1. 2IMA20 Algorithms for Geographic Data
Spring Lecture 4: Segmentation

2. Motivation: Geese Migration
Two behavioural types
stopover
migration flight
Input:
GPS tracks
expert description of behaviour
Goal: Delineate stopover sites of migratory geese

3. Abstract / general purpose questions
Single trajectory
simplification, cleaning
segmentation into semantically meaningful parts
finding recurring patterns (repeated subtrajectories)
Two trajectories
similarity computation
subtrajectory similarity
Multiple trajectories
clustering, outliers
flocking/grouping pattern detection
finding a typical trajectory or computing a mean/median trajectory
visualization

4. Problem
For analysis, it is often necessary to break a trajectory into pieces according to the behaviour of the entity (e.g., walking, flying, …).
Input: A trajectory T, where each point has a set of attribute values, and a set of criteria.
Attributes: speed, heading, curvature…
Criteria: bounded variance in speed, curvature, direction, distance…
Aim: Partition T into a minimum number of subtrajectories (so-called segments) such that each segment fulfils the criteria.
“Within each segment the points have similar attribute values”

5. Criteria-Based Segmentation
Goal: Partition trajectory into a small number of segments
such that a given criterion is fulfilled on each segment

6. Example Trajectory T sampled with equal time intervals
Criterion: speed cannot differ more than a factor 2?

7. Example speed Trajectory T sampled with equal time intervals
Criterion: speed cannot differ more than a factor 2?
Criterion: direction of motion differs by at most 90⁰?
speed

8. Example speed speed Trajectory T sampled with equal time intervals
Criterion: speed cannot differ more than a factor 2?
Criterion: direction of motion differs by at most 90⁰?
speed
speed
heading

9. Example speed speed Trajectory T sampled with equal time intervals
Criterion: speed cannot differ more than a factor 2?
Criterion: direction of motion differs by at most 90⁰?
speed
speed
speed & heading
heading

10. Decreasing monotone criteria
Definition: A criterion is decreasing monotone, if it holds on a segment, it holds on any subsegment.
Examples: disk criterion (location), angular range (heading), speed…
Theorem: A combination of conjunctions and disjunctions of decreasing monotone criteria is a decreasing monotone criterion.

11. Greedy Algorithm Observation:
If criteria are decreasing monotone, a greedy strategy works.

12. Designing greedy algorithms
try to discover structure of optimal solutions: what properties do optimal solutions have ?
what are the choices that need to be made ?
do we have optimal substructure ?
optimal solution = first choice + optimal solution for subproblem
do we have greedy-choice property for the first choice ?
prove that optimal solutions indeed have these properties
prove optimal substructure and greedy-choice property
use these properties to design an algorithm and prove correctness
proof by induction (possible because optimal substructure)

13. lemma obviously holds, so assume OPT does not include ai.
Lemma: Let k be the smallest index such that 𝜏 𝑡 0 , 𝑡 𝑘 fails for a monotone decreasing criterion 𝜏. If there is a solution, then there is an optimal solution including 𝜏 𝑡 0 , 𝑡 𝑘−1 as segment.
Proof. Let OPT be an optimal solution for A. If OPT includes ai then the
lemma obviously holds, so assume OPT does not include ai.
We will show how to modify OPT into a solution OPT* such that
(i) OPT* is a valid solution
(ii) OPT* includes ai
(iii) size(OPT*) ≤ size(OPT)
Thus OPT* is an optimal solution including 𝜏 𝑡 0 , 𝑡 𝑘−1 , and so the lemma holds. To modify OPT we proceed as follows.
standard text you can basically use in proof for any greedy-choice property
quality OPT* ≥ quality OPT
here comes the modification, which is problem-specific

14. Greedy Algorithm Observation:
If criteria are decreasing monotone, a greedy strategy works.
For many decreasing monotone criteria Greedy requires O(n) time, e.g. for speed, heading…

15. Greedy Algorithm Observation:
For some criteria, iterative double & search is faster.
Double & search: An exponential search followed by a binary search.

16. Criteria-Based Segmentation
Boolean or linear combination of decreasing monotone criteria
Greedy Algorithm
incremental in O(n) time or constant-update criteria e.g. bounds on speed or heading
double & search in O(n log n) time for non-constant update criteria
[M.Buchin et al.’11]

17. Motivation: Geese Migration
Two behavioural types
stopover
migration flight
Input:
GPS tracks
expert description of behaviour
Goal: Delineate stopover sites of migratory geese

18. Case Study: Geese Migration
Data
Spring migration tracks
White-fronted geese
4-5 positions per day
March – June
Up to 10 stopovers during spring migration
Stopover: 48 h within radius 30 km
Flight: change in heading <120°
Kees
Adri

19. Comparison
manual
computed

20. Evaluation
Few local differences: Shorter stops, extra cuts in computed segmentation

21. Criteria A combination of decreasing and increasing monotone criteria
stopover
migration flight
Within radius 30km
At least 48h
Change in heading <120°
AND OR

22. Decreasing and Increasing Monotone Criteria
Observation: For a combination of decreasing and increasing monotone criteria the greedy strategy does not always work.
Example: Min duration 2 AND Max speed range 4 5
speed 1

23. Non-Monotone Segmentation
Many Criteria are not (decreasing) monotone:
Minimum time
Standard deviation
Fixed percentage of outliers
For these Aronov et al. introduced the start-stop diagram
Example: Geese Migration

24. Start-Stop Diagram Algorithmic approach
input trajectory compute start-stop diagram compute segmentation

25. Start-Stop Diagram
Given a trajectory T over time interval I = {t0,…,t} and criterion C
The start-stop diagram D is (the upper diagonal half of) the n x n grid,
where each point (i,j) is associated to segment [ti,tj] with
(i,j) is in free space if C holds on [ti,tj]
(i,j) is in forbidden space if C does not hold on [ti,tj]
A (minimal) segmentation of T corresponds to a (min-link) staircase in D
forbidden space
free space
staircase

26. Start-Stop Diagram
A (minimal) segmentation of T corresponds to a (min-link) staircase in D. 12 11 10 9 8 7 6 5 4 3 2 1 6 7 8 5 9 11 10 4 3 12 2 1

27. Start-Stop Diagram
A (minimal) segmentation of T corresponds to a (min-link) staircase in D. 12 11 10 9 8 7 6 5 4 3 2 1 6 7 8 5 9 11 10 4 3 12 2 1

28. Start-Stop Diagram Discrete case:
A non-monotone segmentation can be computed in O(n2) time. 12 11 10 9 8 7 6 5 4 3 2 1 4 5 8 11 1 12 6 9 10 7

29. Start-Stop Diagram Discrete case:
A non-monotone segmentation can be computed in O(n2) time. 12 11 10 9 8 7 6 5 4 3 2 1 4 5 8 11 1 12 6 9 10 7

30. Stable Criteria Definition:
A criterion is stable if and only if 𝑖=𝑜 𝑛 𝑣 𝑖 =𝑂(𝑛) where
𝑣 𝑖 = number of changes of validity on segments [0,i], [1,i], …, [i-1,i]

31. Stable Criteria Definition:
A criterion is stable if and only if 𝑖=𝑜 𝑛 𝑣 𝑖 =𝑂(𝑛) where
𝑣 𝑖 = number of changes of validity on segments [0,i], [1,i], …, [i-1,i]
Observations:
Decreasing and increasing monotone criteria are stable.
A conjunction or disjunction of stable criterion are stable.

32. Compressed Start-Stop Diagram
For stable criteria the start-stop diagram can be compressed by applying run-length encoding.
Examples:
decreasing monotone
increasing monotone 9

33. Computing the Compressed Start-Stop Diagram
For a decreasing criterion consider the algorithm:
ComputeLongestValid(crit C, traj T)
Algorithm:
Move two pointers i,j from n to 0 over the trajectory. For every trajectory index j the smallest index i for which 𝜏 𝑡 𝑖 , 𝑡 𝑗 satisfies the criterion C is stored. 9 i j

34. Computing the Compressed Start-Stop Diagram
For a decreasing criterion ComputeLongestValid(crit C, traj T):
Move two pointers i,j from n to 0
over the trajectory 9 i j
Requires a data structure for segment [ti,tj] allowing the operations isValid, extend, and shorten, e.g., a balanced binary search tree on attribute values for range or bound criteria.
Runs in O(nc(n)) time where c(n) is the time to update & query.
Analogously for increasing criteria.

35. Computing the Compressed Start-Stop Diagram
The start-stop diagram of a conjunction (or disjunction) of two stable criteria is their intersection (or union).
The start-stop diagram of a
negated criteria is its inverse.
The corresponding compressed
start-stop diagrams can be
computed in O(n) time. 9

36. Attributes and criteria
Examples of stable criteria
Lower bound/Upper bound on attribute
Angular range criterion
Disk criterion
Allow an approximate fraction of outliers …

37. Computing the Optimal Segmentation
Observation: The optimal segmentation for [0,i] is either one segment, or an optimal sequence of segments for [0,j<i] appended with a segment [j,i], where j is an index such [j,i] is valid.
Dynamic programming algorithm
for each row from 0 to n
find white cell with min-link
That is, iteratively compute a table S[0,n]
where entry S[i] for row i stores
last: index of last link
count: number of links so far
runs in O(n2) time n
10, 4
8, 3
8, 3
7, 2
0, 1
0, 0+1=1
nil, ∞ S
nil, ∞
nil, ∞
nil, ∞
nil, ∞
nil, 0

38. Computing the Optimal Segmentation
More efficient dynamic programming algorithm for compressed diagrams
Process blocks of white cells using a range query in a binary search tree T (instead of table S) storing
index: row index
last: index of last link
count: number of links so far
augmented by minimal count
in subtree T
[Alewijnse et al.’14]

39. Methodology for augmenting a data structure
Choose an underlying data structure.
Determine additional information to maintain.
Verify that we can maintain additional information for existing data structure operations.
Develop new operations.
R-B tree
min-count[x]
theorem next slide
count(block) using range query

40. Theorem Augment a R-B tree with field f, where f[x] depends only on information in x, left[x], and right[x] (including f[left[x]] and f[right[x]]). Then can maintain values of f in all nodes during insert and delete without affecting O(log n) performance.
When we alter information in x, changes propagate only upward on the search path for x …

41. Computing the Optimal Segmentation
More efficient dynamic programming algorithm for compressed diagrams
Process blocks of white cells using a range query in a binary search tree T (instead of table S) storing
index: row index
last: index of last link
count: number of links so far
augmented by minimal count
in subtree
runs in O(n log n) time T
[Alewijnse et al.’14]

42. Beyond criteria-based segmentation
When is criteria-based segmentation applicable?
What can we do in other cases?

43. Excursion: Movement Models

44. Movement Models
Movement data: sequence of observations, e.g. (xi,yi,ti)
Linear movement realistic?

45. Movement Models
Movement data: sequence of observations, e.g. (xi,yi,ti)
Linear movement realistic?
Space-time Prisms

46. Movement Models
Movement data: sequence of observations, e.g. (xi,yi,ti)
Linear movement realistic?
Space-time Prisms

47. Movement Models
Movement data: sequence of observations, e.g. (xi,yi,ti)
Linear movement realistic?
Space-time Prisms

48. Movement Models
Movement data: sequence of observations, e.g. (xi,yi,ti)
Linear movement realistic?
Space-time Prisms
Random Motion Models (e.g. Brownian bridges)

49. Movement Models
Movement data: sequence of observations, e.g. (xi,yi,ti)
Linear movement realistic?
Space-time Prisms
Random Motion Models (e.g. Brownian bridges)

50. Brownian Bridges - Examples
no bridges

51. Segment by diffusion coefficient

52. Summary
Greedy algorithm for decreasing monotone criteria O(n) or O(n log n) time
[M.Buchin, Driemel, van Kreveld, Sacristan, 2010]
Case Study: Geese Migration
[M.Buchin, Kruckenberg, Kölzsch, 2012]
Start-stop diagram for arbitrary criteria O(n2) time
[Aronov, Driemel, van Kreveld, Löffler, Staals, 2012]
Compressed start-stop diagram for stable criteria O(n log n) time
[Alewijnse, Buchin, Buchin, Sijben, Westenberg, 2014]

53. References References
S. Alewijnse, T. Bagautdinov, M. de Berg, Q. Bouts, A. ten Brink, K. Buchin and M. Westenberg. Progressive Geometric Algorithms. SoCG, 2014.
M. Buchin, A. Driemel, M. J. van Kreveld and V. Sacristan. Segmenting trajectories: A framework and algorithms using spatiotemporal criteria. Journal of Spatial Information Science, 2011.
B. Aronov, A. Driemel, M. J. Kreveld, M. Loffler and F. Staals. Segmentation of Trajectories for Non-Monotone Criteria. SODA, 2013.
M. Buchin, H. Kruckenberg and A. Kölzsch. Segmenting Trajectories based on Movement States. SDH, 2012.