Comparison of Linear Hashing Techniques: LHRBI, MDEH, and PLOP, Study notes of Computer Science

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Presentation on

MULTIDIMENSIONAL

LINEAR HASHING

by B Brent Gordon CMSC 828S, Spring 2005

• Introduction & basic principles
• Linear Hashing with Reversed Bit Interleaving (LHRBI)
• Multidimensional Extendible Hashing (MDEH)
• Comparison of LHRBI and MDEH
• Quantile Hashing
• Piecewise Linear Order Preserving (PLOP)
• Dynamic Z Hashing
• Linear Hashing with Partial Expansions (LHPE)

MULTIDIMENSIONAL LINEAR HASHING

Intro Linear Hashing in general: Create one additional cell at a time May be based on Overflow of bucket or cell, or Storage Utilization Factor Linear Hashing in dimension d > 1 d -dimensional hypervolume is subdivided into d -dimensional cells Cells are linearly ordered, typically in order of creation Linear Hashing

The hashing function must map points in d -space to a unique cell/bucket number Intro 0 1 2 3 4 5 6 7 8 9 10 Buckets d -dimensional hypercube

Intro Total grid partition: number of cells is doubled (But cells have same shape as original only every d th total grid partition (tgp). Example d = 2. Number of cells = 2 Total grid partition 1 Cells are not square

Intro Total grid partition: number of cells is doubled (But cells have same shape as original only every d th total grid partition (tgp). Example d = 2. Number of cells = 4 Total grid partition 2 Cells are square

Intro Total grid partition: number of cells is doubled (But cells have same shape as original only every d th total grid partition (tgp). Example d = 2. Number of cells = 16 Total grid partition 4 Cells are square

Intro Bit Concatonation: (4, 3) 10 = ( 100 , 011 ) 2 ( 100011 ) 2 = (35) 10 y -first Bit Interleaving: (4, 3) 10 = ( 100 , 011 ) 2 ( 011010 ) 2 = (26) 10 y -first Reversed Bit Interleaving: (4, 3) 10 = ( 100 , 011 ) 2 ( 010110 ) 2 = (22) 10 (reverse the previous one)

Bit Operations

LHRBI MAIN POINTS:

Decouple cell overflow from cell split by

having chosen the order of splitting in

advance.

Order preserving.

LHRBI

LHRBI The Specifics:

1. Embedding space-based
2. Always split next smallest-numbered cell. 3. y -first reversed bit interleaving on partition indices
3. Count partition indices by distance from origin 0 0 0

LHRBI

LHRBI

LHRBI

Key Points:

1. Order preserving, i.e., always split next smallest- numbered cell. 2. y -first reversed bit interleaving on partition indices
2. Count partition indices by distance from origin 0 0 0 1 1 2 1 Split cell 0

LHRBI

LHRBI

Key Points:

1. Order preserving, i.e., always split next smallest- numbered cell. 2. y -first reversed bit interleaving on partition indices
2. Count partition indices by distance from origin 0 0 0 1 1 2 1 3 Split cell 1 Finish total grid partition 2

LHRBI

LHRBI

Key Points:

1. Order preserving, i.e., always split next smallest- numbered cell. 2. y -first reversed bit interleaving on partition indices
2. Count partition indices by distance from origin 0 0 0 1 1 2 1 3 4 2 Notice Split cell 1 3 5

LHRBI

LHRBI

Key Points:

1. Order preserving, i.e., always split next smallest- numbered cell. 2. y -first reversed bit interleaving on partition indices
2. Count partition indices by distance from origin 0 0 0 1 1 2 1 3 4 2 Split cell 2 3 5 6