1. Geometric Facility Location Optimization
Boaz Ben-Moshe (Ben-Gurion U.)
11/13/2018

2. Talk outline Geometric Facility Location Optimization
Definition, Motivation
Real life problems: LSRT (consortium)
Theoretical aspects
Expert System: prototype application for LSRT optimal layout design
Open Questions
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3. Definition & Motivation
Geometric Facility Location Optimization (GFLO):
Computational Geometry
Facility Location
Optimization
Application
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4. Definition & Motivation
Real life examples – GFLO problems:
traffic-lights
Air-ports
Shipping: cargo, delivery, etc.
Wireless network
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5. Definition & Motivation
Wireless networks:
LSRT: Large Scale Rural Telephone
telephone & internet service (VoIP).
Input
Clients: schools, pay-phones, etc.
Base station possible location
Parameters, objective function
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6. Definition & Motivation
LSRT elements:
Client:
Base Station:
Network:
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7. Definition & Motivation
LSRT elements:
Client:
Base Station:
Network:
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8. Definition & Motivation
LSRT elements:
Client:
Base Station:
Network:
Microwave  LOS
Satellite
Cable (not applicable for LSRT)
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9. Definition & Motivation
Goal: design an ‘optimal’ LSRT network
Problems of interest:
Locating Base Stations
Frequency Assignment
Connectivity
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10. Definition & Motivation
Problems of interest:
Locating Base Stations:
Guarding like.
Complex objective function.
Frequency Assignment:
Connectivity:
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11. Definition & Motivation
Problems of interest:
Locating Base Stations:
Frequency Assignment:
Conflict free frequency
Connectivity:
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12. Definition & Motivation
Problems of interest:
Locating Base Stations:
Frequency Assignment:
Connectivity:
Smallest set of Relay Stations.
Back to the BS-locator.
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13. Main Obstacles: Huge inputs  simplify & approximation
Formalizing  objective function
NP hardness  efficient Heuristics
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14. Simplifying & Approximating
Visibility Preserving Terrain Simplification: VPTS
Visibility Approximating: Radar
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15. VPTS [BKMN]
Develop a visibility preserving terrain simplification method - VPTS
Should preserve most of the visibility
Should be efficient
Define a visibility-based measure of quality of simplification.
Experiment with VPTS, as well as with other TS methods, using the new quality measure.
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16. Visibility-Preserving TS - Overview
Typically, the view from p is blocked by ridges
Main stages:
Compute the ridge network (a collection of chains of edges of T).
Approximate the ridge network. The ridge network induces a subdivision of the terrain into patches.
Simplify each patch (independently), using one of the standard TS methods.
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17. Defining the ridge network
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18. Approximating the ridge network
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19. Approximating the ridge network
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20. Approximating the ridge network
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21. The main TS algorithm
The (simplified) Ridge Network induces a subdivision of the terrain into regions:
For each region (map[i]) in the subdivision
If map[i] is “big” then recursively apply VPTS to map[i].
Else (map[i] is “small”) simplify map[i] using a “standard” simplification method (such as Garland’s “Terra”).
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22. Tests
Note: every test was repeated 4 times, for each of the 4*20*4 = 320 compressed terrains.
Thus in total about 320*4*4 = 5120 quality of simplification evaluations were done.
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23. Results
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24. Conclusion TS Application. Practical Knowledge: Terrain / Grid.
Accelerating runtime:
7% compress  99.5%
1% compress  98%
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25. Approximating Visibility [BCK]
Given a terrain T and a view point p compute the set of points on the surface of T that are visible from p.
Alternatively: Paint T with two colors (red & blue) s.t. any blue (red) point is visible (invisible) from p.
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26. Project Goals
Develop a new (radar-like) algorithm for computing an approximation of the visible region from a given point.
Implement alternative algorithms.
Experiment: define an Error Measure. Compare our algorithm with the others
Generalize to RF signal strength approximation.
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27. Radar-like generic algorithm
Given Terrain (T), view point (vp), and fixed angle (a=A):
while(int i=0;i<360) {
S1=cross-section(i);
S2=cross-section(i+a);
if(close enough(S1, S2)) {
extrapolate(S1, S2);
a = A; i = min(360, i + a);}
else a = a/2; }
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28. Radar-like: Threshold
Radar-like: 10 deg, low threshold | Radar-like: 10 deg, hi threshold
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29. Radar-like: Pizza slice
Lets look at a specific pizza slice:
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30. Radar-like: Pizza slice
left & right cross-sections  pizza slice.
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31. Error Measure exact radar approx xor
Error value: xor-area / circle-area
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32. Using the Algorithm Generalizing the visibility:
RF: computing approximated radio maps.
Antenna visibility: Locating: MW network.
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33. Theoretical aspects Guarding: 3D terrain  NP Hard (SNP-hard)
Monotone Polygons  ??
Connecting
MST  NP Hard (SNP-hard)
Double Ring
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34. Guarding
Given a set C of clients and a set G of guards (each can guard a subset of C), find the smallest set of guards that together can guard all C.
Models:
C & G can be either finite or an infinite sets.
Guarding is often associated with visibility.
Examples:
The Art Gallery problem.
Guarding a 3D terrain.
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35. Guarding Monotone Chains [BKM]
Monotone Chain (1.5D terrains)
Several versions
unknown bounds: NP-hard?
Constant approximation ?!
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36. Guarding a Monotone Chain
Preliminaries:
convex  concave
basic claims:
partial guarding order
order claim
Independent claim
finite / infinite
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37. Guarding a Monotone Chain
Preliminaries:
convex  concave
basic claims:
finite (discrete):
visibility graph
Infinite ??
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38. Guarding a Monotone Chain
Preliminaries:
convex  concave
Basic claims
finite / infinite
Goal:
Constant factor
Finite (discrete) version
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39. Guarding a Monotone Chain
Frame work:
Division: ‘no-local-guards’
Base Cases
Generalization
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40. Guarding a Monotone Chain
Frame work:
Division: ‘no-local-guards’:
For each vertex v: L(v), R(v) – range(v)
Implies a division (of sub chains) to disjoint ranges (lower bound).
Solve each sub chain independently.
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41. Guarding a Monotone Chain
Frame work:
Division: ‘no-local-guards’
Base Cases:
Sub chain T - does not requires a ‘local guard’.
Case 0: guards must be located on the left (right) of T.
Case 1: guards must be located on T or the left (right).
Case 2: guards can be located anywhere.
Generalization
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42. Guarding a Monotone Chain
Base Cases:
Case 0 (left) - optimal
Case 1: on & left
Case 2: anywhere
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43. Guarding a Monotone Chain
Case 2:
Task: guard A [a,b]
Charging scheme: division or case 1
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44. Guarding a Monotone Chain
Case 1(a):
Task: guard A [a,b]
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45. Guarding a Monotone Chain
Generalization:
Finite  infinite O(n^2)  O(n^4).
Constant factor ?
Hardness (e-approximation) ??
Monotone Polygons ??
Visibility Graphs:
compact representation.
Output sensitivity.
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46. Expert System Main Goal: theory  practice Maps & Geometry & RF model
Input manipulations.
Locating Base Stations.
Connecting Base Stations.
VSAT
Import Export: Opnet (network simulator).
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47. Expert System - basics Maps RF Model (Antenna Patterns) Parameters
Import Export
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48. Expert System - Input Input (Excel): Base Stations Clients GIS info
Objective functions
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49. Expert System - Algorithms
Locating Base Stations:
Several Heuristics
Collection of solutions
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50. Expert System - Algorithms
Frequency Assignment
RF interferences
Several Heuristics
Collection of solutions
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51. Expert System - Algorithms
Frequency Assignment
RF interferences
Several Heuristics
Collection of solutions
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52. Expert System - Algorithms
Connectivity
Minimizing the number of Relay Station
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53. Expert System - Algorithms
Connectivity
Minimizing the number of Relay Station
Single Relay per link: 7 connected components
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54. Expert System - Algorithms
Connectivity
Minimizing the number of Relay Station
Three Relays per link: a single connected component
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55. Low Cost Connectivity [BBBDS]
NP-Hard (MAX SNP) - Reduced from k-set-cover
Heuristics often reach quasi-optimal solutions.
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56. Practical conclusions
Maps formats, simplifications.
Cost bottleneck:
connectivity
Frequency reuse
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57. Open Problems: Facility Location: TS using JPEG (hardware).
Approximating Radio maps.
Connectivity (NLOS diffraction).
Urban, in-door.
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58. Fin http://www.cs.bgu.ac.il/~benmoshe/ Thanks: Matya, Joe, Irena
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