03/10/051 Geometric Facility Location Optimization Class #8, CG in action (applications)

03/10/051

Geometric Facility Location Optimization

Class #8, CG in action (applications)

03/10/052

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

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Ex3 – guarding a polygon

• Questions?

• Implementations: – Classes– Benchmarks (guarding tasks / results)

• GUI?

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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

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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

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Projects

• Others?

• Start ASAP

• Work together: maps, experiments.

• Presentations: class 10-11: when?

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Geometric Facility Location Optimization

CG applications example:LSRT project:

Locating large scale wireless network

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Geometric Facility Location Optimization:• Computational Geometry• Facility Location• Optimization• Application

Definition & Motivation

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


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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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LSRT elements:

• Client:• Base Station:• Network:

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LSRT elements:


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LSRT elements:


– Microwave LOS– Satellite– Cable (not applicable

for LSRT)

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Goal: design an ‘optimal’ LSRT network

Problems of interest:• Locating Base Stations• Frequency Assignment• Connectivity

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Problems of interest:

• Locating Base Stations:– Guarding like.– Complex objective function.

• Frequency Assignment:• Connectivity:

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• Locating Base Stations:• Frequency Assignment:

– Conflict free frequency

• Connectivity:

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• Locating Base Stations:• Frequency Assignment:• Connectivity:

– Smallest set of Relay Stations.

– Back to the BS-locator.

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Main Obstacles:

• Huge inputs simplify & approximation

• Formalizing objective function

• NP hardness efficient Heuristics

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Simplifying & Approximating

• Visibility Preserving Terrain Simplification: VPTS

• Visibility Approximating: Radar

• Radio Maps

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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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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

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Defining the ridge network

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Approximating the ridge network

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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”).

The main TS algorithm

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Quality of Simplification

0.88

0.9

0.92

0.94

0.96

0.98

1

1000500250125

Compression

Qu

ali

ty

GcTin

QSlim

Terra

VPTS

Results

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Conclusion

• TS Application.

• Practical Knowledge: Terrain / Grid.

• Accelerating runtime:

7% compress 99.5%

1% compress 98%

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Farther Research

VPTS using FFT:

• dip tools

• hardware

• ‘fits’ terrains

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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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Radar-like: Pizza slice

left & right cross-sections pizza slice.

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Radar-like: Pizza slice

Lets look at a specific pizza slice:

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Radar-like generic algorithm

1. Given Terrain (T), view point (vp), and fixed angle (a=A):

2. while(int i=0;i<360) {

3. S1=cross-section(i);

4. S2=cross-section(i+a);

5. if(close enough(S1, S2)) {

6. extrapolate(S1, S2);

7. a = A; i = min(360, i + a);}

8. else a = a/2; }

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Radar-like: Threshold

Radar-like: 10 deg, low threshold | Radar-like: 10 deg, hi threshold

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Error Measure

exact radar approx xor

Error value: xor-area / circle-area

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Radar vs’ Naïve sampling

Naïve sampling Radar visibility

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Using the Algorithm• Generalizing the visibility:

– Antenna visibility: Locating: MW network.– RF: computing approximated radio maps.

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Approximating Radio-Maps [ABE]

Generalizing radar-visibility to RF propagation model:– Discrete visibility (boolean) continues– Visibility a long a ray RF sampling

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Approximating Radio-Maps General Frame work:

Sampling Set (SP) Extrapolation DS

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Approximating Radio-Maps

100*100 km elevation-map (of southern Israel) the brighter the higher.

Antenna, 30 km radius.

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Approximating Radio-Maps • Compute two consecutive cross-

sections.

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Approximating Radio-Maps • Compute a sample set along the each

cross-section: using 2D terrain simplification methods.

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Approximating Radio-Maps • Compute the signal strength along the

sample set – using pipe-line method.

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• Compute the distance between the two signal-sections:

• average / max / RMS distance

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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)

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Radio-Maps: resultsMethods:

• Random, Grid, TS

• F-Radar: fixed angle

• S-Radar: sensitive angle

• A-Radar: advance sampling

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Grid Random TS F-Radar S-Radar

• 5000 samples per radio-map

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Grid S-Radar• 5000 samples per radio-map

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Radio-Maps: resultsRun time for the same size sampling.

• The radar is 3-15 times faster than the regular sampling Radio Map methods.

• More accurate.

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Radio-Maps: results

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Radio-Maps: results

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Future Research• Advance interpolation methods

• Interferences

• Urban, sort-range

http://www.cs.bgu.ac.il/~benmoshe/RadioMaps/

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Expert System

Main Goal: theory practice

• Maps & Geometry & RF model

• Locating Base Stations.

• Frequency Allocation.

• Connecting Base Stations.

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Expert System - basics

• Maps

• RF Model (Antenna Patterns)

• Parameters

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Expert System - Input

Input:

• Base Stations• Clients• GIS info • Objective functions

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Expert System - Algorithms

Locating Base Stations:

• Generalization of Set-Cover

• Several Heuristics • Collection of solutions

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Frequency Assignment

Generalization of Conflict Free Coloring

• RF interferences• Several Heuristics • Collection of solutions

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Frequency Assignment

Generalization of Conflict Free Coloring

• RF interferences• Several Heuristics • Collection of solutions

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Connectivity

• Minimizing the number of Relay Station

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Connectivity


• Single Relay per link: 7 connected components

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Connectivity


• Three Relays per link: a single connected component

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Low Cost Connectivity [BBBDS]

• NP-Hard (MAX SNP) - Reduced from k-set-cover

• Heuristics often reach quasi-optimal solutions.

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Low Cost Connectivity

• BS: set of Base Stations

• RS: set of possible location for Relay Stations.

• VG: Visibility Graph (implies by BS & RS)

• Parameters: Height, range.

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Methods (basic):• MST (upper bound)• MST with RS reuse• MST batch• Greedy

Solution 42 RS

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Solution 39 RS

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• When greedy fails?

Solution 37 RS

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Methods (advance):• When greedy fails?• Some how we would

like to choose RS2

• A method to check re-usability of RS

Greedy: 6

Optimum: 5

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Methods (advance):• Check re-usability• Locate a relay at the

best (RS6) location.

• Parameters.• Random / Branch &

Bound search. Solution 4-6 RS

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Methods:• MST: T-MST, S-MST, GI-MST, BR-MST • Advance: GR-MST, B-RSG, H-RSG

Experiment: • 17 High resolution maps.• Ranges: [5000-20000], Height[10-50]• BS:[10-100] , RS:[500-10000]

03/10/0570


Documents

03/10/051 Geometric Facility Location Optimization Class #8, CG in action (applications)