Data Structures:
Range Queries - Space Efficiency
Pooya Davoodi
Aarhus University
PhD Defense
July 4, 2011
Thesis Overview
25
12
14
4
 Range Minimum Queries in Arrays
(ESA 2010, Invited to Algorithmica)
 Path Minima Queries in Trees
(WADS 2011)
6
76
18
98
8
31
79
45
20
7
5
12
43
23
6
4
10
17
84
11
65
62
10
38
7
9
8
58
13
5
40
25
46
4 2 3 10
 Range Diameter Queries in 2D Point Sets
(Submitted to ISAAC 2011)
a
 Succinct 𝑘-ary Trees
c
(TAMC 2011)
b
d e
f
2
Range Minimum Queries
 Database systems
– Lowest average-salary:
Year
Age
1990
1995
2000
2005
2010
60
80,000
85,000
115,000
120,000
118,000
50
81,000
73,000
90,000
100,000
94,000
40
65,000
75,000
86,000
92,000
95,000
30
72,000
91,000
89,000
103,000
102,000
20
35,000
45,000
42,000
50,000
41,000
Minimum:
65,000 at [3,1]
3
Definition
 Input: an array 𝐴 = 1 . . 𝑚 × 1. . 𝑛
(𝑚 ≤ 𝑛)
 Query: where is minimum in 𝑖1 . . 𝑖2 × 𝑗1 . . 𝑗2 ?
𝑛
𝑖1
𝑚
𝑖2
𝑗1
𝑗2
4
Naïve Solution
 Brute force search
 Query time: 𝑂(𝑠 ∙ 𝑠′) time
 Worst case : 𝑂(𝑁) time (𝑁 = 𝑚 ∙ 𝑛)
𝑛
𝑠′
𝑚
𝑠
5
Data Structures
 Preprocess and store some information
 Naïve: store the answers of all queries
Top-Left
1
2
1
2
12
5
8
10
Tabulation
 𝑂(1) query time
 Size of the table:
𝑂(𝑁 2 log 𝑁) bits
Bottom-Right Minimum
(1,1)
(1,1)
(1,1): 12
(1,1)
(1,2)
(1,2): 8
(1,1)
(2,1)
(2,1): 5
(1,1)
(2,2)
(2,1): 5
(2,1)
(2,1)
(2,1): 5
(2,1)
(2,2)
(2,2): 5
(1,2)
(1,2)
(1,2): 8
(1,2)
(2,2)
(1,2): 8
(2,2)
(2,2)
(2,2): 10
6
Space-Efficient Data Structures
Reference
Tabulation
Tarjan et al.
(STOC’84)
Chazelle & Rosenberg
(SoCG’89)
Lewenstein et al.
(CPM’07)
Demaine et al.
(ICALP’09)
Sadakane
(ISAAC’07)
Our Result
(ESA’10)
Our Result
(ESA’10)
Space (bits)
𝑂 𝑁 2 log 𝑁
Query Time
𝑂 1
𝑂 𝑁 log 2 𝑁
𝑂 log 𝑁
𝑂 𝑁 log 𝑁
𝑂 𝛼 𝑁
𝑂 𝑁 log 𝑁
(𝑚 ≤ 𝑛)
Ω 𝑁 log 𝑁
(𝑚 = 𝑛)
𝑂 𝑁
(𝑚 = 1)
Ω 𝑁 log 𝑚
(𝑚 ≤ 𝑛)
𝑂 𝑁 ∙ min 𝑚, log 𝑁
(𝑚 ≤ 𝑛)
2
𝑂 1
𝑂 1
-
𝑂 1
7
1D vs. 2D
 1D: Cartesian Trees
7
 𝑂(log 𝑁) bits per element
5
20
8
(Tarjan et al., STOC’84)
6
10
 𝑂(1) bits per element
(Sadakane, ISAAC’07)
Lowest Common Ancestor
2
7
20
2
10
16
8
5
16
6
 2D: Nothing like Cartesian Trees
Ω(log 𝑚) bits per element 𝑚
(Our Result, ESA’10)
8
Indexing Data Structures
 Popular in Succinct Data Structures
Read-only
Index
Input Array
𝑚×𝑛=𝑁
𝑂(1)
𝑂
𝑂 𝑁 bits
𝑁 log 𝑁 bits
Size of Input
1
𝑐
bits per element
𝑂(1)
𝑂
𝑁
𝑐
bits
𝑂(𝑐 log 2 𝑐)
𝑂
𝑁
𝑐
bits
Ω(𝑐)
Size of Index
Query Time
(Our Results, ESA’10)
9
𝑂(𝑁) bits with 𝑂(1) query time
27 30
2
90 28 15 18
6
13 20 93 54 17 11 16 12
7
74 39 62
61 38 68 10
8
9
5
2
46 23
7
5
20
Cartesian Trees
8
6
16
10
87 98 21
7
20
2
10
8
5
16
6
Cartesian Tree:
log 𝑛
log log 𝑛
log log 𝑚
log 𝑚
Tabulation
Atallah and Yuan (SODA’10)
10
𝑂
 𝑁/𝑐 bits
 𝑁/𝑐 bits
1
bits Per Element
𝑐
𝑂(𝑐 log 2 𝑐) query time
Ω(𝑐) query time
 Proof: 𝑁/𝑐 queries distinguish inputs in Ω(𝑁) time
 ∃query with Ω(𝑐) time
C
1 1 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 1 1 1
1 0 1 1 1 0 1 1 1 1 1 1 1 1 0 1 0 1 1 1
1 1 1 0 1 1 0 1 1 1 1 1 1 0 1 1 1 0 1 1
1 1 1 1 0 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1
0 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0
1 0 1 1 1 1 0 1 1 1 0 1 1 1 1 1 0 1 1 1
11
Outline
 Range Minimum Queries
25
12
14
4
(ESA 2010, Invited to Algorithmica)
 Path Minima Queries
(WADS 2011)
6
76
18
98
8
31
79
45
20
7
5
12
43
23
6
4
10
17
84
11
65
62
10
38
4
42
7
9
8
58
6
3
13
5
40
25
10
 Range Diameter Queries
(Submitted to ISAAC 2011)
12
Path Minima/Maxima Queries
 The most expensive connection between two given nodes?
– between b and k = (c,e)
i
– between e and k = (j,k)
4
7
b
5
a
30 4
c
4
4
6
e
2
f
d
g
j
3
h
10
k
Tree-Topology Networks
 Update(c,e) = 4
Trees with Dynamic Weights
13
Naïve Structures
 Brute Force Search
– Worst case query time: 𝑂(𝑛)
– Update time: 𝑂(1)
b
7
5
a
c
i
4
30
e
6
4 2 3
h
g
f
4
d
j
10
k
 Tabulation
– Query time: 𝑂(1)
– Update time: 𝑂(𝑛2 )
30 4
14
Dynamic Weights
Reference
Query Time
Update Time
Tabulation
𝑂 1
𝑂 𝑛2
Brute Force Search
𝑂 𝑛
𝑂 1
Sleator and Tarjan
(STOC’81)
𝑂 log 𝑛
𝑂
Our Result
(WADS’11)
𝑂 log 𝑛
log 𝑛
log log 𝑛
Comparison
Based
𝑂(log 𝑛)
Optimal: Alstrup et al. (FOCS’98) Optimal: Brodal et al. (SWAT’96)
𝑂
Our Result
(WADS’11)
log 𝑛
log log 𝑛
𝑂
log 𝑛
log log 𝑛
Optimal by conjecture:
Optimal: Alstrup et al. (FOCS’98) Patrascu and Thorup (STOC’06)
RAM
 Reduction from Range Minimum Queries in 1D arrays
𝐴[1]
𝐴[2]
𝐴[3]
𝐴[4]
𝐴[5]
𝐴[6]
15
Dynamic Leaves
Reference
Query
Time
Update Time
Comment
Alstrup and Holm (ICALP’00)
and
Kaplan and Shafrir (ESA’08)
𝑂(1)
𝑂(1)
RAM
Our Results
(WADS’11)
𝑂 𝛼(𝑛)
𝑂(1)
Comparison
based
Optimal: Pettie (FOCS’02)
i
4
b
5
a
7
c
4
d
30
e
6
4 2 3
h
g
f
j
10 4
k
16
Updates with link and cut
i
cut(c,e)
link (d,i,12)
Reference
Sleator and Tarjan
(STOC’81)
Our Results
(WADS’11)
b
5
a
7
30
c
4
e
6
4 2 3
h
g
f
12
d
Query Time
4
Update
Time
𝑂 log 𝑛
𝑂 log 𝑛
Ω(log 𝑛)
𝑂(log 𝑛)
log 𝑛
Ω(
)
log log 𝑛
𝑂(log 𝑐 𝑛)
j
10
k
Comment
Comparison
Based
Cell Probe
Proof: by reduction from connectivity problems in graphs
17
Outline
 Range Minimum Queries
25
12
14
4
(ESA 2010, Invited to Algorithmica)
 Path Minima Queries
(WADS 2011)
6
76
18
98
8
31
79
45
20
7
5
12
43
23
6
4
10
17
84
11
65
62
10
38
4
42
7
9
8
58
6
3
13
5
40
25
10
 Range Diameter Queries
(Submitted to ISAAC 2011)
18
Range Diameter Queries
 Farthest pair of points
A Difficult Problem
19
Known Results
Reference
Query Time
Space
Tabulation
𝑂 log 𝑛
𝑂 𝑛4
Smid et al.
(CCCG’08)
𝑂 log 6 𝑛
𝑂 𝑛2
𝑂
Our Results
(Submitted to ISAAC’11)
𝑛 log 𝑛
𝑂 𝑛
Reduction from Set Intersection
Set Intersection Problem
Cohen and Porat
(2010)
𝑂 log 𝑛
𝑂
𝑛 log 𝑛
𝑂 𝑛2
𝑂 𝑛
Conjecture: Set Intersection problem is difficult
(Patrascu and Roditty, FOCS’10)
20
Set Intersection Queries
Reduction
𝑆1 = 0.5,1,2
𝑆2 = 1,1.5, 3
𝑆3 = 0.2,3
𝑆𝑖 ∩ 𝑆𝑗 = ∅ ?
4
2 1
𝑆1 ∩ 𝑆2 ≠ ∅
𝑆1 ∩ 𝑆3 = ∅
Diameter = 3
𝑆1 ∩ 𝑆2 ≠ ∅
Diameter < 5
𝑆1 ∩ 𝑆3 = ∅
Arithmetic on real numbers
with unbounded precisions
Reference
Query Time
Space
Points in Convex Position
Our Results
(Submitted to ISAAC’11)
𝑂 log 𝑛
𝑂 𝑛 log 𝑛
21
Publications
 Range Minimum Queries
25
12
14
4
(ESA 2010, Invited to Algorithmica)
 Path Minima Queries
12
43
23
6
(WADS 2011)
8
31
79
45
10
17
84
11
20
7
5
6
76
18
98
4
65
62
10
38
7
9
8
58
4
42
13
5
40
25
6
3
10
 Range Diameter Queries
(Submitted to ISAAC 2011)
a
 Succinct 𝑘-ary Trees
b
c de
(TAMC 2011)
f
22