Data Structures:Range Queries - Space Efficiency

Pooya DavoodiAarhus University

PhD DefenseJuly 4, 2011

Thesis Overview Range Minimum Queries in Arrays

(ESA 2010, Invited to Algorithmica)

Path Minima Queries in Trees (WADS 2011)

Range Diameter Queries in 2D Point Sets (Submitted to ISAAC 2011)

Succinct -ary Trees(TAMC 2011)

2

57

44 2 3

4 6

1020

25 12 6 8 10 65 7 1312 43 76 31 17 62 9 514 23 18 79 84 10 8 404 6 98 45 11 38 58 25

c d e

b

f

a

Range Minimum Queries

Database systems– Lowest average-salary:

3

80,000 85,000 115,000 120,000 118,000

81,000 73,000 90,000 100,000 94,000

65,000 75,000 86,000 92,000 95,000

72,000 91,000 89,000 103,000 102,000

35,000 45,000 42,000 50,000 41,000

60

50

40

30

20

1990 1995 2000 2005 2010

Age

Year

Minimum:65,000 at [3,1]

Definition

Input: an array Query: where is minimum in ?

4

()

𝑛

𝑚𝑖1

𝑖2

𝑗1 𝑗2

Naïve Solution

Brute force search Query time: time Worst case : time ()

5

𝑚

𝑛

𝑠 ′

𝑠

Data Structures Preprocess and store some information Naïve: store the answers of all queries

query time Size of the table:

bits6

Top-Left 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

Tabulation12 85 10

1 2

1

2

Reference Space (bits) Query TimeTabulation

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-Efficient Data Structures

7

1D vs. 2D

1D: Cartesian Trees bits per element

(Tarjan et al., STOC’84) bits per element

(Sadakane, ISAAC’07)

2D: Nothing like Cartesian Trees

8

7 20 2 10 8 5 16 6

2

7

20 8

10

6

16

Lowest Common Ancestor

𝑚 bits per element(Our Result, ESA’10)

5

Indexing Data Structures

Popular in Succinct Data Structures

9

IndexRead-only

Input Array

bits

bits

bits

bits

Size of Input Size of Index Query Time

bits per element

(Our Results, ESA’10)

bits with query time

10

27 30 2 90 28 15 18 6

13 20 93 54 17 11 16 12

7 74 39 62 8 5 46 23

61 38 68 10 9 87 98 217 20 2 10 8 5 16 6

2

7

20

5

8

10

6

16

Cartesian Trees

log𝑚

Cartesian Tree:

log log𝑚

log log𝑛

Atallah and Yuan (SODA’10)

Tabulation

log𝑛

bits Per Element bits query time

bits query timeProof: queries distinguish inputs in timequery with time

11

1 1 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 1 1 11 0 1 1 1 0 1 1 1 1 1 1 1 1 0 1 0 1 1 11 1 1 0 1 1 0 1 1 1 1 1 1 0 1 1 1 0 1 11 1 1 1 0 1 1 0 1 1 1 1 0 1 1 1 1 0 1 10 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 01 0 1 1 1 1 0 1 1 1 0 1 1 1 1 1 0 1 1 1

C

Outline Range Minimum Queries

(ESA 2010, Invited to Algorithmica)

Path Minima Queries(WADS 2011)

Range Diameter Queries(Submitted to ISAAC 2011)

12

57

44 2

3

4 6

1020

25 12 6 8 10 65 7 1312 43 76 31 17 62 9 514 23 18 79 84 10 8 404 6 98 45 11 38 58 25

Path Minima/Maxima Queries The most expensive connection between two given nodes?

– between b and k = (c,e)– between e and k = (j,k)

Update(c,e) = 413

Tree-Topology Networks

30

a

b

d

ce

f gh

k

j

i

5

7

44 2

3

4

6

10

4

Trees with Dynamic Weights

4

Naïve Structures

Brute Force Search– Worst case query time: – Update time:

Tabulation– Query time: – Update time:

14

a

b

d

c e

f gh

k

j

i

5

7

44 2

3

46

10

30

30

Reduction from Range Minimum Queries in 1D arrays

Reference Query Time Update Time

Tabulation

Brute Force SearchSleator and Tarjan

(STOC’81)Our Result(WADS’11)

Our Result(WADS’11)

Dynamic Weights

15

𝐴 [1] 𝐴 [2] 𝐴 [3] 𝐴 [4 ] 𝐴 [5] 𝐴 [6]

Optimal: Brodal et al. (SWAT’96)

Optimal by conjecture:Patrascu and Thorup (STOC’06)Optimal: Alstrup et al. (FOCS’98)

ComparisonBased

RAM

Optimal: Alstrup et al. (FOCS’98)

Dynamic Leaves

16

Reference Query Time Update Time Comment

Alstrup and Holm (ICALP’00)and

Kaplan and Shafrir (ESA’08)RAM

Our Results(WADS’11)

Comparisonbased

Optimal: Pettie (FOCS’02)

a

b

d

c e

f gh

k

j

i

5

7

44 2

3

46

10

30

4

Updates with link and cut

17

a

b

d

c e

f gh

k

j

i

5

7

44 2

3

46

10

30

link (d,i,12)

cut(c,e)

12

Reference Query Time Update Time Comment

Sleator and Tarjan(STOC’81)

ComparisonBased

Our Results(WADS’11) Cell Probe

Proof: by reduction from connectivity problems in graphs

Outline Range Minimum Queries

(ESA 2010, Invited to Algorithmica)

Path Minima Queries(WADS 2011)

Range Diameter Queries(Submitted to ISAAC 2011)

18

57

44 2

3

4 6

1020

25 12 6 8 10 65 7 1312 43 76 31 17 62 9 514 23 18 79 84 10 8 404 6 98 45 11 38 58 25

Range Diameter Queries

Farthest pair of points

19

A Difficult Problem

Known Results

20

Cohen and Porat(2010)

Set Intersection Problem

Conjecture: Set Intersection problem is difficult(Patrascu and Roditty, FOCS’10)

Reference Query Time Space

Tabulation

Smid et al.(CCCG’08)

Our Results(Submitted to ISAAC’11) Reduction from Set Intersection

Set Intersection QueriesReduction

21

?

124

Diameter = 3 𝑆1∩𝑆2≠∅

Diameter < 5 𝑆1∩𝑆3=∅

Arithmetic on real numberswith unbounded precisions

Points in Convex Position

Our Results(Submitted to ISAAC’11)

Reference Query Time Space

Publications Range Minimum Queries

(ESA 2010, Invited to Algorithmica)

Path Minima Queries(WADS 2011)

Range Diameter Queries(Submitted to ISAAC 2011)

Succinct -ary Trees(TAMC 2011)

22

57

44 2

3

4 6

1020

25 12 6 8 10 65 7 1312 43 76 31 17 62 9 514 23 18 79 84 10 8 404 6 98 45 11 38 58 25

c d e

b

f

a