computational movement analysis lecture 1: similarity joachim gudmundsson
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
Computational Movement Analysis
Lecture 1: SimilarityJoachim 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
53 4748 16
25
285
12 38
31 30
6
Our model
Input: Locations sampled over timeAssumptions:
a) Straight-line movement between consecutive samplesb) Constant velocity
0
53 4748 16
25
285
12 38
31 30
6
Brownian bridge movement model
Input: Locations sampled over timeOther 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 DailyWolves Finland 16 4 hours
Elks Sweden 25 30 minReindeer Finland 19 8 hoursRed deer Germany 7 4 hoursLeopards South
Africa32 Daily
Mountain Goats
USA 32 3 hours
Objects Number
Frequency
Football 23 25HzSupermarke
tlarge 100Hz
Mobile phones
huge 5 min
Trajectory data 2011
Applications: Behavioural ecology
› Migration patterns› “Popular places”› Home region› …
Applications: Behavioural ecology
› Migration patterns› “Popular places”› Home range› Flock movements› …
Applications: Behavioural ecology
[Joint work: Georgina Wilcox, Kevin buchin, Maike Buchin,
Ran Nathan and Anael Engel]
• Interaction between jackdaws? • Effect of external factors ?
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, 1995Yi, Jagadish & Faloutsos, 1998Gaffney & Smyth, 1999Kalpakis, Gada & Puttagunta, 2001Vlachos, Gunopulos & Kollios, 2002Gunopoulos, 2002Mamoulis, Cao, Kollios, Hadjieleftheriou, Tao & Cheung, 2004Chen, Ozsu & Oria, 2005Nanni & Pedreschi, 2006Van Kreveld & Luo, 2007Gaffney, Robertson, Smyth, Camargo & Ghil, 2007Lee, Han & Whang, 2007Buchin, Buchin, van Kreveld & Luo, 2009Agarwal, Aranov, van Kreveld, Löffler & Silveira, 2010Sankaraman, 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
2D
10D
Similar?
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
6D
Not similar
Similarity measures: Hausdorff
Input: Two polygonal curves P and Q in Rd
Distance: Hausdorff distance
H(PQ) = maxpP minqQ |p-q|
dH(P,Q) = max [H(PQ), H(QP)]
Similarity measures: Hausdorff
P P
Q Q
Input: Two polygonal curves P and Q in Rd
Distance: Hausdorff distance
H(PQ) = maxpP minqQ |p-q|
dH(P,Q) = max [H(PQ), H(QP)]
Similarity measures: Hausdorff
Drawbacks: Fail to capture the distance between curves.
P
Q
Input: Two polygonal curves P and Q in Rd
Distance: Hausdorff distance
H(PQ) = maxpP minqQ |p-q|
dH(P,Q) = max [H(PQ), H(QP)]
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
Input: Two polygonal chains P=p1, … , pn and Q=q1, … , qm in Rd.
The Fréchet distance between P and Q is:
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.
Fréchet Distance
(P,Q) =
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.
Discrete Fréchet Distance
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
Q
P
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
Q
P
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
Q
P
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
Q
P
r
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
Q
P
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
Q
Pr
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
Q
Pr
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
Q
P
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
Q
P
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
Q
P
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
Q
P
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
Q
P
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
Q
P
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
Q
P
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
Q
P
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
Q
P
Compute the Fréchet distance
Input: Two polygonal chains P=p1, … , pn and Q=q1, … , qm in Rd.
Decision problem: F(P, Q) ≤ r ?
Q
P
Freespace diagram of P and Qfree spac
e
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?
Q
P
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?
Recall that it has to be continuous, non-decreasing, q1 p1 and qm pn. Q
P
Compute the Fréchet distance
› q1 p1
Q
P
Compute the Fréchet distance
› q1 p1
› qm pn
Q
P
Compute the Fréchet distance
› q1 p1
› qm pn
› must lie in the free space
Q
P
Not valid!
Compute the Fréchet distance
› q1 p1
› qm pn
› must lie in the free space
› continuous
Q
P
Compute the Fréchet distance
Q
P
› q1 p1
› qm pn
› must lie in the free space
› continuous
› non-decreasing
Compute the Fréchet distance
› q1 p1
› qm pn
› must lie in the free space
› continuous
› non-decreasing
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.
Q
P
Free Space Diagram
Decision version Is the Fréchet distance between two paths at
most r?
r
P
P
Q
Q
Free Space Diagram
Decision version Is the Fréchet distance between two paths at
most r?
r
P
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.
(q1,p1)
(qm, pn)
r
P
Q
P
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.
(q1,p1)
(qm, pn)
r
P
Q
P
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.
(q1,p1)
(qm, pn)
r
P
Q
P
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.
(q1,p1)
(qm, pn)
r
P
Q
P
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.
(q1,p1)
(qm, pn)
r
P
Q
P
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.
(q1,p1)
(qm, pn)
r
P
Q
P
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.
(q1,p1)
(qm, pn)
r
P
Q
P
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.
(q1,p1)
(qm, pn)
r
P
Q
P
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.
(q1,p1)
(qm, pn)
r
P
Q
P
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
Q
P
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
Q
P
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
Q
P
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
Q
P
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
Q
P
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
Q
P
Decision algorithm: free space algorithm
Algorithm: 1. Compute Free Space diagram
O(mn) time
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).
Critical points
Decision algorithm: compute path
(q1,p1)
(qm,pn)
P
Q
Algorithm: 1. Compute Free Space diagram mn cells O(mn) time 2. Compute a non-xy-decreasing path from (q1,p1) to (qm,pn).
Decision algorithm: compute path
(q1,p1)
(qm,pn)
P
Q
Algorithm: 1. Compute Free Space diagram mn cells O(mn) time 2. Compute a non-xy-decreasing path from (q1,p1) to (qm,pn).
Decision algorithm: compute path
Algorithm: 1. Compute Free Space diagram mn cells O(mn) time 2. Compute a non-xy-decreasing path from (q1,p1) to (qm,pn).
(q1,p1)
(qm,pn)
P
Q
Decision algorithm: compute path
Algorithm: 1. Compute Free Space diagram mn cells O(mn) time 2. Compute a non-xy-decreasing path from (q1,p1) to (qm,pn).
(q1,p1)
(qm,pn)
P
Q
Decision algorithm: compute path
Algorithm: 1. Compute Free Space diagram mn cells O(mn) time 2. Compute a non-xy-decreasing path from (q1,p1) to (qm,pn).
(q1,p1)
(qm,pn)
P
Q
Decision algorithm: compute path
Algorithm: 1. Compute Free Space diagram mn cells O(mn) time 2. Compute a non-xy-decreasing path from (q1,p1) to (qm,pn).
(q1,p1)
(qm,pn)
P
Q
Decision algorithm: compute path
Algorithm: 1. Compute Free Space diagram mn cells O(mn) time 2. Compute a non-xy-decreasing path from (q1,p1) to (qm,pn).
(q1,p1)
(qm,pn)
P
Q
Decision algorithm: compute path
Algorithm: 1. Compute Free Space diagram mn cells O(mn) time 2. Compute a non-xy-decreasing path from (q1,p1) to (qm,pn).
(q1,p1)
(qm,pn)
P
Q
Decision algorithm: compute path
Algorithm: 1. Compute Free Space diagram mn cells O(mn) time 2. Compute a non-xy-decreasing path from (q1,p1) to (qm,pn).
(q1,p1)
(qm,pn)
P
Q
Decision algorithm: compute path
Algorithm: 1. Compute Free Space diagram mn cells O(mn) time 2. Compute a non-xy-decreasing path from (q1,p1) to (qm,pn). Build network O(mn) time.
(q1,p1)
(qm,pn)
P
Q
Decision algorithm: compute path
Algorithm: 1. Compute Free Space diagram mn cells O(mn) time 2. Compute a non-xy-decreasing path from (q1,p1) to (qm,pn). Build network O(mn) time. Find a path O(mn) time.
(q1,p1)
(qm,pn)
P
Q
Compute the Fréchet distance
Theorem: Given two polygonal curves P, Q and r0 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 r0 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) eventsCase 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) eventsCase 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) eventsCase 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) eventsCase 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 theparameter 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 ConjectureSODA, 2014Time: O(n2 (log n)1/2 (loglog n)3/2)
R. van Oostrum and R. Veltkamp. Parametric search made practical. CGTA, 2004