Engineering a Sorted List Data
Structure for 32 Bit Keys
Roman Dementiev
Lutz Kettner
Jens Mehnert
Peter Sanders
MPI für Informatik,
Saarbrücken
Introduction
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The power of integer keys helps in
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2
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?
R. Dementiev et al.: A Sorted List Data Structure for 32 Bit Keys
(ALENEX'04)
Motivation

van Emde Boas (vEB) search trees [van Emde Boas77,MehlhornNaeher90]:
operation
comparison based
van Emde Boas
insert, delete, search
O(log n)
O(log K)
O(c + log n)
O(c + log K)
range query
3
n – number of elements
K – bit width of keys
c – size of the output

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
R. Dementiev et al.: A Sorted List Data Structure for 32 Bit Keys
(ALENEX'04)
Direct vEB Implementation
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vEB tree maintains set M  0...2K  1
Recursive definition:
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|M|=1 or K=1: store directly,
otherwise let K’ = K/2: store minM,maxM,
K'
 top: store {x div 2 : x  M } (top recursion)
K'
K'
 boti: store {x mod 2 : x  M , x div 2  i}
(bottom recursion) use hash table
K’ bit
vEB
top
hash table
K’ bit
vEB
boti
4
R. Dementiev et al.: A Sorted List Data Structure for 32 Bit Keys
(ALENEX'04)
Improvement 1

Replace top data structure with a bit pattern hierarchy
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3132
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3132
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K’ bit
vEB
top
4095
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65535
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hash table
K’ bit
vEB
boti
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R. Dementiev et al.: A Sorted List Data Structure for 32 Bit Keys
(ALENEX'04)
Improvement 2
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Break recursion
when K=8
3 levels max.
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3132
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3132
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65535
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Level 1 – root
Bits 31-16
hash table
Level 2
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Bits 15-8
single elements
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Level 2
Bits 15-8
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K’ bit
vEB
boti
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hash table
hash table
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Level 3
Bits 7-0
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hash table
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R. Dementiev et al.: A Sorted List Data Structure for 32 Bit Keys
(ALENEX'04)
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Improvement 3
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Replace root hash table
with an array
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3132
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4095
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3132
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63
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65535
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0
65535
0 0 0 0 0
hasharray
table
0 0 0 0
0 0 0 0 0 0 0 0 0
0 0
0 0 0 0
0
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0
Level 1 – root
Bits 31-16
0 0
Level 2
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Bits 15-8
single elements
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Level 2
Bits 15-8
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hash table
hash table
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Level 3
Bits 7-0
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hash table
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R. Dementiev et al.: A Sorted List Data Structure for 32 Bit Keys
(ALENEX'04)
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Range Query Support
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Link elements
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3132
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4095
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3132
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65535
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65535
0 0 0 0 0
0 0 0 0
0 0 0 0 0 0 0 0 0
array
0 0
0 0 0 0
0
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0
Level 1 – root
Bits 31-16
0 0
Level 2
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Bits 15-8
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Level 2
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Bits 15-8
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hash table
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hash table
Bits 7-0
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hash table
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R. Dementiev et al.: A Sorted List Data Structure for 32 Bit Keys
(ALENEX'04)
Level 3
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Example: Locate Operation
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return handle of min x  M : y  x
Function locate(y:N):ElementHandle
if y > maxM then return 
i := y[16..31]
if top[i]=null or y>maxMi then return minMtop.locate(i)
if Mi={x} then return x
j := y[8..15]
if ri[j]=null or y > maxMij then return minMi,top(i).locate(j)
if Mij={x} then return x
return rij[topij.locate(y[0..7])]

9
At most 9 comparisons for any input sizes
R. Dementiev et al.: A Sorted List Data Structure for 32 Bit Keys
(ALENEX'04)
// no larger element
// index into root table top
// look in the next L2 table
// single element case
// key for L2 table at Mi
// look in the next L3 table
// single element case
// L3 table access
Locate Performance
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R. Dementiev et al.: A Sorted List Data Structure for 32 Bit Keys
(ALENEX'04)
Construction
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R. Dementiev et al.: A Sorted List Data Structure for 32 Bit Keys
(ALENEX'04)
Deletion
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R. Dementiev et al.: A Sorted List Data Structure for 32 Bit Keys
(ALENEX'04)
Hard Inputs
 225 
M  {2 i,2 i  255 : i  0.. | M | / 2},   
, queries for 256j  128 for random j  0.. | M | / 2
| M | 
8
13
8
R. Dementiev et al.: A Sorted List Data Structure for 32 Bit Keys
(ALENEX'04)
Conclusions and Future Work


Integer search trees can outperform comp. based search
data struct.
Future work:
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Support multi-set functionality
Other key lengths (up to 38 bits)
Reduce space consumption
Find real inputs
Port it to the LEDA library
R. Dementiev et al.: A Sorted List Data Structure for 32 Bit Keys
(ALENEX'04)