Using the A Algorithm to Find Optimal Sequences for Area Aggregation
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1 Using the A Algorithm to Find Optimal Sequences for Area Aggregation Dongliang Peng 1, Alexander Wolff 1, Jan-Henrik Haunert 2 1 Chair of Computer Science I, University of Würzburg, Germany 2 Institute of Geodesy and Geoinformation, University of Bonn, Germany
2 Map Generalization is about selecting and representing information on a map.
3 Map Generalization is about selecting and representing information on a map. Typical generalization operators (ESRI 1996): Elimination Simplification Aggregation Collapse Typification Exaggeration Classifi. and symboli. Displacement Refinement
4 Map Generalization is about selecting and representing information on a map. Typical generalization operators (ESRI 1996): Elimination Simplification Aggregation Collapse Typification Exaggeration Classifi. and symboli. Displacement Refinement
5 Research Problem
6 Research Problem
7 Research Problem
8 Research Problem
9 Research Problem
10 Research Problem
11 Research Problem
12 Research Problem
13 Research Problem Given aggregation costs: What is an optimal sequence?
14 Standard Approach for Dynamic Maps Users zoom in and out to read digital maps
15 Standard Approach for Dynamic Maps Users zoom in and out to read digital maps Multi-Representation Databases (MRDB) support Levels of Detail (LODs) (Hampe et al. 2004)
16 Standard Approach for Dynamic Maps Users zoom in and out to read digital maps Multi-Representation Databases (MRDB) support Levels of Detail (LODs) (Hampe et al. 2004) zoom out (Google Maps)
17 Standard Approach for Dynamic Maps Users zoom in and out to read digital maps Multi-Representation Databases (MRDB) support Levels of Detail (LODs) (Hampe et al. 2004) Expect continuous change! zoom out (Google Maps)
18 Outline Introduction Methodology Case Study Concluding Remarks
19 Preliminaries
20 Preliminaries Patch p i : a set of land-cover areas whose union is connected p 1 p 2 p 1 p 5 p 4 p 3 p 5 p 6 p 7 p 6
21 Preliminaries Patch p i : a set of land-cover areas whose union is connected Region R i : an area on the goal map R 1 R 2
22 Preliminaries Patch p i : a set of land-cover areas whose union is connected Region R i : an area on the goal map Aggregate the smallest patch with one of its neighbours at each step R 1 R 2
23 Integrate Aggregation Sequences according to the order of smallest areas input output input sequence of the pink region input output input sequence of the green region
24 Integrate Aggregation Sequences according to the order of smallest areas input output input sequence of the pink region input output input sequence of the green region input output output output integrated sequence of the two regions input
25 Integrate Aggregation Sequences according to the order of smallest areas input output input sequence of the pink region input output input sequence of the green region input output output output integrated sequence of the two regions input
26 Subdivision Subdivision P t,i : a set of patches subdividing a single region R P start = P 1,1 P 2,1 P 2,2 P 2,3 P 2,4 P 3,1 P 3,2 P 3,3 P 3,4 P 3,5 P goal = P 4,1
27 Number of Subdivisions layer 1 layer 2k + 1 n = 2k patch
28 Number of Subdivisions layer 2 layer 1 layer 2k + 1 n = 2k patch 2k patches
29 Number of Subdivisions layer 2 layer k + 1 layer 1 layer 2k + 1 n = 2k patch 2k patches k + 1 patches
30 Number of Subdivisions layer 2 layer k + 1 layer 1 layer 2k + 1 n = 2k patch k + 1 patches At ) layer k + 1, remove k of the 2k intermediate sticks: 2 k = 2 (n 1)/2 subdivisions ( 2k k 2k patches
31 Number of Subdivisions layer 2 layer k + 1 layer 1 layer 2k + 1 n = 2k patch k + 1 patches At ) layer k + 1, remove k of the 2k intermediate sticks: 2 k = 2 (n 1)/2 subdivisions ( 2k k 2k patches The number of subdivions is at least exponential in the number n of polygons in layer k + 1.
32 A Algorithm A best-first search algorithm to find a path from a source node s to a target node t s t
33 A Algorithm A best-first search algorithm to find a path from a source node s to a target node t Cost function: F (u) = g(u) + h(u) g(u): exact cost of s-u path h(u): estimated cost of shortest u-t path s g(u) u h(u) t
34 A Algorithm A best-first search algorithm to find a path from a source node s to a target node t Cost function: F (u) = g(u) + h(u) g(u): exact cost of s-u path h(u): estimated cost of shortest u-t path s g(u) u h(u) t Guarantees a shortest path if h(u) never overestimates.
35 Shortest-Path Algorithms Find shortest path from pink cell to blue cell (Amit s A Pages) Dijkstra s algorithm:
36 Shortest-Path Algorithms Find shortest path from pink cell to blue cell (Amit s A Pages) Dijkstra s algorithm: A algorithm:
37 A Algorithm source target
38 A Algorithm Estimation: min #steps to reach target source target
39 A Algorithm Estimation: min #steps to reach target Step no. Cost 1 3 source 0+3= target
40 A Algorithm Step no. Cost 1 0+3= 3 source Estimation: min #steps to reach target = target
41 A Algorithm Step no. Cost 1 0+3= 3 source Estimation: min #steps to reach target = 1+2= target
42 A Algorithm Step no. Cost 1 0+3= 3 source Estimation: min #steps to reach target = 1+2= 3+2= target
43 A Algorithm Step no. Cost 1 0+3= 3 source Estimation: min #steps to reach target = 1+2= 3+2= = target
44 A Algorithm Step no. Cost 1 0+3= 3 source Estimation: min #steps to reach target = 1+2= 3+2= = 6 3+1= target
45 A Algorithm Step no. Cost 1 0+3= 3 source Estimation: min #steps to reach target = 1+2= 3+2= = = = 4 target
46 A Algorithm Step no. Cost 1 0+3= 3 source Estimation: min #steps to reach target = 1+2= 3+2= = = = 4 target
47 Formalizing a Pathfinding
48 Formalizing a Pathfinding Each subdivision is represented as a node
49 Formalizing a Pathfinding Each subdivision is represented as a node Find a shortest path with respect to cost functions
50 Cost Function Type change f type (P s 1,i, P s,j ) = A p A R Cost(T (p),t (q)) T max
51 Cost Function Type change f type (P s 1,i, P s,j ) = A p A R Cost(T (p),t (q)) T max
52 Cost Function Type change f type (P s 1,i, P s,j ) = A p A R Cost(T (p),t (q)) T max Shape (non-compactness) f shape (P s,k ) = 1 min p P s,k c p n 2
53 Cost Function Type change f type (P s 1,i, P s,j ) = A p A R Cost(T (p),t (q)) T max Shape (non-compactness) f shape (P s,k ) = 1 min p P s,k c p n 2
54 Exact Cost and Estimated Cost s g(u) u h(u) t
55 Exact Cost and Estimated Cost Exact cost g(p t,i ) = (1 λ) t j=2 f type(p j 1,ij 1, P j,ij )+λ t 1 j=2 f shape(p j,ij ) s g(u) u h(u) t
56 Exact Cost and Estimated Cost Exact cost g(p t,i ) = (1 λ) t j=2 f type(p j 1,ij 1, P j,ij )+λ t 1 j=2 f shape(p j,ij ) Estimated cost h(p t,i ) = (1 λ)h type (P t,i ) + λh shape (P t,i ) s g(u) u h(u) t
57 Estimated Cost h type (P t,i ) = n 1 s=t f type(p s,is, P s+1,is+1 ). We assume: Each patch immediately gets the target type. t = 2 t = 3 t = 4 t = 5
58 Estimated Cost h type (P t,i ) = n 1 s=t f type(p s,is, P s+1,is+1 ). We assume: Each patch immediately gets the target type. t = 2 t = 3 t = 4 t = 5 h shape (P t,i ) = n 1 s=t C(P s,j s ), The more the edges remain, the smaller the estimated compactness C(P s,js ) is We assume: The boundary with the fewest edges is removed n int (P 2,j2 ) = 17 n int (P 3,j3 ) = 15 n int (P 4,j4 ) = 12 n int (P 5,j5 ) =
59 Overestimation Try finding a path by exploring M nodes. If we do not find one, then we overestimate and try exploring M nodes again. Overestimate by increasing k {0, 1,... }: F (P t,i ) = g(p t,i ) + 2 k h(p t,i )
60 Overestimation Try finding a path by exploring M nodes. If we do not find one, then we overestimate and try exploring M nodes again. Overestimate by increasing k {0, 1,... }: F (P t,i ) = g(p t,i ) + 2 k h(p t,i ) With a larger k, a path seems more expensive (or longer), thus may not be explored
61 Overestimation Try finding a path by exploring M nodes. If we do not find one, then we overestimate and try exploring M nodes again. Overestimate by increasing k {0, 1,... }: F (P t,i ) = g(p t,i ) + 2 k h(p t,i ) With a larger k, a path seems more expensive (or longer), thus may not be explored Not optimal anymore if k > 0!
62 Outline Introduction Methodology Case Study Concluding Remarks
63 Case Study Environment C# (using the.net Framework 4.0) ArcObjects SDK 10.2 Windows 7, 3.3 GHz dual core CPU, 8 GB RAM Time measure: Stopwatch (a class in C#)
64 Case Study Environment C# (using the.net Framework 4.0) ArcObjects SDK 10.2 Windows 7, 3.3 GHz dual core CPU, 8 GB RAM Time measure: Stopwatch (a class in C#) Data Source: scale 1 : 50,000; 5,448 patches; place Buchholz in der Nordheide ; from ATKIS (Das Amtliche Topographisch-Kartographische Informationssystem), processed by Haunert (2009, pp ) Target: scale 1 : 250,000; 730 patches; generalized from the source by Haunert and Wolff (2010)
65 Data 5 km 5,448 patches, scale 1 : 50, patches, scale 1 : 250,000
66 Legend 2101: settlement area fg 2112: industrial area fg 2114: construction area fg 2201: sport facility fg 2202: leisure facility fg 2213: cemetery fg 2230: golf course fg 2301: mining, pit, quarry fg 3103: square fg 3302: airport, air strip fg 4101: farmland fg 4102: grassland fg 4103: garden fg 4104: heath fg 4105: swamp fg 4107: wood, forest fg 4108: bosk fg 4109: specialized crop fg 4111: wet ground fg 5112: lake, barrierlake, pond fg The 20 land-cover types appearing in our data
67 Type Distance object built up area 2000 settlement 3000 traffic 2101: settlement area 2352: fence 3101: street 3543: antenna natural spaces 4000 vegetation 5000 waters 4101: farmland 4203: hedge 5101: river, stream 5321: shore protection
68 Type Distance Cost(fence, street) = 4 object built up area natural spaces 2000 settlement 3000 traffic 4000 vegetation 5000 waters 2101: settlement area 2352: fence 3101: street 3543: antenna 4101: farmland 4203: hedge 5101: river, stream 5321: shore protection
69 Result λ = 0.5, M = 200,000
70 Result λ = 0.5, M = 200,000 Found optimal solutions for 692 regions out of 734
71 Result λ = 0.5, M = 200,000 Found optimal solutions for 692 regions out of 734 Explored arcs and nodes
72 Result λ = 0.5, M = 200,000 Found optimal solutions for 692 regions out of 734 Explored arcs and nodes Running time: 36 min
73 Result λ = 0.5, M = 200,000 Found optimal solutions for 692 regions out of 734 Explored arcs and nodes Running time: 36 min Largest overestimation factor: 512
74 Cost in detail ID n m K i R type R shape Time (s)
75 Cost in detail ID n m K i R type R shape Time (s) R type = g type (P goal )/h type (P start ) Good estimation for type change
76 Cost in detail ID n m K i R type R shape Time (s) R type = g type (P goal )/h type (P start ) Good estimation for type change R shape = g shape (P goal )/h shape (P start ) Poor estimation for shape (non-compactness)..
77 K 543 = 512 (20) (17) (13) (9) (5)
78 K 543 = 512 (20) (17) (13) (9) (5) K 436 = 8 (27) (22) (17) (12) (7)
79 K 543 = 512 (20) (17) (13) (9) (5) K 436 = 8 (27) (22) (17) (12) (7) K 77 = 1 (17) (14) (11) (8) (5)
80 Inappropriate Step
81 Inappropriate Step
82 Inappropriate Step
83 Inappropriate Step Prefer!
84 Animation Thick blue polyline represents a patch to be aggregated, and thin blue polyline represents an aggregated result.
85 Animation Thick blue polyline represents a patch to be aggregated, and thin blue polyline represents an aggregated result. In Adobe Acrobat Pro, set Advance every to 1 second, then watch animation with Full Screen Mode.
86 source: 92 patches target: 6 patches
87 source: 92 patches target: 6 patches
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171 source: 92 patches target: 6 patches
172 source: 92 patches target: 6 patches
173 source: 92 patches target: 6 patches
174 Outline Our Example Problem Methodology Case Study Concluding Remarks
175 Concluding Remarks Tried integer linear programming (ILP), A solved more instances to optimality than ILP
176 Concluding Remarks Tried integer linear programming (ILP), A solved more instances to optimality than ILP Look for better estimations: faster and find optimal solutions for more regions
177 Concluding Remarks Tried integer linear programming (ILP), A solved more instances to optimality than ILP Look for better estimations: faster and find optimal solutions for more regions Have not considered keeping important land-cover areas for a longer time
178 Concluding Remarks Tried integer linear programming (ILP), A solved more instances to optimality than ILP Look for better estimations: faster and find optimal solutions for more regions Have not considered keeping important land-cover areas for a longer time Have not considered aggregating two patches of different types into a patch of a higher-level type (e.g., river + lake waters)
179 Concluding Remarks Tried integer linear programming (ILP), A solved more instances to optimality than ILP Look for better estimations: faster and find optimal solutions for more regions Have not considered keeping important land-cover areas for a longer time Have not considered aggregating two patches of different types into a patch of a higher-level type (e.g., river + lake waters) Thank you!
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