Engineering%20a%20Sorted%20List%20Data%20Structure%20for%2032%20Bit%20Keys

1
Engineering a Sorted List Data Structure for 32
Bit Keys

• Roman Dementiev
• Lutz Kettner
• Jens Mehnert
• Peter Sanders
• MPI für Informatik,
• Saarbrücken

2
Introduction

• The power of integer keys helps in
• Sorting (radix MSB,LSB)
• Priority queues (radix heaps)
• Static search trees
• Dictionaries (hash tables)
• Faster both in theory and practice
• What about dynamic search data structures?

3
Motivation

• van Emde Boas (vEB) search trees van Emde
  Boas77,MehlhornNaeher90
• Small K, large n ? vEB are faster ?
• NO, their direct implementations are 2-8 times
  slower than comp. based trees Wenzel92,here
• Here a tuned vEB data structure that outperforms
  comp. based implementations

operation comparison based van Emde Boas
insert, delete, search O(log n) O(log K)
range query O(c log n) O(c log K)
n number of elements K bit width of keys c
size of the output
4
Direct vEB Implementation

• vEB tree maintains set
• Recursive definition
• M1 or K1 store directly,
• otherwise let K K/2 store minM,maxM,
• top store (top
  recursion)
• boti store
  
• (bottom recursion) use hash table

K bit vEB top
hash table
K bit vEB boti
5
Improvement 1

• Replace top data structure with a bit pattern
  hierarchy

K bit vEB top
hash table
K bit vEB boti
6
Improvement 2

• Break recursion
• when K8
• 3 levels max.

hash table
7
Improvement 3
63
0

• Replace root hash table
• with an array

31
32
63
0
4095




31
32
0
63
65535




8
Range Query Support
63
0

• Link elements

31
32
63
0
4095




31
32
0
63
65535




Level 1 root Bits 31-16
Level 2 Bits 15-8
Level 2 Bits 15-8
Level 3 Bits 7-0
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Example Locate Operation

• return handle of
• Function locate(yN)ElementHandle
• if y gt maxM then return ? // no larger element
• i y16..31 // index into root table top
• if topinull or ygtmaxMi then return
  minMtop.locate(i) // look in the next L2 table
• if Mix then return x // single element case
• j y8..15 // key for L2 table at Mi
• if rijnull or y gt maxMij then return
  minMi,top(i).locate(j) // look in the next L3
  table
• if Mijx then return x // single element
  case
• return rijtopij.locate(y0..7) // L3 table
  access
• At most 9 comparisons for any input sizes

10
Locate Performance
11
Construction
12
Deletion
13
Hard Inputs
14
Conclusions and Future Work

• Integer search trees can outperform comp. based
  search data struct.
• Future work
• Support multi-set functionality
• Other key lengths (up to 38 bits)
• Reduce space consumption
• Find real inputs
• Port it to the LEDA library