U. Glaeser

FIGURE 32.2 Extendible hashing with block size B = 3. The keys are indicated in italics. For convenience of
exposition, the hash address of a key consists of its binary representation. For example, the hash address of key 4 is
“…000100” and the hash address of key 44 is “…0101100”. (a) The hash table after insertion of the keys 4, 23, 18,
10, 44, 32, 9. (b) Insertion of the key 76 into table location 100 causes the block with local depth 2 split into two
blocks with local depth 3. (c) Insertion of the key 20 into table location 100 causes a block with local depth 3 to split
into two blocks with local depth 4. The directory doubles in size and the global depth d is incremented from 3 to 4.
range of N. The challenge is to develop dynamic EM structures that can adapt smoothly to widely varying
values of N.
EM hashing methods fall into one of two categories: directory methods and directoryless methods. Fagin
et al. [79] proposed a directory scheme called extendible hashing, illustrated in Fig. 32.2: The directory,
for a given d ≥ 0, consists of a table (array) of 2 d pointers. Each item is assigned to the table location
corresponding to the d least significant bits of its hash address. The value of d, called the global depth,
is set to the smallest value for which each table location has at most B items assigned to it. Each table
location contains a pointer to a block where its items are stored. Thus, a lookup takes two I/Os: one to
access the directory and one to access the block storing the item. If the directory fits in internal memory,
only one I/O is needed. Several table locations may have many fewer than B assigned items, and for
purposes of minimizing storage utilization, they can share the same disk block for storing their items by
using a local depth smaller than the global depth. When new items are inserted and deleted, the blocks
can overflow or underflow, and the local depths and global depth are changed accordingly.
The expected number of disk blocks required to store the data items is asymptotically n/ln 2 ≈ n/0.69;
that is, the blocks tend to be about 69% full [147]. At least Ω(n/B) blocks are needed to store the directory.
Flajolet [86] showed on the average that the directory uses Θ(N 1/B n/B) = Θ(N 1+1/B /B 2 ) blocks, which
can be superlinear in N asymptotically; however, for practical values of N and B, the N 1/B term is a small
constant, typically less than 2, and directory size is within a constant factor of optimal.
The resulting directory is equivalent to the leaves of a perfectly balanced trie [124], in which the search
path for each item is determined by its hash address, except that hashing allows the leaves of the trie to
© 2002 by CRC Press LLC
2
000 000
001
010
011
100
101
110
111
3
1
3
local depth
4 44 32
001
18 18
23 9
10
010
011
100
101
110
111
global depth d = 3 global depth d = 3
(a) (b)
3
3
1
3
3
32
23 9
4 44 76
10
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
3
3
1
4
3
4
global depth d
= 4
(c)
32
18
23 9
4 20
10
44 76