Kmr slides

A K-Main Routes Approach to
Spatial Network Activity Summarization
Authors:
Dev Oliver
Shashi Shekhar
James M. Kang
Renee Bousselaire
Abdussalam Bannur

Outline
 Motivation
 Problem Statement
 Contributions
 Validation
 Analytical
 Experimental
 Case Studies
 Summary and Future Work

Motivation: Crime Analysis (application domain)
 Crime hotspot
 Area of concentrated crime
Street Place
Neighborhood
**J. E. Eck et. al. Mapping Crime: Understanding Hot Spots. US National Inst. of Justice (http://www.ncjrs.gov/pdffiles1/nij/209393.pdf), 2005.
“Most clustering algorithms will show areas of concentration even when a line
is the most appropriate dimension.” – National Institute of Justice**
Star Tribune, January 26, 2011

Examples of Linear Patterns
Linear patterns resulting from deforestation in Brazil
http://en.wikipedia.org/wiki/Deforestation_in_Brazil
Linear patterns of crime in a major US
city

Motivation: Environmental Criminology (scientific domain)
 Spatial theories in Environmental Criminology
1L.E. Cohen et al., Social change and crime rate trends: A routine activity approach, American sociological review, 1979.
2P. L. Brantingham et al., Environmental Criminology, Waveland Press, 1990.
 Routine Activity Theory1
 Crime location related to criminal’s
frequently visited areas
 Crime Pattern Theory2
 Based on spatial model
 Nodes (e.g. home, work,
entertainment),
 Paths (e.g. routes between
nodes),
 Edges
 Crime locations close to edges
 Near criminal’s activity
boundaries where residents may
not recognize him/her
Source: Rossmo, Kim (2000). Geographic Profiling. Boca Raton, FL: CRC Press.
http://www.popcenter.org/learning/60steps/index.cfm?stepNum=16
 Network based summarization adds value to Environmental Criminology
 Assist with large scale verification of real-world data matching theories
 Opportunities to develop hypotheses for new theory formulation

Other Domains
Accident Analysis and PreventionDisaster Relief

Motivation Problem Contributions Validation Summary
Key Concepts
 Activity
 Object of interest located at node or edge
 Summary path
 A path chosen by KMR to summarize activities
 Activity coverage
 Total number of activities of a path or set of paths
 Active node
 A node having n ≥ 1 activities or joined by an edge
having n ≥ 1 activities e.g., A, B, C, D, E
 Inactive node
 A node having n = 0 activities and joined by edges
all having n = 0 activities e.g., F
 Active node ratio
 Total # active nodes/Total # nodes
 e.g., 5/6
Each edge has a weight of 1

Problem Statement
 Given
 A spatial network G = (N, E)
 A set of activities, A and their
locations (e.g. a node or edge)
 A set of Paths, P
 K (Number of routes)
 Edge weights
 Find
 A cardinality k subset P′ of P, i.e.,
a subset P′⊆ P with |P′| = k
 Objective
 Maximize the activity coverage
(AC) by P′
 Constraints
 1 ≤ k ≤ |P|.
k = 2
Edge Weights
are 1
Given P = the set of Shortest Paths

Challenges
 Measures of interestingness
 Activity coverage, average distance, etc
 Computational Complexity
 Choose(N,2) paths, given N nodes
 Exponential number of k subsets of paths

Related Work
Network Summarization by Grouping/Clustering
Clumping (Okabe), e.g.
NT-VCM (Shiode)
Max. Subgraph, e.g.
path, tree (Buchin)
Multiple routesZero or One routes
Our Work

Contributions
 K-Main Routes (KMR) algorithm
 Finds a set of k routes to group activities
 New design decisions added
 Network Voronoi Activity assignment
 Divide and Conquer Summary path recomputation
 Spatial network activity summarization is shown to be NP-complete.
 Analytically demonstrate correctness of design decisions and show cost
analysis
 Experimental evaluation of the various algorithms
 Performance evaluated using synthetic and real world datasets
 Case study comparing KMR with geometry based summarization

K-Main Routes (KMR) Algorithm
 K-Main Routes Algorithm
 Select k paths as initial summary paths
 Repeat
1. Form k clusters by assigning each activity
to its closest summary path
2. Recompute summary path of each cluster
 Until summary paths do not change
 Design Decisions
 Inactive node pruning
 Network Voronoi Activity assignment
 Divide and Conquer Summary path
recomputation
P = the set of Shortest Paths, K=2

Design Decision: Inactive Node Pruning
 Only consider paths between active nodes
 Optimal solution will still be in this set
Given the set of shortest paths
• 20 shortest paths calculated and stored versus 30

Design Decision: Network Voronoi (NV) Activity Assignment
 Goals
 Form k clusters by assigning each activity to its closest summary path
 Improve execution time of current assignment strategy
 Example (execution trace) Next
K-Main Routes Algorithm
Select k shortest paths as initial summary paths
Repeat
1. Form k clusters by assigning each activity
to its closest summary path
Until summary paths do not change
Repeat
1. Network Voronoi Activity Assignment

A B C D
E F G H
1
2
3 4
5 6
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9
10
X
DISTANCEFROM
Open:
ACTIVITIES
1 2 3 4 5 6 7 8 9 10
A
E
D
H
AE
DH
Closed:
Activity
Active Node
Inactive Node
Virtual Node
Summary Path
Edge weight = 1
Edge weight = 0
Closed Node
X A E
∞
0
∞
∞
∞
∞∞
∞
∞
0
0
0
D
0
H
X

A B C D
E F G H
1
2
3 4
5 6
7 8
9
10
X
DISTANCEFROM
Open:
ACTIVITIES
1 2 3 4 5 6 7 8 9 10
A
E
D
H
AE
DH
Closed:
Activity
Active Node
Inactive Node
Virtual Node
Summary Path
Edge weight = 1
Edge weight = 0
Closed Node
A E
∞
0
∞
∞∞0
0
0
D
0
H
X1
B
1 < 0?
0
0
A
0
0

A B C D
E F G H
1
2
3 4
5 6
7 8
9
10
X
DISTANCEFROM
Open:
ACTIVITIES
1 2 3 4 5 6 7 8 9 10
A
E
D
H
AE
DH
Closed:
Activity
Active Node
Inactive Node
Virtual Node
Summary Path
Edge weight = 1
Edge weight = 0
Closed Node
E
∞
0
∞
∞
0
0
0
D
0
H
X1
B
0
0
A
F
1
0
0
0 0
0 0
E
0 0

A B C D
E F G H
1
2
3 4
5 6
7 8
9
10
X
DISTANCEFROM
Open:
ACTIVITIES
1 2 3 4 5 6 7 8 9 10
A
E
D
H
AE
DH
Closed:
Activity
Active Node
Inactive Node
Virtual Node
Summary Path
Edge weight = 1
Edge weight = 0
Closed Node
0
∞
∞0
0
0
D
0
H
X1
B
0
0
A
F
1
0
0
0 0
0 0
E1
C
0 0
0 0
1 < 0?
0 0
0 0
D

A B C D
E F G H
1
2
3 4
5 6
7 8
9
10
X
DISTANCEFROM
Open:
ACTIVITIES
1 2 3 4 5 6 7 8 9 10
A
E
D
H
AE
DH
Closed:
Activity
Active Node
Inactive Node
Virtual Node
Summary Path
Edge weight = 1
Edge weight = 0
Closed Node
0
∞
0
0
0
0
H
X1
B
0
0
A
F
1
0
0
0 0
0 0
E1
C
0 0
0 0
0 0
0 0
D
1
G
H
0 0

A B C D
E F G H
1
2
3 4
5 6
7 8
9
10
X
DISTANCEFROM
Open:
ACTIVITIES
1 2 3 4 5 6 7 8 9 10
A
E
D
H
AE
DH
Closed:
Activity
Active Node
Inactive Node
Virtual Node
Summary Path
Edge weight = 1
Edge weight = 0
Closed Node
0
0
0
0
0
X1
B
0
0
A
F
1
0
0
0 0
0 0
E1
C
0 0
0 0
0 0
0 0
D
1
G
H
2 < 1?
1
1
1
1
2 < 1?
B
0 0

A B C D
E F G H
1
2
3 4
5 6
7 8
9
10
X
DISTANCEFROM
Open:
ACTIVITIES
1 2 3 4 5 6 7 8 9 10
A
E
D
H
AE
DH
Closed:
Activity
Active Node
Inactive Node
Virtual Node
Summary Path
Edge weight = 1
Edge weight = 0
Closed Node
0
0
0
0
0
X1
0
0
A
F
1
0
0
0 0
0 0
E1
C
0 0
0 0
0 0
0 0
D
1
G
H
1
1
1
1
B
2 < 1?
F
0 0

A B C D
E F G H
1
2
3 4
5 6
7 8
9
10
X
DISTANCEFROM
Open:
ACTIVITIES
1 2 3 4 5 6 7 8 9 10
A
E
D
H
AE
DH
Closed:
Activity
Active Node
Inactive Node
Virtual Node
Summary Path
Edge weight = 1
Edge weight = 0
Closed Node
0
0
0
0
0
X1
0
0
A
1
0
0
0 0
0 0
E1
C
0 0
0 0
0 0
0 0
D
1
G
H
1
1
1
1
B F
1 1
1 1
C
2 < 1?
0 0

 Network Voronoi Activity Assignment algorithm
Input: Graph G = (N, E), a set of Activities A, a set of k Summary Paths, S
Output: A set of k clusters formed by assigning all ai ∈A to one si ∈S, where dist(ai, si) ≤
dist(ai, sj) and sj ∈S and sj ≠ si
1. Open ← all nodes ∈ S, Closed ← Ø
2. Tnodes ← all nodes ∈ S,
3. Tactivities ← activities on si ∈S
4. repeat
5. nc ← next node ∈ Open
6. remove nc from Open
7. Closed ← nc
8. X ← neighbors of nc
9. foreach xi ∈ X
10. if xi ∉ Tnodes and xi ∉ Closed
11. Tnodes ← xi
12. xi.prev ← nc,
13. xi.dist ← dist(xi, nc) + nc.dist
14. xi.sp ← nc.sp
15. else if xi ∈Tnodes
16. update xi if new dist < xi.dist
17. if xi ∉ Open
18. Open ← xi
19. Y ← activities on edge {nc, xi}
20. foreach yi ∈ Y
21. if yi ∉ Tactivities
22. Tactivities ← yi
23. yi.prev ← nc
24. yi.dist ← xi.dist
25. yi.sp ← xi.sp
26. else
27. update yi if new dist < yi.dist
28. until all active nodes ∈ Closed
29. return currentClusters

Design Decision: Divide and Conquer Summary PAth
REcomputation
 Goals
 Recompute the summary path of each cluster
 Improve execution time of current recomputation strategy
 Example (execution trace) Next
Repeat
Repeat
2. Divide and Conquer Summary path
Recomputation Design Decision

Design Decision: Divide and Conquer Summary PAth
REcomputation
 Summary Path Recomputation Algorithm
Input: Graph G = (N, E), a set of Clusters, C
Output: A set of summary paths, S where si ∈S has max coverage for ci ∈ C and si ∈ ci
1. nextClusters ← Ø
2. foreach ci ∈ C
3. X ← active nodes of ci
4. maxP ← Ø
5. foreach xi ∈ X
6. foreach xj ∈ X
7. if (i ≠ j)
8. cP ← getSP(xi, xj)
9. if (maxP = Ø)
10. maxP ← cP
11. if (maxP.activities < cP.activities)
12. maxP ← cP
13. if (maxP ≠ ci.summaryPath
14. nextClusters ← maxP
15. else
16. nextClusters ← ci.summaryPath
17. return nextClusters
A B C D
E F G H
1
2
3 4
5 6
7 8
9
10
Activity
Active Node
Inactive Node
Summary Path
Edge weights are 1
Cluster

Validation
 Analytical
 Cost analysis explaining computational savings
 Experimental
 Comparative analysis of KMR with various design decisions
 Performed on real and synthetic data
 Network voronoi activity assignment and divide and conquer summary path
recomputation saves computational costs
 Savings increase with number of nodes, routes, activities and active node ratio
 Case studies
 Qualitatively shows the usefulness of network based summarization on Crime
data

Analytical Evaluation: Computational Analysis
 KMR Execution Time = Number of Iterations × (Activity Assignment
Cost + Summary Path Recomputation Cost)
 TKMR = I × ([K × |A| × cost(ai,ci)] + [K × dc × |N|2])
 TKMR_I = I × ([K × |A| × cost(ai,ci)] + [K × dc × (|N| × r)2])
 TKMR_IAS = I × ([|E| + |N|×log |N|] + [K × dc × (|N|/K × r)2])
I = Number of Iterations
K = Number of Clusters
A = Set of activities
cost(ai, ci) = Cost of calculating the distance between activity ai and cluster ci
dc = Cost of looking up a path
N = Set of Nodes
E = Set of Edges
r = active node ratio, 0 ≤ r ≤ 1

Experimental Evaluation
• Goal: Comparative analysis
• Candidates: KMR with various design decisions
• KMR_I – KMR with inactive node pruning
• KMR_IV – KMR with inactive node pruning and Network voronoi activity assignment
• KMR_ID – KMR with Divide and conquer summary path recomputation
• KMR_IVD – KMR with all three design decisions
• Measure: CPU time (Unix time command)
• Platform: Mac Pro, 2 x Xeon Quad Core 2.26 GHz, 16 GB RAM
• Variables: #Nodes, #Routes, #Activities, Active Node Ratio
• Fixed Parameters: unit edge length
• Datasets: Synthetic and Real (Haiti Earthquake)
Real Dataset
Analysis
#Nodes
#Routes
Java-based Simulator
KMR_I KMR_IV
Candidates
Variables
#Activities
Active Node
Ratio
Measures
Synthetic Dataset
KMR_ID KMR_IVD

Data Description and Characteristics
 Synthetic Data
 2010 Census TIGER/Line® Shapefiles used for road network
 Activities randomly assigned to each edge
 Real-world data: Haiti Data Set
 Geospatial and Temporal Dataset describing recent events post-disaster
 Dataset collected from Jan 12, 2010 to March 23, 2010
 1,677 records
 Characteristics
 Attributes
• Incident Title (e.g., “Food, Water, Tents needed…”)
• Incident Date and Time
• Location (City, port name)
• Category (numeric category)
• Latitude/Longitude
 Sources
 Crisis Map of Haiti - http://haiti.ushahidi.com/
 OpenStreetMap - http://www.openstreetmap.org/

Effect of Number of Nodes
Synthetic Data Set
Number of Activities = 1200
Active Node Ratio = 0.2
K = 2
Trends:
 Voronoi Activity assignment and divide and conquer summary path recomputation saves comp. costs
 Savings increase with number of nodes
Real Data Set
K = 2

Effect of Number of Routes, K
Synthetic Data Set
Number of Nodes = 1000
Real Data Set
Trends:
 Savings increase with number of routes

Effect of Number of Activities
Synthetic Data Set
K = 2
Trends:
 Savings increase with number of activities

Effect of Active Node Ratio
Synthetic Data Set
K = 2
Trends:
 Savings increase with active node ratio

Input (a set of crime incidents, k=5) KMR Output
Crimestat K-Means (Euclidean distance) Crimestat K-Means (Network distance)
Case Study: Crime Analysis

Summary
 Spatial network activity summarization was shown to be NP-complete.
 K-Main Routes (KMR) algorithm and its design decisions described
 Inactive node pruning
 Network Voronoi Activity assignment
 Divide and Conquer Summary path recomputation
 Analytically demonstrated correctness of design decisions and cost analysis
showed
 Experimental evaluation
 Performance evaluated using synthetic and real world datasets
 Case study comparing KMR with geometry based summarization

Acknowledgements
 Members of the Spatial Database and Spatial Data Mining Research Group, University of
Minnesota, Twin-Cities.
 This work was supported by grants from USARMY and USDOD.
 Thank you for your time! Any questions or comments?

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