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Blooming Trees: Space-Efficient Structures for Data Representation. Author: Domenico Ficara, Stefano Giordano, Gregorio Procissi, Fabio Vitucci Publisher: ICC 2008 Presenter: Yu-Ping Chiang Date: 2009/05/20. Outline. Blooming Tree Lookup Insert Delete Optimized Blooming Tree - PowerPoint PPT Presentation
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1
Blooming Trees: Space-Efficient Structures
for Data Representation
Author: Domenico Ficara, Stefano Giordano, Gregorio Procissi, Fabio Vitucci
Publisher: ICC 2008Presenter: Yu-Ping ChiangDate: 2009/05/20
2
Outline Blooming Tree
Lookup InsertDelete
Optimized Blooming TreeLookup InsertDelete
Simulations
3
Blooming Tree
B0
B1
B2
B3
0
1 1 1 10 0 0 0
1 2 1 1 1 1
1 1 1 10 0 0 0
1 1 1 1 1 10 0
0 0 0
0 0
1
1 1 1
3 items 2 items1 item
item
Bit string
HASH FUNCTION
3 bits 1 bit 1 bit
index
index
index
4
Blooming Tree n items, k0 hash functions, L+2 layers
Layer0 (B0) : m = nk0/ln2 bits
Layer1~L (B1~BL) : bits/block ( b=1 in following examples ) Block numbers is modified
LayerL+1 (BL+1) : Composed c-bits counters
Hash function k0 hash functions log m + L*b bits output
log m bit for layer0 B bits for layer1~layerL+1
B0
B1
B2
B3
0
1 1 1 10 0 0 0
1 2 1 1 1 1
1 1 1 10 0 0 0
1 1 1 1 1 10 0
0 0 0
0 0
1
1 1 1
3 items 2 items1 item
B0
B1
B2
B3
0
1 1 1 10 0 0 01 1 1 10 0 0 0
1 2 1 1 1 11 2 1 1 1 1
1 1 1 10 0 0 01 1 1 10 0 0 0
1 1 1 1 1 10 01 1 1 1 1 10 0
0 0 0
0 0
1
1 1 1
3 items 2 items1 item
b2
5
Blooming Tree - lookup Algorithm:
Using first log m bits as layer0 index. Compute a popcount on layer i, that gives index of the couple in layer i+1. Checking the bit string output by hash function, the bit for layer i.
0 for first bit. 1 for second bit.
If processing bit is 0, result NOT FOUND. Otherwise continue search in next layer. Time complexity:
k0 [ hash + L ( popcount + 2 * check ) ]
B0
B1
B2
B3
1 1 1 10 0 0 0
1 2 1 1 1 1
1 1 1 10 0 0 0
1 1 1 1 1 10 0
item1 bit string: 01010hash
6
Blooming Tree - lookup Algorithm:
Using first log m bits as layer0 index. Compute a popcount on layer i, that gives index of the couple in layer i+1. Checking the bit string output by hash function, the bit for layer i.
0 for first bit. 1 for second bit.
If processing bit is 0, result NOT FOUND. Otherwise continue search in next layer. Time complexity:
k0 [ hash + L ( popcount + 2 * check ) ]
B0
B1
B2
B3
1 1 1 10 0 0 0
1 2 1 1 1 1
1 1 1 10 0 0 0
1 1 1 1 1 10 0
1
item1 bit string: 01010hash
7
Blooming Tree - lookup Algorithm:
Using first log m bits as layer0 index. Compute a popcount on layer i, that gives index of the couple in layer i+1. Checking the bit string output by hash function, the bit for layer i.
0 for first bit. 1 for second bit.
If processing bit is 0, result NOT FOUND. Otherwise continue search in next layer. Time complexity:
k0 [ hash + L ( popcount + 2 * check ) ]
B0
B1
B2
B3
1 1 1 10 0 0 0
1 2 1 1 1 1
1 1 1 10 0 0 0
1 1 1 1 1 10 0
0
1
item1 bit string: 01010hash
Match !!
8
Blooming Tree - lookup Algorithm:
Using first log m bits as layer0 index. Compute a popcount on layer i, that gives index of the couple in layer i+1. Checking the bit string output by hash function, the bit for layer i.
0 for first bit. 1 for second bit.
If processing bit is 0, result NOT FOUND. Otherwise continue search in next layer. Time complexity:
k0 [ hash + L ( popcount + 2 * check ) ]
B0
B1
B2
B3
1 1 1 10 0 0 0
1 2 1 1 1 1
1 1 1 10 0 0 0
1 1 1 1 1 10 0
item2 bit string: 10000hash
NOT FOUND !!
9
Blooming Tree - insert Algorithm:
Using first log m bits as layer0 index. In layer1~layerL+1, using popcount of layer0~layerL and bit for each layer as inde
x. If bit in layer I already set (means COLLOSION), directly set bit in layer i+1.
else, allocate a new block and insert it into original layer i+1 blocks. Increase count at layer L+1.
Time complexity: k0 [ hash + L ( popcount + shift + bitset ) ]
B0
B1
B2
B3
0 1 0 00 0 0 0
10
1 0
1
item1 bit string: 01010hash
allocate a new block(2^b bits)allocate a new block(2^b bits)
allocate a new block(2^b bits)
10
Blooming Tree - insert Algorithm:
Using first log m bits as layer0 index. In layer1~layerL+1, using popcount of layer0~layerL and bit for each layer as inde
x. If bit in layer I already set (means COLLOSION), directly set bit in layer i+1.
else, allocate a new block and insert it into original layer i+1 blocks. Increase count at layer L+1.
Time complexity: k0 [ hash + L ( popcount + shift + bitset ) ]
B0
B1
B2
B3
1 1 0 00 0 0 0
1 1
10
1 1 0
1 0
0
item2 bit string: 00101hash
allocate a new block(2^b bits)allocate a new block(2^b bits)allocate a new block(2^b bits)
11
Blooming Tree - insert Algorithm:
Using first log m bits as layer0 index. In layer1~layerL+1, using popcount of layer0~layerL and bit for each layer as inde
x. If bit in layer I already set (means COLLOSION), directly set bit in layer i+1.
else, allocate a new block and insert it into original layer i+1 blocks. Increase count at layer L+1.
Time complexity: k0 [ hash + L ( popcount + shift + bitset ) ]
B0
B1
B2
B3
1 1 1 10 0 0 0
2 1
10
1 1 0
1 0
0
item3 bit string: 00101hash
Collision occur
12
Blooming Tree - delete Algorithm:
Trace to the last layer, decrease count. If counter isn’t equal to 0, terminal processing.
else, remove the block and checking upper layer if there only this item in the block, if yes, remove that block too.recursive processing upper layers.
B0
B1
B2
B3
1 1 1 10 0 0 0
2 1
10
1 1 0
1 0
item1 bit string: 00100hash
1
10
Remove empty block
0
13
Blooming Tree - delete Algorithm:
Trace to the last layer, decrease count. If counter isn’t equal to 0, terminal processing.
else, remove the block and checking upper layer if there only this item in the block, if yes, remove that block too.recursive processing upper layers.
B0
B1
B2
B3
1 1 1 10 0 0 0
2 1
10
1 1 0
1 0
item2 bit string: 01001hash
0
1
14
Outline Blooming Tree
Lookup InsertDelete
Optimized Blooming TreeLookup InsertDelete
Simulations
15
Optimized Blooming Tree
B0
B1
B2
B3
1 1 1 10 0 0 0
1 2 1 1
1 0 0 10 0 0 0
1 1 1 1
01
1 1
1 10 0
0 1
3 items 2 items1 item
01100 01 1
bitmap Hash substrings
1 1
16
Optimized Blooming Tree - lookup Algorithm:
Access B0 Checking bitmap
If there’s 1 in bitmap, directly compare last L*b bits of hashed bit string, and terminate processing.
Else, lookup method is same as previous defined. Recursively repeat at each level.
B0
B1
B2
B3
1 1 1 10 0 0 0
1 2 1 1
1 0 0 10 0 0 0
1 1 1 1
3 items 2 items1 item
01100 01 1
bitmap Hash substrings
B0
B1
B2
B3
1 1 1 10 0 0 01 1 1 10 0 0 0
1 2 1 1
1 0 00 0 10 0 0 0
1 1 1 1
3 items 2 items1 item
01100 01 1
bitmap Hash substrings
01100 01 1
bitmap
0110 01100 01 10 01 1
bitmap Hash substrings
17
Optimized Blooming Tree - lookup
1 2 1 1
B0
B1
B2
B3
1 1 1 10 0 0 0
1 0 0 10 0 0 0
1 1 1 1
01100 01 1
bitmap Hash substrings
Algorithm: Access B0 Checking bitmap
If there’s 1 in bitmap, directly compare last L*b bits of hashed bit string, and terminate processing.
Else, lookup method is same as previous defined. Recursively repeat at each level.
item1 bit string : 10001hash
Popcount = 3Popcount = 2
18
Optimized Blooming Tree - insert Without collision
Add a zero-block Set bit string and hash substring
B0
B1
B2
B3
1 1 1 10 0 0 0
1 2 1 1
1 0 0 10 0 0 0
1 1 1 1
01100 01 1
bitmap Hash substrings
item1 bit string : 01101hash
1
0 0
0110
Hash substrings
0 11 1
bitmap
0 01
19
Optimized Blooming Tree - insert With collision
Set corresponding branches
B0
B1
B2
B3
1 1 1 10 0 0 0
1 2 1 1
1 0 10 0 0
1 1 1 1
00 01100 01 1
bitmap Hash substrings
item2 bit string : 01001hash
10
00 00 1 0
11
00 010 1 1 0
010 00 1
20
Optimized Blooming Tree - delete
B0
B1
B2
B3
1 1 1 10 0 0 0
1 2 1 1
1 0 10 0 0
1 1 1 1
item2 bit string : 01001hash
00
11
00 010 1 1 0
010 00 1
0
0 1 0
0
21
Outline Blooming Tree
Lookup InsertDelete
Optimized Blooming TreeLookup InsertDelete
Simulations
22
Simulation Size comparison
23
Simulation Build on NP Intel IXP2800
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