Computational Movement
Analysis
Lecture 1: Similarity
Joachim Gudmundsson
Aim
Goal: Extract and make sense of information from
trajectories (GPS, mobile, video, rfid…).
Provide fundamental software tools to simplify and
support qualitative analysis.
Our model
Input: Locations sampled over time
0
31
30
5
6
28
12
38
25
53
48
47
16
Our model
Input: Locations sampled over time
Assumptions:
a) Straight-line movement between consecutive samples
b) Constant velocity
0
31
30
5
6
28
12
38
25
53
48
47
16
Brownian bridge movement model
Input: Locations sampled over time
Other models? Brownian bridge
Brownian motion conditioned under starting and ending position
Probability that object
visits region A
Our model
Input: a sequence of (x,y,t) - coordinates and a time stamp
for example GPS data, mobile phone, radar, …
Additional information
Input: a sequence of (x, y, t, z, temperature, speed, direction…)
Trajectory data 2005
Animal
Country
Number
Frequency
Wolves
Spain
15
Daily
Wolves
Finland
16
4 hours
Elks
Sweden
25
30 min
Reindeer
Finland
19
8 hours
Red deer
Germany
7
4 hours
Leopards
South
Africa
32
Daily
Mountain
Goats
USA
32
3 hours
Trajectory data 2011
Objects
Number
Frequency
Football
23
25Hz
Supermarket
large
100Hz
Mobile
phones
huge
5 min
Applications: Behavioural ecology
› Migration patterns
› “Popular places”
› Home region
›…
Applications: Behavioural ecology
› Migration patterns
› “Popular places”
› Home range
› Flock movements
›…
Applications: Behavioural ecology
• Interaction between jackdaws?
• Effect of external factors ?
[Joint work: Georgina Wilcox,
Kevin buchin, Maike Buchin,
Ran Nathan and Anael Engel]
Applications: Behavioural ecology
Behavioural ecology
Short term:
- Annual migration behaviour
- Automatically detect activity
- Home range
Long term:
- Foraging search strategies
- Factors that may influence animal movement
- How changes in the environment affects wildlife
Applications: Sport
Data
Accuracy: 100 mm
Frequency: 25Hz
[Joint work: Thomas Wolle and many others]
Applications: Sport
Tools
•
•
•
to support sports analysts and coaches
Evaluate a players ability to pass (receive, execute…)
Evaluate a players ability to move into open areas
Evaluate a players ability to follow instructions
• Detect frequent movement patterns
• Detect correlations between players’ movement
• Detect frequent passing patterns
• Analyse turn-overs
• Which events change the intensity of the game?
• …
Applications: Hurricanes
Goal:
Early detection of hurricanes that may hit land
Applications: Transport
› Collision prevention
› Maximize unloading/loading
Applications: Shopping behaviour
› Detect movement patterns
› Change structure of shop to
increase sales?
Examples of general problems
General problems
Repetitive patterns
Popular places
Trajectory segmentation
Trajectory simplification
Trajectory clustering
Subtrajectory clustering
Movement patterns
• Single file
• Formations
• Leadership
• Meetings
• …
•
•
•
•
•
•
•
Seminar schedule
Lecture 1: Similarity
Lecture 2: Curve clustering
Lecture 3: Simplification
Lecture 4: Movement patterns
Lecture 5: Trajectory segmentation
Fundamental tools: clustering
When are two trajectories similar?
In most cases the spatial similarity
is much more important than the
temporal aspect.
Similarity of curves…measurements
Agrawal, Lin, Sawhney & Shim, 1995
Alt & Godau, 1995
Yi, Jagadish & Faloutsos, 1998
Gaffney & Smyth, 1999
Kalpakis, Gada & Puttagunta, 2001
Vlachos, Gunopulos & Kollios, 2002
Gunopoulos, 2002
Mamoulis, Cao, Kollios, Hadjieleftheriou, Tao & Cheung, 2004
Chen, Ozsu & Oria, 2005
Nanni & Pedreschi, 2006
Van Kreveld & Luo, 2007
Gaffney, Robertson, Smyth, Camargo & Ghil, 2007
Lee, Han & Whang, 2007
Buchin, Buchin, van Kreveld & Luo, 2009
Agarwal, Aranov, van Kreveld, Löffler & Silveira, 2010
Sankaraman, Agarwal, Molhave, Pan and Boedihardjo, 2013
...
Similarity measures
Input: Two polygonal curves P and Q of size m in Rd
Approach: Map curve into a point in dm-dimensional space

2D
10D
Similarity measures
Input: Two polygonal curves P and Q of size m in Rd
Approach: Map curve into a point in dm-dimensional space

Similar?
10D
2D
Similarity measures
Input: Two polygonal curves P and Q of size m in Rd
Approach: Map curve into a point in dm-dimensional space
Drawbacks: P and Q must have same number points
Only takes the vertices into consideration

2D
Not
similar
6D
Similarity measures: Hausdorff
Input: Two polygonal curves P and Q in Rd
Distance: Hausdorff distance
H(PQ) = maxpP minqQ |p-q|
dH(P,Q) = max [H(PQ), H(QP)]
Similarity measures: Hausdorff
Input: Two polygonal curves P and Q in Rd
Distance: Hausdorff distance
H(PQ) = maxpP minqQ |p-q|
dH(P,Q) = max [H(PQ), H(QP)]
P
Q
P
Q
Similarity measures: Hausdorff
Input: Two polygonal curves P and Q in Rd
Distance: Hausdorff distance
H(PQ) = maxpP minqQ |p-q|
dH(P,Q) = max [H(PQ), H(QP)]
P
Q
Drawbacks: Fail to capture the distance between curves.
Similarity measures: Fréchet
Input: Two polygonal curves P and Q in Rd
Other interesting measures are:
• Dynamic Time Warping (DTW)
• Longest Common Subsequence (LCSS)
• Model-Driven matching (MDM)
Drawback: Non-metrics (triangle inequality does not hold)
which makes clustering complicated.
Similarity measure: Fréchet Distance
Fréchet Distance measures the similarity of two curves.
Dog walking example
- Person is walking his dog (person on one curve and the dog on other)
- Allowed to control their speeds but not allowed to go backwards!
- Fréchet distance of the curves: minimal leash length necessary for both
to walk the curves from beginning to end
Fréchet Distance
Fréchet Distance measures the similarity of two curves.
Dog walking example
- Person is walking his dog (person on one curve and the dog on other)
- Allowed to control their speeds but not allowed to go backwards!
- Fréchet distance of the curves: minimal leash length necessary for both
to walk the curves from beginning to end
Fréchet Distance
Input: Two polygonal chains P=p1, … , pn and Q=q1, … , qm in Rd.
The Fréchet distance between P and Q is:
𝛿𝐹 (P,Q) = 𝛼:[0,1]→𝑃
inf
max |𝑃(𝛼
𝑡 ∈[0,1]
𝑡 ) − 𝑄(𝛽 𝑡 )|
𝛽:[0,1]→𝑄
where  and  range over all continuous non-decreasing
reparametrizations.
Note that (0)=p1, (1)=pn, (0)=q1 and (1)=qm.
Well-suited for the comparison of curves since it takes the
continuity of the curves into account.
Discrete Fréchet Distance
The discrete Fréchet distance considers only positions of the leash
where its endpoints are located at vertices of the two polygonal
curves and never in the interior of an edge.
May give large errors!
Compute the continuous Fréchet distance
Input: Two polygonal chains P=p1, … , pn and Q=q1, … , qm in Rd.
Decision problem: F(P, Q) ≤ r ?
P
Q
P
Q
Compute the Fréchet distance
Input: Two polygonal chains P=p1, … , pn and Q=q1, … , qm in Rd.
Decision problem: F(P, Q) ≤ r ?
P
Q
P
Q
Compute the Fréchet distance
Input: Two polygonal chains P=p1, … , pn and Q=q1, … , qm in Rd.
Decision problem: F(P, Q) ≤ r ?
P
Q
P
Q
Compute the Fréchet distance
Input: Two polygonal chains P=p1, … , pn and Q=q1, … , qm in Rd.
Decision problem: F(P, Q) ≤ r ?
P
P
Q
r
Q
Compute the Fréchet distance
Input: Two polygonal chains P=p1, … , pn and Q=q1, … , qm in Rd.
Decision problem: F(P, Q) ≤ r ?
P
Q
P
Q
Compute the Fréchet distance
Input: Two polygonal chains P=p1, … , pn and Q=q1, … , qm in Rd.
Decision problem: F(P, Q) ≤ r ?
P
r
Q
P
Q
Compute the Fréchet distance
Input: Two polygonal chains P=p1, … , pn and Q=q1, … , qm in Rd.
Decision problem: F(P, Q) ≤ r ?
P
r
Q
P
Q
Compute the Fréchet distance
Input: Two polygonal chains P=p1, … , pn and Q=q1, … , qm in Rd.
Decision problem: F(P, Q) ≤ r ?
P
Q
P
Q
Compute the Fréchet distance
Input: Two polygonal chains P=p1, … , pn and Q=q1, … , qm in Rd.
Decision problem: F(P, Q) ≤ r ?
P
Q
P
Q
Compute the Fréchet distance
Input: Two polygonal chains P=p1, … , pn and Q=q1, … , qm in Rd.
Decision problem: F(P, Q) ≤ r ?
P
Q
P
Q
Compute the Fréchet distance
Input: Two polygonal chains P=p1, … , pn and Q=q1, … , qm in Rd.
Decision problem: F(P, Q) ≤ r ?
P
Q
P
Q
Compute the Fréchet distance
Input: Two polygonal chains P=p1, … , pn and Q=q1, … , qm in Rd.
Decision problem: F(P, Q) ≤ r ?
P
Q
P
Q
Compute the Fréchet distance
Input: Two polygonal chains P=p1, … , pn and Q=q1, … , qm in Rd.
Decision problem: F(P, Q) ≤ r ?
P
Q
P
Q
Compute the Fréchet distance
Input: Two polygonal chains P=p1, … , pn and Q=q1, … , qm in Rd.
Decision problem: F(P, Q) ≤ r ?
P
Q
P
Q
Compute the Fréchet distance
Input: Two polygonal chains P=p1, … , pn and Q=q1, … , qm in Rd.
Decision problem: F(P, Q) ≤ r ?
P
Q
P
Q
Compute the Fréchet distance
Input: Two polygonal chains P=p1, … , pn and Q=q1, … , qm in Rd.
Decision problem: F(P, Q) ≤ r ?
P
Q
P
Q
Compute the Fréchet distance
Input: Two polygonal chains P=p1, … , pn and Q=q1, … , qm in Rd.
Decision problem: F(P, Q) ≤ r ?
Freespace diagram of P and Q
P
free
space
Q
Compute the Fréchet distance
Input: Two polygonal chains P=p1, … , pn and Q=q1, … , qm in Rd.
Decision problem: F(P, Q) ≤ r ?
How do we find a reparametrization
between P and Q that realises the
Fréchet distance r?
P
Q
Compute the Fréchet distance
Input: Two polygonal chains P=p1, … , pn and Q=q1, … , qm in Rd.
Decision problem: F(P, Q) ≤ r ?
How do we find a reparametrization
between P and Q that realises the
Fréchet distance r?
P
Recall that it has to be continuous,
non-decreasing, q1  p1 and qm  pn.
Q
Compute the Fréchet distance
› q1  p1
P
Q
Compute the Fréchet distance
› q1  p1
› qm  pn
P
Q
Compute the Fréchet distance
› q1  p1
› qm  pn
› must lie in the free space
Not valid!
P
Q
Compute the Fréchet distance
› q1  p1
› qm  pn
› must lie in the free space
› continuous
P
Q
Compute the Fréchet distance
› q1  p1
› qm  pn
› must lie in the free space
› continuous
› non-decreasing
P
Q
Compute the Fréchet distance
› q1  p1
› qm  pn
› must lie in the free space
P
› continuous
› non-decreasing
Q
 If there exists an xy-non-decreasing path inside the freespace
from the bottom left to the top right corner of the diagram then
the Fréchet distance is at most r.
Free Space Diagram
Decision version
Is the Fréchet distance between two paths at most r?
P
P
Q
r
Q
Free Space Diagram
Decision version
Is the Fréchet distance between two paths at most r?
P
r
Q
Free Space Diagram
If there exists a non-decreasing path in the free space from (q1,p1)
to (qm,pn) which is non-decreasing in both coordinates, then curves
P and Q have Fréchet distance at most r.
P
(qm, pn)
P
Q
r
(q1,p1)
Q
Free Space Diagram
If there exists a non-decreasing path in the free space from (q1,p1)
to (qm,pn) which is non-decreasing in both coordinates, then curves
P and Q have Fréchet distance at most r.
P
(qm, pn)
P
Q
r
(q1,p1)
Q
Free Space Diagram
If there exists a non-decreasing path in the free space from (q1,p1)
to (qm,pn) which is non-decreasing in both coordinates, then curves
P and Q have Fréchet distance at most r.
P
(qm, pn)
P
Q
r
(q1,p1)
Q
Free Space Diagram
If there exists a non-decreasing path in the free space from (q1,p1)
to (qm,pn) which is non-decreasing in both coordinates, then curves
P and Q have Fréchet distance at most r.
P
(qm, pn)
P
Q
r
(q1,p1)
Q
Free Space Diagram
If there exists a non-decreasing path in the free space from (q1,p1)
to (qm,pn) which is non-decreasing in both coordinates, then curves
P and Q have Fréchet distance at most r.
P
(qm, pn)
P
Q
r
(q1,p1)
Q
Free Space Diagram
If there exists a non-decreasing path in the free space from (q1,p1)
to (qm,pn) which is non-decreasing in both coordinates, then curves
P and Q have Fréchet distance at most r.
P
(qm, pn)
P
Q
r
(q1,p1)
Q
Free Space Diagram
If there exists a non-decreasing path in the free space from (q1,p1)
to (qm,pn) which is non-decreasing in both coordinates, then curves
P and Q have Fréchet distance at most r.
P
(qm, pn)
P
Q
r
(q1,p1)
Q
Free Space Diagram
If there exists a non-decreasing path in the free space from (q1,p1)
to (qm,pn) which is non-decreasing in both coordinates, then curves
P and Q have Fréchet distance at most r.
P
(qm, pn)
P
Q
r
(q1,p1)
Q
Free Space Diagram
If there exists a non-decreasing path in the free space from (q1,p1)
to (qm,pn) which is non-decreasing in both coordinates, then curves
P and Q have Fréchet distance at most r.
P
(qm, pn)
P
Q
r
(q1,p1)
Q
Decision algorithm: free space diagram
Algorithm:
1. Compute Free Space diagram
O(mn) time
A square and an ellipsoid can intersect in at most 8 points.
(critical points)
The free space in a cell is a convex region.
 One cell can be constructed in time O(1).
Compute the Fréchet distance
One cell can be constructed in time O(1). How?
Note that we do not need the exact shape. We only need the
intersection points between the cell boundary and the ellipsoid.
P
Q
P
Q
Compute the Fréchet distance
One cell can be constructed in time O(1). How?
Note that we do not need the exact shape. We only need the
intersection points between the cell boundary and the ellipsoid.
P
Q
P
Q
Compute the Fréchet distance
One cell can be constructed in time O(1). How?
Note that we do not need the exact shape. We only need the
intersection points between the cell boundary and the ellipsoid.
P
Q
P
Q
Compute the Fréchet distance
One cell can be constructed in time O(1). How?
Note that we do not need the exact shape. We only need the
intersection points between the cell boundary and the ellipsoid.
P
Q
P
Q
Compute the Fréchet distance
One cell can be constructed in time O(1). How?
Note that we do not need the exact shape. We only need the
intersection points between the cell boundary and the ellipsoid.
P
Q
P
Q
Compute the Fréchet distance
One cell can be constructed in time O(1). How?
Note that we do not need the exact shape. We only need the
intersection points between the cell boundary and the ellipsoid.
P
Q
P
Q
Decision algorithm: free space algorithm
Algorithm:
1. Compute Free Space diagram
O(mn) time
Critical points
A square and an ellipsoid can intersect in at most 8 points.
The free space in a cell is a convex region.
 One cell can be constructed in time O(1).
Decision algorithm: compute path
Algorithm:
1. Compute Free Space diagram
mn cells  O(mn) time
(qm,pn)
2. Compute a non-xy-decreasing path
from (q1,p1) to (qm,pn).
P
(q1,p1)
Q
Decision algorithm: compute path
Algorithm:
1. Compute Free Space diagram
mn cells  O(mn) time
(qm,pn)
2. Compute a non-xy-decreasing path
from (q1,p1) to (qm,pn).
P
(q1,p1)
Q
Decision algorithm: compute path
Algorithm:
1. Compute Free Space diagram
mn cells  O(mn) time
(qm,pn)
2. Compute a non-xy-decreasing path
from (q1,p1) to (qm,pn).
P
(q1,p1)
Q
Decision algorithm: compute path
Algorithm:
1. Compute Free Space diagram
mn cells  O(mn) time
(qm,pn)
2. Compute a non-xy-decreasing path
from (q1,p1) to (qm,pn).
P
(q1,p1)
Q
Decision algorithm: compute path
Algorithm:
1. Compute Free Space diagram
mn cells  O(mn) time
(qm,pn)
2. Compute a non-xy-decreasing path
from (q1,p1) to (qm,pn).
P
(q1,p1)
Q
Decision algorithm: compute path
Algorithm:
1. Compute Free Space diagram
mn cells  O(mn) time
(qm,pn)
2. Compute a non-xy-decreasing path
from (q1,p1) to (qm,pn).
P
(q1,p1)
Q
Decision algorithm: compute path
Algorithm:
1. Compute Free Space diagram
mn cells  O(mn) time
(qm,pn)
2. Compute a non-xy-decreasing path
from (q1,p1) to (qm,pn).
P
(q1,p1)
Q
Decision algorithm: compute path
Algorithm:
1. Compute Free Space diagram
mn cells  O(mn) time
(qm,pn)
2. Compute a non-xy-decreasing path
from (q1,p1) to (qm,pn).
P
(q1,p1)
Q
Decision algorithm: compute path
Algorithm:
1. Compute Free Space diagram
mn cells  O(mn) time
(qm,pn)
2. Compute a non-xy-decreasing path
from (q1,p1) to (qm,pn).
P
(q1,p1)
Q
Decision algorithm: compute path
Algorithm:
1. Compute Free Space diagram
mn cells  O(mn) time
(qm,pn)
2. Compute a non-xy-decreasing path
from (q1,p1) to (qm,pn).
Build network O(mn) time.
P
(q1,p1)
Q
Decision algorithm: compute path
Algorithm:
1. Compute Free Space diagram
mn cells  O(mn) time
(qm,pn)
2. Compute a non-xy-decreasing path
from (q1,p1) to (qm,pn).
Build network O(mn) time.
Find a path O(mn) time.
P
(q1,p1)
Q
Compute the Fréchet distance
Theorem: Given two polygonal curves P, Q and r0 one can decide
in O(mn) time, whether (P,Q)  r.
How can we use the algorithm to compute the Fréchet distance?
Binary search on r?
But what do we do binary search on?
Compute the Fréchet distance
Theorem: Given two polygonal curves P, Q and r0 one can decide
in O(mn) time, whether (P,Q)  r.
How can we use the algorithm to compute the Fréchet distance?
In practice: Determine r bit by bit
 O(mn log “accuracy”) time.
Compute the Fréchet distance
Assume r=0 and the let r grow.
 The free space will become larger and larger.
Aim: Determine the smallest r such that it contains a montone
path from (q1,p1) to (qm,pn).
This can only occur when:
1. r is minimal with (q1,p1) to (qm,pn) in the free space,
2. r is minimal when a new vertical or horizontal passage opens
up between two adjacent cells in the free space.
3. r is minimal when a new vertical or horizontal passage opens
up between two non-adjacent cells in the free space.
Compute the Fréchet distance
1. r is minimal with (q1,p1) to (qm,pn) in the free space,
2. r is minimal when a new vertical or horizontal passage opens up
between two adjacent cells in the free space.
3. r is minimal when a new vertical or horizontal passage opens up
between two non-adjacent cells in the free space.
Case 1 is easy to test in constant time.
Compute the Fréchet distance
1. r is minimal with (q1,p1) to (qm,pn) in the free space,
2. r is minimal when a new vertical or horizontal passage opens up
between two adjacent cells in the free space.
3. r is minimal when a new vertical or horizontal passage opens up
between two non-adjacent cells in the free space.
Case 1 is easy to test in constant time.
Case 2:
Compute the Fréchet distance
1. r is minimal with (q1,p1) to (qm,pn) in the free space,
2. r is minimal when a new vertical or horizontal passage opens up
between two adjacent cells in the free space.
3. r is minimal when a new vertical or horizontal passage opens up
between two non-adjacent cells in the free space.
Case 1 is easy to test in constant time.
Case 2:
Compute the Fréchet distance
1. r is minimal with (q1,p1) to (qm,pn) in the free space,
2. r is minimal when a new vertical or horizontal passage opens up
between two adjacent cells in the free space.
3. r is minimal when a new vertical or horizontal passage opens up
between two non-adjacent cells in the free space.
Case 1 is easy to test in constant time.
Case 2:
Compute the Fréchet distance
1. r is minimal with (q1,p1) to (qm,pn) in the free space,
2. r is minimal when a new vertical or horizontal passage opens up
between two adjacent cells in the free space.
3. r is minimal when a new vertical or horizontal passage opens up
between two non-adjacent cells in the free space.
Case 1 is easy to test in constant time.
Case 2:
Number of events?
Compute the Fréchet distance
1. r is minimal with (q1,p1) to (qm,pn) in the free space,
2. r is minimal when a new vertical or horizontal passage opens up
between two adjacent cells in the free space.
3. r is minimal when a new vertical or horizontal passage opens up
between two non-adjacent cells in the free space.
Case 1 is easy to test in constant time.
Case 2:
Number of events: O(mn)
Compute the Fréchet distance
1. r is minimal with (q1,p1) to (qm,pn) in the free space,
2. r is minimal when a new vertical or horizontal passage opens up
between two adjacent cells in the free space.
3. r is minimal when a new vertical or horizontal passage opens up
between two non-adjacent cells in the free space.
Case 1 is easy to test in constant time.
Case 2: O(mn) events
Case 3:
Compute the Fréchet distance
1. r is minimal with (q1,p1) to (qm,pn) in the free space,
2. r is minimal when a new vertical or horizontal passage opens up
between two adjacent cells in the free space.
3. r is minimal when a new vertical or horizontal passage opens up
between two non-adjacent cells in the free space.
Case 1 is easy to test in constant time.
Case 2: O(mn) events
Case 3:
Compute the Fréchet distance
1. r is minimal with (q1,p1) to (qm,pn) in the free space,
2. r is minimal when a new vertical or horizontal passage opens up
between two adjacent cells in the free space.
3. r is minimal when a new vertical or horizontal passage opens up
between two non-adjacent cells in the free space.
Case 1 is easy to test in constant time.
Case 2: O(mn) events
Case 3: Number of events?
Compute the Fréchet distance
1. r is minimal with (q1,p1) to (qm,pn) in the free space,
2. r is minimal when a new vertical or horizontal passage opens up
between two adjacent cells in the free space.
3. r is minimal when a new vertical or horizontal passage opens up
between two non-adjacent cells in the free space.
Case 1 is easy to test in constant time.
Case 2: O(mn) events
Case 3: O(m2n+mn2)
Compute the Fréchet distance
Case 1: easy to test in constant time.
Case 2: O(mn) possible events
Case 3: O(m2n+mn2) possible events
Perform binary search on the events  O((m2n+mn2) log mn) time.
Compute the Fréchet distance
Case 1: easy to test in constant time.
Case 2: O(mn) possible events
Case 3: O(m2n+mn2) possible events
Improvement: Use parametric search [Meggido’83, Cole’88]
 O(mn log mn) time.
Parametric search
The decision version F(r) of the Fréchet distance depends on the
parameter r, and it is monotone in r.
Aim: Find minimum r such that F(r) is true.
Our algorithm As can solve F(r) in Ts=O(mn) time.
Meggido’83 and Cole’88 (simplified):
If there exists a parallel algorithm Ap that can solve F(r) in Tp time
using P processors then there exists a sequential algorithm with
running time O(PTp+TpTs+Ts log P).
For our problem we get:
O(mn log mn) time using a parallel sorting network
[n processors and O(log n) time]
Compute the Fréchet distance
Case 1: easy to test in constant time.
Case 2: O(mn) possible events
Case 3: O(m2n+mn2) possible events
Improvement: Use parametric search [Meggido’83, Cole’88]
 O(mn log mn) time.
Summary: Lecture 1
Many different similarity measures.
We use the Fréchet distance!
Theorem: Given two polygonal curves P, Q and r  0 one can decide
in O(mn) time, whether (P,Q)  r.
Theorem: The Fréchet distance between two polygonal curves P and
Q can be computed in O(mn log mn) time.
Open problems
1. Can the Fréchet distance be computed in subquadratic time or
is there an (n2) lower bound?
2. Approximate the Fréchet distance in subquadratic time?
3. Special cases in subquadratic time?
If the curves are c-packed then a (1+ )-approximation of the
Fréchet distance can be computed in O(cn/+cn log n).
similar results for low density.
[Driemel, Har-Peled and Wenk’12]
Open problems
1. Can the Fréchet distance be computed in subquadratic time or
is there an (n2) lower bound?
2. Approximate the Fréchet distance in subquadratic time?
3. Special cases in subquadratic time?
If the curves are c-packed then a (1+ )-approximation of the
Fréchet distance can be computed in O(cn/+cn log n).
similar results for low density.
[Driemel, Har-Peled and Wenk’12]
Part of P inside any ball B is at
most c times the radius of B.
References
H. Alt and M. Godau.
Computing the Fréchet distance between two polygonal curves.
IJCGA, 1995.
K. Buchin, M. Buchin, W. Meulemans and W. Mulzer.
Four Soviets Walk the Dog - with an Application to Alt's Conjecture
SODA, 2014
Time: O(n2 (log n)1/2 (loglog n)3/2)
R. van Oostrum and R. Veltkamp.
Parametric search made practical.
CGTA, 2004