Dynamic Perfect Hashing

In computer science, dynamic perfect hashing is a programming technique for resolving collisions in a hash table data structure.Fredman, M. L., Komlós, J., and Szemerédi, E. 1984. Storing a Sparse Table with 0(1) Worst Case Access Time. J. ACM 31, 3 (Jun. 1984), 538-544 http://portal.acm.org/citation.cfm?id=1884#Dietzfelbinger, M., Karlin, A., Mehlhorn, K., Meyer auf der Heide, F., Rohnert, H., and Tarjan, R. E. 1994
"Dynamic Perfect Hashing: Upper and Lower Bounds"
.
SIAM J. Comput. 23, 4 (Aug. 1994), 738-761.
http://portal.acm.org/citation.cfm?id=182370

While more memory-intensive than its hash table counterparts, this technique is useful for situations where fast queries, insertions, and deletions must be made on a large set of elements.

Details

Static case

FKS Scheme

The problem of optimal static hashing was first solved in general by Fredman, Komlós and Szemerédi. In their 1984 paper, they detail a two-tiered hash table scheme in which each bucket of the (first-level) hash table corresponds to a separate second-level hash table. Keys are hashed twice—the first hash value maps to a certain bucket in the first-level hash table; the second hash value gives the position of that entry in that bucket's second-level hash table. The second-level table is guaranteed to be collision-free (i.e. perfect hashing) upon construction. Consequently, the look-up cost is guaranteed to be O(1) in the worst-case.

In the static case, we are given a set with a total of entries, each one with a unique key, ahead of time.
Fredman, Komlós and Szemerédi pick a first-level hash table with size $s = 2(x-1)$ buckets.

To construct, entries are separated into buckets by the top-level hashing function, where $s = 2(x-1)$. Then for each bucket with entries, a second-level table is allocated with $k^2$ slots, and its hash function is selected at random from a universal hash function set so that it is collision-free (i.e. a perfect hash function) and stored alongside the hash table. If the hash function randomly selected creates a table with collisions, a new hash function is randomly selected until a collision-free table can be guaranteed. Finally, with the collision-free hash, the entries are hashed into the second-level table.

The quadratic size of the $k^2$ space ensures that randomly creating a table with collisions is infrequent and independent of the size of , providing linear amortized construction time. Although each second-level table requires quadratic space, if the keys inserted into the first-level hash table are uniformly distributed, the structure as a whole occupies expected $O(n)$ space, since bucket sizes are small with high probability.

The first-level hash function is specifically chosen so that, for the specific set of unique key values, the total space used by all the second-level hash tables has expected $O(n)$ space, and more specifically $T < s + 4 \cdot x$.
Fredman, Komlós and Szemerédi showed that given a universal hashing family of hash functions, at least half of those functions have that property.

Dynamic case

Dietzfelbinger et al. present a dynamic dictionary algorithm that, when a set of n items is incrementally added to the dictionary, membership queries always run in constant time and therefore $O(1)$ worst-case time, the total storage required is $O(n)$ (linear), and $O(1)$ expected amortized insertion and deletion time (amortized constant time).

In the dynamic case, when a key is inserted into the hash table, if its entry in its respective subtable is occupied, then a collision is said to occur and the subtable is rebuilt based on its new total entry count and randomly selected hash function. Because the load factor of the second-level table is kept low $1/k$, rebuilding is infrequent, and the amortized expected cost of insertions is $O(1)$. Similarly, the amortized expected cost of deletions is $O(1)$.

Additionally, the ultimate sizes of the top-level table or any of the subtables is unknowable in the dynamic case. One method for maintaining expected $O(n)$ space of the table is to prompt a full reconstruction when a sufficient number of insertions and deletions have occurred. By results due to Dietzfelbinger et al., as long as the total number of insertions or deletions exceeds the number of elements at the time of last construction, the amortized expected cost of insertion and deletion remain $O(1)$ with full rehashing taken into consideration.

The implementation of dynamic perfect hashing by Dietzfelbinger et al. uses these concepts, as well as lazy deletion, and is shown in pseudocode below.

Pseudocode implementation

Locate

function Locate(''x'') is
''j'' := h(x)
if (position h_j(''x'') of subtable ''T_j'' contains ''x'' (not deleted))
return (''x'' is in ''S'')
end if
else
return (''x'' is not in ''S'')
end else
end

Insert

During the insertion of a new entry ''x'' at ''j'', the global operations counter, ''count'', is incremented.

If ''x'' exists at ''j'', but is marked as deleted, then the mark is removed.

If ''x'' exists at ''j'' or at the subtable ''T_j'', and is not marked as deleted, then a collision is said to occur and the ''j''^th bucket's second-level table ''T_j'' is rebuilt with a different randomly selected hash function ''h_j''.

function Insert(''x'') is
''count'' = ''count'' + 1;
if (''count'' > ''M'')
FullRehash(''x'');
end if
else
''j'' = h(''x'');
if (Position h_''j''(x) of subtable ''T_j'' contains ''x'')
if (''x'' is marked deleted)
remove the delete marker;
end if
end if
else
''b_j'' = ''b_j'' + 1;
if (''b_j'' <= ''m_j'')
if position h_''j''(''x'') of ''T_j'' is empty
store ''x'' in position h_''j''(''x'') of ''T_j'';
end if
else
Put all unmarked elements of ''T_j'' in list ''L_j'';
Append ''x'' to list ''L_j'';
''b_j'' = length of ''L_j'';
repeat
''h_j'' = randomly chosen function in ''H_sj'';
until ''h_j'' is injective on the elements of ''L_j'';
for all ''y'' on list ''L_j''
store ''y'' in position h_''j''(''y'') of ''T_j'';
end for
end else
end if
else
''m_j'' = 2 * max;
''s_j'' = 2 * ''m_j'' * (''m_j'' - 1);
if the sum total of all s_j ≤ 32 * ''M''² / ''s''(''M'') + 4 * ''M''
Allocate ''s_j'' cells for ''T_j'';
Put all unmarked elements of ''T_j'' in list ''L_j'';
Append ''x'' to list ''L_j'';
''b_j'' = length of ''L_j'';
repeat
''h_j'' = randomly chosen function in ''H_sj'';
until ''h_j'' is injective on the elements of ''L_j'';
for all ''y'' on list ''L_j''
store ''y'' in position h_''j''(''y'') of ''T_j'';
end for
end if
else
FullRehash(''x'');
end else
end else
end else
end else
end

Delete

Deletion of ''x'' simply flags ''x'' as deleted without removal and increments ''count''. In the case of both insertions and deletions, if ''count'' reaches a threshold ''M'' the entire table is rebuilt, where ''M'' is some constant multiple of the size of S at the start of a new ''phase''. Here ''phase'' refers to the time between full rebuilds. Note that here the -1 in "Delete(''x'')" is a representation of an element which is not in the set of all possible elements ''U''.

function Delete(''x'') is
''count'' = ''count'' + 1;
''j'' = h(''x'');
if position h_j(''x'') of subtable ''Tj'' contains ''x''
mark ''x'' as deleted;
end if
else
return (x is not a member of S);
end else
if (''count'' >= ''M'')
FullRehash(-1);
end if
end

Full rebuild

A full rebuild of the table of ''S'' first starts by removing all elements marked as deleted and then setting the next threshold value ''M'' to some constant multiple of the size of ''S''. A hash function, which partitions ''S'' into ''s''(''M'') subsets, where the size of subset ''j'' is ''s_j'', is repeatedly randomly chosen until:

$\sum_ s_j \le \frac + 4M.$

Finally, for each subtable ''T_j'' a hash function ''h_j'' is repeatedly randomly chosen from ''H_sj'' until ''h_j'' is injective on the elements of ''T_j''. The expected time for a full rebuild of the table of ''S'' with size ''n'' is O(''n'').

function FullRehash(''x'') is
Put all unmarked elements of ''T'' in list ''L'';
if (''x'' is in ''U'')
append ''x'' to ''L'';
end if
''count'' = length of list ''L'';
''M'' = (1 + ''c'') * max;
repeat
h = randomly chosen function in ''H_s(M)'';
for all ''j'' < ''s''(''M'')
form a list ''L_j'' for h(''x'') = ''j'';
''b_j'' = length of ''L_j'';
''m_j'' = 2 * ''b_j'';
''s_j'' = 2 * ''m_j'' * (''m_j'' - 1);
end for
until the sum total of all s_j ≤ 32 * ''M''² / ''s''(''M'') + 4 * ''M''
for all ''j'' < ''s''(''M'')
Allocate space ''s_j'' for subtable ''T_j'';
repeat
''h_j'' = randomly chosen function in ''H_sj'';
until ''h_j'' is injective on the elements of list ''L_j'';
end for
for all ''x'' on list ''L_j''
store ''x'' in position h_''j''(''x'') of ''T_j'';
end for
end

See also

* Perfect hashing

References

{{DEFAULTSORT:Dynamic Perfect Hashing
Hashing
Search algorithms