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Geometric Facility Location Optimization

Geometric Facility Location Optimization. Class #8, CG in action (applications). Class 8 - agenda:. Projects: Who does what, when (presentation dates). Cross-projects cooperation CG in action: some applications Visibility, Connectivity. Simplification, approximation.

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Geometric Facility Location Optimization

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  1. Geometric Facility Location Optimization Class #8, CG in action (applications)

  2. Class 8 - agenda: • Projects: • Who does what, when (presentation dates). • Cross-projects cooperation • CG in action: some applications • Visibility, Connectivity. • Simplification, approximation. • Facility Location Optimization

  3. Ex3 – guarding a polygon • Questions? • Implementations: • Classes • Benchmarks (guarding tasks / results) • GUI?

  4. Projects • List: • Yossi – Guarding 2.5D Terrains • Itay – Watersheds, water flow simulating • Liad – Terrain Simplification (fft ...) • Yael, Ben – Unit disc cover problems • Yoav – Visibility Graph 1.5 Terrain • ?? – Vehicle Routing Problem

  5. Projects • List: • Ronit, Inbar, Anat - Tetrix original problem • Boris, Amir – Tetrix (scheduler  frame builder) • Elior – Tetrix (breaking the packets) • Yonni – Tetrix – Scheduler • Eldan, Ilan – Terrain guarding • Yossi - Terrain guarding

  6. Projects • Others? • Start ASAP • Work together: maps, experiments. • Presentations: class 10-11: when?

  7. Geometric Facility Location Optimization CG applications example: LSRT project: Locating large scale wireless network

  8. Definition & Motivation • Geometric Facility Location Optimization: • Computational Geometry • Facility Location • Optimization • Application

  9. Definition & Motivation • Real life examples – problems: • traffic-lights • Air-ports • Shipping: cargo, delivery, etc. • Wireless networks

  10. 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

  11. Definition & Motivation LSRT elements: • Client: • Base Station: • Network:

  12. Definition & Motivation LSRT elements: • Client: • Base Station: • Network:

  13. Definition & Motivation LSRT elements: • Client: • Base Station: • Network: • Microwave  LOS • Satellite • Cable (not applicable for LSRT)

  14. Definition & Motivation Goal: design an ‘optimal’ LSRT network Problems of interest: • Locating Base Stations • Frequency Assignment • Connectivity

  15. Definition & Motivation Problems of interest: • Locating Base Stations: • Guarding like. • Complex objective function. • Frequency Assignment: • Connectivity:

  16. Definition & Motivation Problems of interest: • Locating Base Stations: • Frequency Assignment: • Conflict free frequency • Connectivity:

  17. Definition & Motivation Problems of interest: • Locating Base Stations: • Frequency Assignment: • Connectivity: • Smallest set of Relay Stations. • Back to the BS-locator.

  18. Main Obstacles: • Huge inputs  simplify & approximation • Formalizing  objective function • NP hardness  efficient Heuristics

  19. Simplifying & Approximating • Visibility Preserving Terrain Simplification: VPTS • Visibility Approximating: Radar • Radio Maps

  20. 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.

  21. Visibility-Preserving TS - Overview 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. Typically, the view from p is blocked by ridges

  22. Defining the ridge network

  23. Approximating the ridge network

  24. Approximating the ridge network

  25. Approximating the ridge network

  26. 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”).

  27. Results

  28. Conclusion • TS Application. • Practical Knowledge: Terrain / Grid. • Accelerating runtime: 7% compress  99.5% 1% compress  98%

  29. Farther Research VPTS using FFT: • dip tools • hardware • ‘fits’ terrains

  30. 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.

  31. Radar-like: Pizza slice left & right cross-sections  pizza slice.

  32. Radar-like: Pizza slice Lets look at a specific pizza slice:

  33. 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; }

  34. Radar-like: Threshold Radar-like: 10 deg, low threshold | Radar-like: 10 deg, hi threshold

  35. Error Measure exact radar approx xor Error value: xor-area / circle-area

  36. Radar vs’ Naïve sampling Naïve sampling Radar visibility

  37. Using the Algorithm • Generalizing the visibility: • Antenna visibility: Locating: MW network. • RF: computing approximated radio maps.

  38. Approximating Radio-Maps [ABE] Generalizing radar-visibility to RF propagation model: • Discrete visibility (boolean) continues • Visibility a long a ray RF sampling

  39. Approximating Radio-Maps General Frame work: Sampling Set (SP) Extrapolation DS

  40. Approximating Radio-Maps 100*100 km elevation-map (of southern Israel) the brighter the higher. Antenna, 30 km radius.

  41. Approximating Radio-Maps • Compute two consecutive cross-sections.

  42. Approximating Radio-Maps • Compute a sample set along the each cross-section: using 2D terrain simplification methods.

  43. Approximating Radio-Maps • Compute the signal strength along the sample set – using pipe-line method.

  44. Approximating Radio-Maps • Compute the distance between the two signal-sections: • average / max / RMS distance

  45. Approximating Radio-Maps Putting it all together: • Sensitive Radar algorithm • Sensitive 2D Simplification • Robust distance norm Fine Tuning: • None grid sampling (2D) • Parameters (terrain independent)

  46. Radio-Maps: results Methods: • Random, Grid, TS • F-Radar: fixed angle • S-Radar: sensitive angle • A-Radar: advance sampling

  47. Approximating Radio-Maps Grid Random TS F-Radar S-Radar • 5000 samples per radio-map

  48. Approximating Radio-Maps Grid S-Radar • 5000 samples per radio-map

  49. Radio-Maps: results Run time for the same size sampling. • The radar is 3-15 times faster than the regular sampling Radio Map methods. • More accurate.

  50. Radio-Maps: results

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