmath5 steps .    clearly , insertion and
lookup both take at most @xmath5 probes , so the time complexity of cascade hash table is o(1 ) .    specially , when @xmath10 , it s the ordinary ( limited ) double - hashing scheme . when @xmath11(one probe every level )
, it s the  multilevel adaptive hashing " scheme presented by paper@xcite .
experiments show that it is not the best configuration .
are all occupied ) ]
by experiments on random data sets , we get the result as in table [ result1 ] , [ result2 ] and [ result3 ] .    .experiment result 1 [ cols="<,^,^,^,^,^,^ " , ]     from these tables , we can see that in a certain level of a multiple level hash table , the load factor decreases drastically .
take three - level hash table for example , when hash tables are  full"(not really full , but we can not insert the coming new item into it in limited probes ) , @xmath12 is around @xmath13 , @xmath14 is @xmath15 , but @xmath16 is only around @xmath17 .
so if a new item comes , the _ crisis rate_(the probability that the item ca nt find an empty room in any hash table ) is not bigger than @xmath18 .
given a one level hash table , assume it has a load factor of @xmath19 , then it will take at least @xmath20 probes on average to obtain the same small crisis rate(which ensures a high load factor ) as 3-level cascade hash table .
but cascade hash table just makes @xmath5 probes meanwhile .    to our surprise
, the space efficiency of @xmath3-level hash table does nt increase monotonically with @xmath3 . at @xmath21 ,
the space efficiency hits the climax , then falls down at @xmath11 .
in this paper , we introduce a series of hash table schemes  cascade hash tables .
it uses @xmath3 levels of hash tables ; in every level , we use limited double hashing to make probes .
smaller hash tables work as fail - safes of bigger hash tables . roughly speaking ,
different tables are similar with sieves with holes in different shapes .
we hope no object escape through these sieves . with more sieves ,
the chance that an object is screened in some level is bigger .
the idea is simple , but its performance exceeds the ordinary one level hash table dramatically when there are more than @xmath22 levels . by choosing @xmath21 , it s much better than the hash scheme proposed in paper@xcite .
obviously , if we permit a larger total probe count , we can achieve higher load factor .
but the average speed will be slower .
so a user can choose an appropriate configuration which balances best between speed and space efficiency to him .
12 andrei z. broder , anna r. karlin .
multilevel adaptive hashing . in proceedings of the 1st acm - siam symposium on discrete algorithms ( soda ) , pp .
43 - 53 , 1990 .
bill mcquain .
hashing , slides for class cs 2604 .
http://courses.cs.vt.edu/~cs2604/summerii_2005/notes/c10.hashing.pdf thomas h. cormen , charles e. leiserson , ronald l. rivest and clifford stein .
introduction to algorithms , second edition . the mit press 
hash table " entry in wikipedia .
in this paper , the author proposes a series of multilevel double hashing schemes called cascade hash tables . 
 they use several levels of hash tables . in each table 
 , we use the common double hashing scheme . 
 higher level hash tables work as fail - safes of lower level hash tables . by this strategy 
 , it could effectively reduce collisions in hash insertion . 
 thus it gains a constant worst case lookup time with a relatively high load factor(@xmath0 ) in random experiments . 
 different parameters of cascade hash tables are tested .