1. Mining the Most Influential k-Location Set from Massive Trajectories
Y. Li, J. Bao, Y. Li, Y. Zheng, Y. Wu, Z. Gong
Presented by
Mi Tian, Dhaval Deepark Dholakia, Deepan Ashok Sanghavi
9/21/2018

2. Motivating Scenarios Advertisement Placement Resource Allocation
Place multiple billboards within a region
To be seen by more unique people
Resource Allocation
Place the charging stations for EVs
To serve more unique vehicles
Chained Business Location Selection
Collective location selection, e.g., Starbucks, KFC
To sever more people
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3. The Goal Find k targets to cover the most unique trajectories
Two spots to serve gas to the most cars?
Two spots to attract the most customers?
Two spots to best observe bird migration
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4. Term Definition
Trajectory (tr): A sequence of GPS locations of a moving object through time.
Location (v): A meaningful location on the map. Depends on the problem, can be a road intersection, a grid, a stay point, and etc.
Spatial Network (G): A graph G = (V, E) formed by Locations and their connections.
Coverage (Tr(v)): All trajectories passing a specific location. A B D C E
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5. Problem Definition Location-based Max-Cover Query Input: Output:
Trajectories: T = { 𝑡𝑟 1 , 𝑡𝑟 2 , …, 𝑡𝑟 𝑛 }
Road network: G={E, V}
Integer k
Spatial region R
Output:
Select k vertices in the spatial region R, which covers the maximum number of unique trajectories.
Objective
Minimize response time & improve system frequency
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6. Naïve Solution A B 2 D C E Question: find the k-set where k = 2?
Let’s count coverage:
Tr (n1) = {A, B}  2 tr
Tr (n2) = {B, C, D}  3 tr
Tr (n3) = {C, D, E}  3 tr
How many different combos? C3
Which combo is the best?
Tr (n2) + Tr (n3) = 6 is the biggest number, is it the right answer?
It’s about unique trajectories A B 2 D C E
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7. Naïve Solution (cont.) A B D C E k
Question: find the k-set where k = 2?
Let’s count unique trajectories:
Tr (n1, n2) = {A, B, C, D}  4 tr
Tr (n2, n3) = {B, C, D, E}  4 tr
Tr (n1, n3) = {A, B, C, D, E}  5 tr
n1 & n3 is the winner, why?
n2 & n3 had overlapped trajectories {C, D}
Again, it’s about unique trajectories
But, does this method scale?
NO  Cn A B D C E k
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8. Challenges Cn Lots of computation (NP-Hard) Dynamic requests
Massive trajectories
Large n and k 
Dynamic requests
User can pick any region
User can discard certain answers and ask for recalculate
Software needs to be interactive (answer quickly)
The goal of this paper is to Make It Fast, and
Optimal for small or medium region
Near optimal for large region Cn k
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9. System Overview
k, region selection
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10. Pre-Processing Map-Matching Inverted Index Building
Map raw trajectory data to road networks
Tr={(lat1, lng1, t1), (lat2, lng2, t2), … (latn, lngn, tn)} -> {v1, v2, … vn}
Trajectory-Vertex Index
Inverted Index Building
Vertex - Trajectory
V1={trj1, trj2, … , trjn}
Vertex-Vertex Index Building
Shared trajectories between two vertices
Spatial Indexing
Index vertex location
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11. Optimal Solution Cn 11 k Compare (k = 2): Better than all other (G, G)
Naïve Approach 
Group-based Pruning Approach
Group the vercies
Estimate coverage upper bound
for every (G, G) combination
Sort (G, G) by upper bound (high to low)
For each (G, G)
Find best k-set vertices
with most unique coverage
Could Stop Early!
Compare (k = 2):
Naïve: C (9, 2) = 36
Group-base Pruning:
3 * 3 (G) + 3 * 3 (V) = 15 (reduced!) Cn k
Better than all other (G, G) 11
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12. Greedy Solution： Overview
Optimal solution
Not scale to large k and spatial region R
Efficient when we need multiple round interactions from users.
Greedy solution
A good approximation to optimal solution (1-1/e) approximation
Idea is very simple
Main algorithm framework
First Step: Select a set of candidate vertices in spatial region R
A k-round selection process
Selection  the vertex covers most un-covered trajectories
Updating  remove the covered trajectories
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13. Greedy Solution： Basic Algorithm
E.g., k = 2
Current selection: { 𝑣 1 }
{ 𝑣 1 , 𝑣 4 }
Current covered trajectories: { 𝑡𝑟 1 , 𝑡𝑟 2 , 𝑡𝑟 3 , 𝑡𝑟 4 , 𝑡𝑟 5 , 𝑡𝑟 6 }
𝑡𝑟 7 , 𝑡𝑟 8 , 𝑡𝑟 9 }
𝑣 1 : 𝑡𝑟 1 , 𝑡𝑟 2 , 𝑡𝑟 3 , 𝑡𝑟 4 , 𝑡𝑟 5 , 𝑡𝑟 6
𝑣 2 : 𝑡𝑟 1 , 𝑡𝑟 2 , 𝑡𝑟 3 , 𝑡𝑟 8 , 𝑡𝑟 10
𝑣 3 : 𝑡𝑟 4 , 𝑡𝑟 5 , 𝑡𝑟 6 , 𝑡𝑟 7 , 𝑡𝑟 9
𝑣 4 : 𝑡𝑟 1 , 𝑡𝑟 7 , 𝑡𝑟 8 , 𝑡𝑟 9
𝑣 5 : 𝑡𝑟 1 , 𝑡𝑟 3 , 𝑡𝑟 5
𝑣 6 : 𝑡𝑟 2 , 𝑡𝑟 4 , 𝑡𝑟 6
𝑡𝑟 1 : 𝑣 1 , 𝑣 2 , 𝑣 4 , 𝑣 5
𝑡𝑟 2 : 𝑣 1 , 𝑣 2 , 𝑣 6
𝑡𝑟 3 : 𝑣 1 , 𝑣 2 , 𝑣 5
𝑡𝑟 4 : 𝑣 1 , 𝑣 3 , 𝑣 6
𝑡𝑟 5 : 𝑣 1 , 𝑣 3 , 𝑣 5
𝑡𝑟 6 : 𝑣 1 , 𝑣 3 , 𝑣 6
𝑣 1  6
𝑣 2  5
𝑣 3  5
𝑣 4  4
𝑣 5  3
𝑣 6  3
𝑣 1  6
𝑣 2  3
𝑣 3  5
𝑣 4  3
𝑣 5  2
𝑣 6  2
𝑣 1  6
𝑣 2  4
𝑣 3  5
𝑣 4  3
𝑣 5  2
𝑣 6  3
𝑣 1  6
𝑣 2  2
𝑣 3  2
𝑣 4  3
𝑣 5  0
𝑣 6  0
Vertex coverage state
Vertex coverage count
Trajectories-vertex index
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14. Greedy Solution： Basic Algorithm
Performance analysis
The dominant cost is updating phase
Scan the trajectory-vertex index “one-by-one”
Can not scale to large trajectory dataset
𝑣 1  6
𝑣 2  5
𝑣 3  5
𝑣 4  4
𝑣 5  3
𝑣 6  3
𝑣 1  6
𝑣 2  2
𝑣 3  2
𝑣 4  3
𝑣 5  0
𝑣 6  0
Trajectory-vertex index
Is it possible to update the coverage of each node by batch ?
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15. Greedy Solution ： Partition Index Batch Updating Algorithm
Main intuition
To minimize the trajectory scan operation
Update the coverage values by batch
Main Techniques
Smart Update Decision
Index Partition
Workload-based Optimization
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16. Greedy Solution ： Partition Index Batch Updating Algorithm
In this way, we always scan less number of trajectories for updating.
Smart Update Decision
Utilize vertex-vertex index
Two cases
Case 1： Major Coverage Overlap – Apply the basic updating method
Case 2： Minor Coverage Overlap – Subtract and add back
Vertex
Coverage table
Trajectory-vertex
Index
Vertex
Coverage table
Scan tr4 and tr5 to update
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17. Greedy Solution ： Partition Index Batch Updating Algorithm
Index Partition
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18. Greedy Solution ： Partition Index Batch Updating Algorithm
Index Partition
Why and How?
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19. Greedy Solution ： Partition Index Batch Updating Algorithm
Workload-based Optimization
Selective Indexing
To store vertex-vertex index takes |V| * |V| space, (Tianjing Road has 900k vertices)
To make partitions p -> p*|V|*|V| (impossible to store in the memory)
Workload-based Partition
Not possible to cluster the trajectories based on similarities (|N|*|N| similarity computing)
Observation
Many vertices are selected in a sequence
Many of the vertices will not be selected
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20. Experiments Dataset Road Networks (Tianjing) Trajectories (Taxies)
99,007 vertices and 133,726 road segments
covers a 123 × 187 km2 spatial region with a total length of 32,487 km
Trajectories (Taxies)
3,501 taxicabs from Tianjin in 61 days.
It contains 4,509,519 trajectories
average sampling rate is 24:05 seconds per point
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21. Trajectory Distributions in Tianjin
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22. Ubuntu Machine
Intel Core 6-Cores (12-Threads) i7-3930K 3.2GHz and 16GBytes of main memory
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23. Basic Updating Algorithm vs Partition Index Batch Updating (or PIBU) Algorithm
scanned trajectories (bars) and the processing time (lines) for vertex selection iteration.
Aim: Mine 10-location set
PIBU is 5.02 times faster
Basic Algorithm : 905, 623 trajectories
PIBU : 87, 330 trajectories
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24. Basic Updating Algorithm vs Partition Index Batch Updating (or PIBU) Algorithm
the processing time (lines) and the total scanned trajectories (bars) for the two approaches, with different k values
PIBU is 3.9 times faster
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25. Processing time (lines) and the number scanned trajectories (bars) versus the query region sizes
PIBU 3.8 times faster
Processing time (lines) and the number scanned trajectories (bars) by varying the size of trajectories datasets
PIBU 3.2 times faster
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26. Case Study: Advertisement Placement
Task: Put three billboards in New York City (NYC) for promotion
Dataset: Location based social networking check ins dataset
Divided the city into equal sized grids
(a) graph : multiple check ins by same users
(b) graph : Overlapped users in selected areas
(c0 graph : Result of paper solution
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27. Case Study: Charging Station Placement
Aim domain constraints:
Space for parking
POI categories locations
Two location should be far enough
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28. Conclusion Most influential k-location set mining problem
Covers Optimal and Approximate solutions
Optimal solution works of small regions and k values
Approximate works better on large regions and k values
2 case studies:
Billboard placement in NYC based on location-based social network data
EV charging station placement in Beijing
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29. Further Extensions Weighted Location Weighted Trajectories
What if the prices for selecting the locations are different
Weighted Trajectories
What if each person (trajectory) has different profile, and you want to make advertisement for different items
Spatio-temporal Selection
What about the bar, night clubs? They just care about people travel at night..
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30. Thanks Q A
input
hidden
output
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