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Space-Efficient Range Reporting for
Categorical Data
Yakov Nekrich
Department of Computer Science
University of Chile
PODS 2012
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Outline
O Colored Range Reporting
O Three-sided Color Reporting in Linear Space
O Path-range Trees
O Three-sided Color Reporting for O(B*log2n)
Points
O Colored Range Reporting in Two Dimensions
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Colored Range Reporting
O Definition
O Each point p in a set S is assigned a color
col(p)
O For a query rectangle Q
O Report the colors of all points that occur in Q
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Colored Range Reporting
O Contributions
O First I/O-efficient data structures
O Achieve optimal query costs and almost the
same space usage as the corresponding data
structures for regular range reporting queries
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Outline
O Colored Range Reporting
O Three-sided Color Reporting in Linear Space
O Path-range Trees
O Three-sided Color Reporting for O(B*log2n)
Points
O Colored Range Reporting in Two Dimensions
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Lemma 1
O There exists an data structure D
O D stores two-dimensional points and uses
O(N/B) space
O For any three-sided range Q = [a, b] × [0, c]
O D can report all results in O((N/B)δ + (K/B))
I/Os for any δ > 0
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Lemma 1
1/δ
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(N/B)δ
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Lemma 1
O For every node v (containing Nv points)
O Maintain a tree Lv
O Uses space O(Nv/B)
O Range search cost in
(logBNv + K/B) I/Os
O O(logBN + K/B) I/Os
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Lemma 1
O Space used
O O ((1 / δ ) * (N / B)) = O(N/B)
O I/O cost ([a, b] × [0, c])
O [a, b] can identify O((N/B) δ) nodes on every
level
O Total cost O ((1 / δ ) * (N/B) δlogBN + (K / B))
= O((N/B) δlogBN + (K/B))
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Lemma2
O There exists an data structure D
O D contains colored one-dimensional points
and uses O(N/B) space
O For any one-dimensional range Q = [a, b]
O D can report all distinct colors of points in
O ((N/B) δlogBN + (K/B)) I/Os
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Lemma2
[2, 6]
[2, 6] ×
[0, 2)
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Lemma 3
O There exists O(N/B) space data structure D
O D answers three-sided color range reporting
queries for points on an N × N grid in O
((N/B) δlogBN + (K/B)) I/Os
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Lemma 3
O Sweep a horizontal line h in +y direction
O When h hits a point p
O Add p to Structure E in Lemma 2
O Since points lie on N × N grid
O Q = [a, b] ×[0, c] cost
N * O ((N/B) δlogBN + (K/B)) I/Os
= O ((N/B) δlogBN + (K/B)) I/Os
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Outline
O Colored Range Reporting
O Three-sided Color Reporting in Linear Space
O Path-range Trees
O Three-sided Color Reporting for O(B*log2n)
Points
O Colored Range Reporting in Two Dimensions
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Structure
π(w, la)
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πr(w, la)
πl(w, lb)
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Structure
O ymin(v, F)
O For a color v and a set of points F
O ymin(v, F) denote the point with the smallest
y-coordinate among all points p ∈ F with col(p)
=v
O Yl(u, v) (Yr (u, v))
O ymin(c, πl(u, v)) (ymin(c, πr(u, v))
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Structure
O Identity levels of the tree Ni
O logB + f(i) log logN for f(i) = 2i and
i = 1, … , log loglogN(N/B)
O O(B logf(i)N) points in each node
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Structure
O For each node v in Ni
O For each ancestor u of v we store the lists
Ll(u, v) = Yl(u, v)[1, …, mi]
Lr(u, v) = Yr(u, v)[1, …, mi]
where mi = B logf(i)-1N
O For i >= 2 store all points from S(v) in a
O(|Sv|/B) space data structure D(v)
O For each v ∈ N1 are stored in a structure D’(v)
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Queries
O Q = [a, b] × [0, c]
O Identity la and lb
O Identity the lowest common ancestor w of la
and lb
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Queries
O Find the ancestor vi of lb that belongs to Ni
O Traverse the lists Ll(w, vi) until a point pi, pi.y > c
O If pi is not found, increment i and proceed in the
next node vi
O If Found
O Report colors of all points p∈Ll(w, vi), p.y < c.
O Answer the query Q = [a, b] × [0, c] to data
structure D(vi)
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Space Usage
O D(v) cost
O v ∈ Ni, v contains O(B*logf(i)N) points
O O(N/(B*logf(i)N)) nodes in Ni
O Total log loglogN(N/B) levels
O D(v) total cost O((N/B) log log(N/B)) blocks
O Lr(u, v) cost
O Each list Lr(u, v) contains B logf(i)-1N points
O O(logN) lists for each v in Ni
O Lr(u, v) cost O((N/B) log log(N/B))
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I/O cost
O Traversing all Ll(w, vt)
O K >=|Ll(w, vt-1)| = O(B logf(t-1)-1N)
O |Ll(w, vt-1)| >= ∑1,t-2 |Ll(w, vt)|
O K = Ω(∑1,t-1 |Ll(w, vt)|)
O Total cost O(K/B) I/Os
O Cost in D(v)
O O(((B logf(t)N)/B)1/4+K’/B) K’ <= K
O (B logf(t)N)/B)1/4=O(logf(t-1)-1N)
=O(|Ll(w, vt-1)|/B) for t > 1
O Total cost O(K/B) I/Os
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Outline
O Colored Range Reporting
O Three-sided Color Reporting in Linear Space
O Path-range Trees
O Three-sided Color Reporting for O(B*log2N)
Points
O Colored Range Reporting in Two Dimensions
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Lemma 5
O Let S be a set of r = O(B log2N) points on an
N * N grid
O There exists a O((|S|/B) log logN) space
data structure D
O Cost in O((log logN)δ + K/B) I/Os
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Lemma 6
O Let S be a set of r = O(B log2N) points on an
N * N grid
O There exists a O((|S|/B) log logNlog(3)N)
space data structure D
O Cost in O(K/B) I/Os
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Outline
O Colored Range Reporting
O Three-sided Color Reporting in Linear Space
O Path-range Trees
O Three-sided Color Reporting for O(B*log2N)
Points
O Colored Range Reporting in Two Dimensions
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N×N to U×U
O Rank space technique
O Scaling and Related Techniques for Geometry
Problems
O Query cost increases in O(loglogBU) I/Os
O Space usage remains unchanged
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Three-sided to Two
Dimensions
O Construct a range tree Ty on the y
coordinates
O In each node v of Ty, define path range trees
P1(v) and P2(v) that support three-sided
queries
O To answer a query [a, b] × [c, d], just
answer either S(w1)∩([a, b]×[c, +∞]) or
S(w2) ∩([a, b]×[0, d])
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Three-sided to Two
Dimensions
O Query cost is the same as the cost of three
sided queries
O Space usage is increases by O(log N) factor
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