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Master presentation of Mike Argyriou in Technological University of Crete about Branch and-bound nearest neighbor searching over unbalanced trie-structured overlays.
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Branch-and-bound nearest neighbor searching over unbalanced trie-structured overlays
Master’s Thesis Presentation Technical University of Crete
4.2.2013
Author: Michail Argyriou Supervisor: Ass’t Prof. Vasilis Samoladas
1998
1999
2000
2001
2002
Centralized Semi-distributed Fully-distributed
1999
2000
2001 2001 DHTs 2001
P2P Evolution
2
Distributed Hash Table (DHT)
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PGrid • Rectangular queries support • Peers only on leaves • High-dimensional queries support with space filling curves
VBI • Height-balanced search tree limitation
GRaSP
• No height-balanced search tree limitation • Abstract types of data and queries • Data: point, rectangular • Queries: point, 3-sided, n-d rectangular
DHT Frameworks Evolution
2003:
2006:
2008:
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Nearest neighbor search
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Given a distributed data set how can we find the k most similar data to a query?
“k-Nearest Neighbor Search”
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Applications
GIS Distributed Databases
Statistical Classification
Recommendation Systems
Cluster analysis Similarity Scores
7
Related Work
1. Naïve algorithm: Central peer collects data and performs k-NN searching
2. K-nn search algorithm over CAN
3. Distributed quad-based index each quadtree block is uniquely identified by its centroid mapped to Chord k-NN search algorithm
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Contents
GRaSP
k-NN
Evaluation
Conclusions
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GRaSP
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Hierarchical space partition:
GRaSP Building the trie ...
Peer p joins
Finds a bootstrapping peer q
Space region s(q) splits into s(q0) and s(q1)
1
2
3 11
GRaSP Space Partition
Before
Volume-balanced
Before
Data-balanced
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GRaSP Space Partition for a 3-sided query
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GRaSP Space Partition for a 3-sided query
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GRaSP Space Partition for a 3-sided query
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GRaSP Data Insertion
We insert a key k into all peers who own regions that contain k
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GRaSP Routing Tables
Each peer knows a peer in each complementary subtrie ...
0100 = 1 0100 = 00 0100 = 011 0100 = 0101
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GRaSP Routing
“In order to route a message from peer p to peer q, the message is forwarded from p to a neighbor peer included in a known subtrie closer
to peer q. From r it is recursively forwarded to q.”
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Contents
GRaSP
k-NN
Evaluation
Conclusions
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Searching Algorithm Branch-and-bound algorithm
Priority queue PQ of candidate peers holding answer better than the k-th answer found so far Fringe
1. Branch Step: expand PQ 2. Bound Step: prune PQ
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Searching Algorithm Parallel Searching vs Iterative Searching
Parallel Searching requires huge message state!
Iterative Searching prunes larger regions of the data space!
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Searching Algorithm
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Searching Algorithm Branch-and-bound algorithm
1? d(q,s(1)) < d(q,a) 00? d(q,s(1)) > d(q,a) 011? d(q,s(1)) > d(q,a) 0101? d(q,s(1)) < d(q,a)
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Latency Complexity Theorem
Latency = |T|O(logn)
Support Set T:
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Latency Complexity Theorem Proof
Peers visited:
Peers in T:
|T| peers
Find peer in the complementary subtrie: O(logn)
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Contents
GRaSP
k-NN
Evaluation
Conclusions
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Performance Evaluation Taking into account number of dimensions
Low Medium High
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Performance Evaluation Metrics
• Data Fairness Index • Latency • Max Throughput • Fringe Size (mean, max)
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Low Medium High
Low dimensions
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Low dimensions Workloads
• Greece, data-balanced partition, k=1/10/100
• Greece, volume-balanced partition, k=1
Datasets
• Synthetic queries • For a network size of n peers we asked n/3
queries
Querysets
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Low dimensions Which space partition is the best?
Volume-balanced
Data-balanced
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Low dimensions Which space partition is the best?
Greece ...
Volume-balanced partition Data-balanced partition
Data FI vs
Space Partition
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Greece, k=1 ...
Volume-balanced partition Data-balanced partition
Low dimensions Which space partition is the best?
Latency vs
Space Partition
34
Greece, k=1 ...
Volume-balanced partition Data-balanced partition
Low dimensions Which space partition is the best?
Fringe Size vs
Space Partition
35
Greece, k=1 ...
Volume-balanced partition Data-balanced partition
Low dimensions Which space partition is the best?
Max Throughput vs
Space Partition
36
Low dimensions Which space partition is the best?
Volume-balanced
Data-balanced
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Low dimensions k ?
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Low dimensions How is the size of the fringe
affected?
Fringe Size vs k
k=10
Greece, data-balanced partition ...
k=100 k=1 39
k=10
Low dimensions How is the latency affected?
Greece, data-balanced partition ...
k=100 k=1
Latency vs k
40
Low dimensions How is the Max. Throughput affected?
k=10
Greece, data-balanced partition ...
k=100 k=1
Max Throughput vs k
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Low dimensions … efficient routing!
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Low Medium High
Medium dimensions
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Medium dimensions Workloads
• Uniform, volume-balanced partition, k=1 • ColorMoments, data-balanced partition,
k=1
Datasets
• Synthetic queries • For a network size of n peers we asked
n/3 queries
Querysets
44
Medium dimensions How is the size of the fringe
affected?
45
Medium dimensions How is the size of the fringe
affected?
ColorMoments, data-balanced, k=1 46
Medium dimensions How is the size of the fringe
affected?
Uniform, volume-balanced, k=1 Uniform, volume-balanced, k=1 Uniform, volume-balanced, k=1
Max. Fringe Size Mean Fringe Size 47
Medium dimensions Data Fairness Index
48
Medium dimensions Data Fairness Index
ColorMoments, data-balanced, k=1 49
Medium dimensions Data Fairness Index
Uniform, volume-balanced, k=1 50
Medium dimensions Latency
51
Medium dimensions Latency
ColorMoments, data-balanced, k=1 52
Medium dimensions Latency
Uniform, volume-balanced, k=1 53
Medium dimensions Latency
Latency is high but near to the optimum!
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Medium dimensions Max. Throughput
55
Medium dimensions Max. Throughput
ColorMoments, data-balanced, k=1 56
Medium dimensions Max. Throughput
Uniform, volume-balanced, k=1 57
Medium dimensions … not efficient routing but near optimum!
It's still good enough for practical
applications!
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Low Medium High
High dimensions
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High dimensions Curse of dimensionality
“When the dimensionality increases, the volume of the space
increases so fast that the available data becomes sparse.”
60
Contents
GRaSP
k-NN
Evaluation
Conclusions
61
Conclusions
API
Searching (k-NN)
Trie Data Ins/Rem
Space Partition
Query Types Data Types
Metric Space 62
Future Work
Approximate k-NN searching for high
dimensions
Redundancy
63
THANK YOU
QUESTIONS ?
64
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