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1. Computational Movement AnalysisLecture 5: Segmentation, Popular Places and Regular PatternsJoachim Gudmundsson
2. Problem Input: x1, … , xn: moving entities in the plane k: a positive integer r: a positive real value An axis aligned square σ of side length r is a popular place if at least k entities visit it. σ σ is a popular place for k  5
3. σ Problem: Two models Continuous model: σ is a popular place if it is intersected by polylines from at least k different entities. Discrete model: σ is a popular place in the discrete model if it contains input points from at least k different entities. T1 T2
4. Results Continuous model: maximum number of visitors O(2n2) (2n2) Discrete model: maximum number of visitors O(n log n) (n log n)
5. What should we output? Detect if there exists a popular place? Report the popular place with the maximum number of visitors? Report “all” popular places? …
6. Continuous model – report all
7. Continuous model – report all Problem: Given a set of polygons, with holes, find a point that stabs a maximum number of polygons.
8. Continuous model – report all Idea: Use a sweep-line algorithm
9. Continuous model – report all Complexity: O(2)
10. Continuous model – report all Straight forward approach (almost): Running time:O(t2n2 log tn) Space: O(t2n2) Apply topological sweep by Edelsbrunner and Guibas: Running time:O(t2n2) Space: O(t2n)
11. Continuous model – report all Find the maximum “colour depth” using topological sweep by Edelsbrunner & Guibas.
12. Continuous model – report all Improvement (incl. topological sweep [Edelbsbrunner & Guibas’89]) Running time:O(t2n2) Space: O(tn) Non-trivial to implement due to the degenerate cases. The problem has an (2n2) lower bound.
13. Continuous model – report all n=32 t=1000 Time: 5 seconds Compression + finding all popular places
14. Results Discrete model: maximum number of visitors O(n log n) (n log n) Continuous model: maximum number of visitors O(2n2) (2n2)
15. 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”
16. Example Trajectory T sampled with equal time intervals.
17. Example Trajectory T sampled with equal time intervals. Top right:speed cannot differ more than a factor 2
18. Example Trajectory T sampled with equal time intervals. Bottom left:direction of motion differs by at most 90◦
19. Example Trajectory T sampled with equal time intervals. Bottom right:both criteria are used in conjunction (speed and direction)
20. Problem Observation: Many trajectories span over several activities. Goal: Segment a trajectory into subtrajectories according to its behaviour. We will only consider segmenting trajectories at vertices.
21. Criteria-Based Segmentation Goal: Partition trajectory into a small number of segments such that a given criterion is fulfilled on each segment each segment are uniform e.g. heading angular range
22. Criteria-Based Segmentation Goal:Partition trajectory into a small number of segments such that a given criterion is fulfilled on each segment Criteria: heading, speed, location, curvature, sinuosity … and combinations of these e.g., describe movement characteristics
23. Greedy Algorithm 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.
24. Greedy Algorithm Observation: If criteria are decreasing monotone, a greedy strategy works.
25. 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…
26. Greedy Algorithm Observation: For some criteria, iterative double & search is faster. Double & search: An exponential search followed by a binary search.
27. Criteria-Based Segmentation Boolean or linear combination of decreasing monotone criteria Greedy Algorithm incremental in O(n) time or constant-update criteriae.g. bounds on speed or heading double & search in O(n log n) time for non-constant update criteria e.g. staying within some radius [Buchin et al.’11]
28. 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 2ANDMax speed range 4 5 speed 1
29. Case Study: Geese Migration Goal: Delineate stopover sites of migratory geese Two behavioural types stopover migration flight Input: GPS tracks expert description of behaviour
30. 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
31. Comparison manual computed
32. Evaluation Few local differences: Shorter stops, extra cuts in computed segmentation
33. Criteria A combination of decreasing and increasing monotone criteria stopover migration flight Within radius 30km At least 48h Change in heading <120° AND OR
34. Non-Monotone Segmentation Many Criteria are not (decreasing) monotone: Minimum time Standard deviation Fixed percentage of outliers Example: Geese Migration For these Aronov et al. introduced the start-stop diagram
35. Start-Stop Diagram Algorithmic approach: input trajectory compute start-stop diagram compute segmentation
36. 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 spaceif 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) staircasein D forbidden space free space staircase
37. Start-Stop Diagram A (minimal) segmentation of T corresponds to a (min-link) staircasein D. 12 11 10 9 8 7 6 5 4 3 2 1 7 6 8 5 9 10 11 4 12 3 2 1 2 3 4 5 6 7 8 9 10 11 12 123456789101112 1
38. Start-Stop Diagram A (minimal) segmentation of T corresponds to a (min-link) staircasein D. 12 11 10 9 8 7 6 5 4 3 2 1 7 6 8 5 9 10 11 4 12 3 2 1 2 3 4 5 6 7 8 9 10 11 12 123456789101112 1
39. 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 8 5 11 12 1 6 9 10 7 1 2 3 4 5 6 7 8 9 10 11 12 123456789101112
40. 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 8 5 11 12 1 6 9 10 7 1 2 3 4 5 6 7 8 9 10 11 12 123456789101112 [Aronov et al.13]
41. Stable Criteria Definition: A criterion is stableif and only if where = number of changes of validity on segments [0,i], [1,i],…, [i-1,i]
42. Stable Criteria Definition: A criterion is stableif and only if where = 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.
43. 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 22
44. Computing the Compressed Start-Stop Diagram For a decreasing criterion consider the algorithm: ComputeLongestValid(critC, trajT) Algorithm: Move two pointersi,j from n to 0 over the trajectory. For every trajectory index j the smallest index i for which [I,j] satisfies the criterion C is stored. 9 i j 22
45. 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 i j 9 Requires a data structure for segment [i,j] 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. 22
46. 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 22
47. Attributes and criteria Examples of stable criteria Lower bound/Upper bound on attribute Angular range criterion Disk criterion Allow a fraction of outliers … 22
48. 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 S 0 22
49. Computing the Optimal Segmentation More efficient dynamic programming algorithm for compressed diagrams Process blocks of white cellsusing 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(nlog n)time T [Alewijnse et al.’14] 22
50. Summary Greedy algorithm for decreasing monotone criteria O(n) or O(n log n) time [Buchin, Driemel, van Kreveld, Sacristan, 2010] Case Study: Geese Migration [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]
51. Regular visits Given a query region, does an object (animal, vehicle, …) regularly visit the region? Regular behaviour? Every day? Every month? Annual migration? Regular? 75% of all Saturdays I play football. Regular?
52. Motivation Time start end outside inside
53. Motivation Time start end Length = 5 offset period length
54. Motivation Time start end Length = 9 Visit 78% of the times offset period length
55. Motivation Time start end period offset period length
56. Simplest case Given period offset and period length. period length Time 0 1 1 0 1 1 1 0 1 1 0 period offset period offset Problem: Given a bitstring S of length n and a constant 0<c<1, find the longest subsequence S’ of S such that there are at least c|S’| 1s in S’.
57. Longest Dense Substring problem S = 100011100110 i f(i) =  si Maximize b - a s.t. f(b) - f(a)  c(b-a) g(i)= f(i) – ci Maximize b - a s.t. g(b) - g(a)  0 j=1 1 6 g(i) f(i) 0 3 -1 0 6 9 3 12 6 9 3 12
58. Longest Dense Substring problem S = 100011100110 Maximize b - a s.t. g(b)  g(a) g(b) must lie on the URE g(a) must lie on the LLE g(i)= f(i) – ci Maximize b - a s.t. g(b) - g(a)  0 1 URE 0 1 g(i) 0 -1 LLE -1 0 6 9 3 12 0 6 9 3 12
59. Longest Dense Substring problem Longest Dense Substring can be computed in O(n) time Compressed case: 11100010100000011100000  3,3,1,1,1,6,3,5 size n  size k Compressed LDS can be computed in O(k) time
60. Given period length Given period length, but no period offset Time
61. Given period length Time
62. Given period length Time Find the longest vertical segment A such that at least a fraction c of the horizontal segments it intersects are “inside”. A offset
63. Given period length Theorem: Given a set of intervals I, a constant 0<c1 and a period length p the longest c-dense repetitive pattern over all offsets can be computed in O(n3/2log2 n) time using O(n log n) space. Time
64. General case What if neither period offset nor period length is known? (no two consecutive visits in the “same region”) Observation: The longest pattern can be translated and scaled such that one of its “visits” coincides with the start point of an interval and one “visit” coincides the start/end point of an interval.
65. General case Theorem: There are O(n2) pairs of distances. Each distance may generate n/c period lengths, thus O(n3/c) possible period lengths in total. This also fixes the offset! The longest c-dense repetitive pattern for each period length can be found in O(n) time, thus O(n4/c) time in total. Can be improved to O(n7/2log3 n). [Djordjevic et al.’10]
66. Summary Theorem: Offset & period length O(n) Period length O(n3/2log2 n) Period length O(n log n) - approximate Nothing O(n7/2log3 n)
67. Open problems Approximate version of the general version? Improve running times? Lower bounds?
68. Problem Input:A Euclidean graph G in the plane PreprocessG into a data structure such that given a polygonal query curve Q and a positive constant  report all subcurves  Gfor which (,Q) ≤  G  Q ( does not need to start and end at a vertex of G)
69. Previous results Assume  is given as part of the input Input: polygonal path P of size n and a constant >0 Query: segment Q (length >6) Q P de Berg et al.’11: (Counting) 6-distance approximation Preprocessing O(n2 + s polylog n) Space O(s polylog n) Query time O(n/s polylog n), for n<s<n2
70. Our results A geometric graph G is c-packed if the total length of all the edges of G inside any ball is at most c times the radius of the ball. Input: O(1)-packed tree T of size n and a constant >0 Query: polygonal path Q of size m (each segment >2) T Gudmundsson and Smid: (3+)-distance approximation Preprocessing O(n polylog n) Space O(n polylog n) Query time O(m polylog n)
71. Our results A geometric graph G is c-packed if the total length of all the edges of G inside any ball is at most c times the radius of the ball. Input: O(1)-packed tree T of size n and a constant >0 Query: polygonal path Q of size m (each segment >2) T Q Gudmundsson and Smid: (3+)-distance approximation Preprocessing O(n polylog n) Space O(n polylog n) Query time O(m polylog n)
72. Our results Pros: Extends to trees (instead of paths) Query path instead of query segment Approximation improved from 6 to (1+) for query segments (3+) for query paths Improved preprocessing time Cons: Input must be O(1)-packed If G is a tree the edges must be “long” Q G
73. References M. Benkert, B. Djordjevic, J. Gudmundsson and T. Wolle. Finding popular places. IJCGA, 2010. 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. B. Djordjevic, J. Gudmundsson, A. Pham and T. Wolle. Detecting Regular Visit Patterns. Algorithmica, 2011.
74. References B. A. Burton. Searching a bitstream in linear time for the longest substring of any given density, Algorithmica, 2011. B. Djordjevic and J. Gudmundsson. Detecting areas visited regularly. COCOON 2010. M. de Berg, A. F. Cook IV and J. Gudmundsson. Fast Frechet Queries. CGTA, 2013. J. Gudmundsson and M. Smid. Frechetqueries in geometric trees. ESA, 2013.