Formulation of the . . . Towards Precise . . . Optimal Placement of . . . Towards Optimal Placement Estimating the Effect . . . Precise Formulation of . . . of Bio-Weapon Detectors Solution Towards More . . . Chris Kiekintveld 1 and Octavio Lerma 2 Fuzzy Techniques May . . . Home Page 1 Department of Computer Science Title Page 2 Computational Sciences Progtram University of Texas at El Paso ◭◭ ◮◮ El Paso, TX 79968, USA ◭ ◮ cdkiekintveld@utep.edu lolerma@episd.edu Page 1 of 14 Go Back Full Screen Close Quit
Formulation of the . . . 1. Formulation of the Practical Problem Towards Precise . . . Optimal Placement of . . . • Biological weapons are difficult and expensive to de- Estimating the Effect . . . tect. Precise Formulation of . . . • Within a limited budget, we can afford a limited num- Solution ber of bio-weapon detector stations. Towards More . . . Fuzzy Techniques May . . . • It is therefore important to find the optimal locations Home Page for such stations. Title Page • A natural idea is to place more detectors in the areas with more population. ◭◭ ◮◮ ◭ ◮ • However, such a commonsense analysis does not tell us how many detectors to place where. Page 2 of 14 • To decide on the exact detector placement, we must Go Back formulate the problem in precise terms. Full Screen Close Quit
Formulation of the . . . 2. Towards Precise Formulation of the Problem Towards Precise . . . Optimal Placement of . . . • The adversary’s objective is to kill as many people as Estimating the Effect . . . possible. Precise Formulation of . . . • Let ρ ( x ) be a population density in the vicinity of the Solution location x . Towards More . . . Fuzzy Techniques May . . . • Let N be the number of detectors that we can afford Home Page to place in the given territory. Title Page • Let d 0 be the distance at which a station can detect an outbreak of a disease. ◭◭ ◮◮ ◭ ◮ • Often, d 0 = 0 – we can only detect a disease when the sources of this disease reach the detecting station. Page 3 of 14 • We want to find ρ d ( x ) – the density of detector place- Go Back ment. Full Screen � • We know that ρ d ( x ) dx = N. Close Quit
Formulation of the . . . 3. Optimal Placement of Sensors Towards Precise . . . Optimal Placement of . . . • We want to place the sensors in an area in such a way Estimating the Effect . . . that Precise Formulation of . . . – the largest distance D to a sensor Solution – is as small as possible. Towards More . . . Fuzzy Techniques May . . . • It is known that the smallest such number is provided Home Page by an equilateral triangle grid: Title Page h ✛ ✲ ◭◭ ◮◮ r r r r ❆ ✁ ❆ ✁ ❆ ✁ ❆ ◭ ◮ ❆ ❆ ❑ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ❆ ❆ h ❆ ✁ ❆ ✁ ❆ ✁ ❆ ❆ ❆❆ Page 4 of 14 ❆ ✁ ❆ ✁ ❆ ✁ ❆ ❯ ❆ ❆ ✁ ❆ ✁ ❆ ✁ r r r ✁ ❆ ✁ ❆ ✁ ❆ Go Back ✁ ❆ ✁ ❆ ✁ ❆ ✁ ✁ ❆ ✁ ❆ ✁ ❆ ✁ Full Screen ✁ ❆ ✁ ❆ ✁ ❆ ✁ ✁ ❆ ✁ ❆ ✁ ❆ ✁ r r r r Close Quit
Formulation of the . . . For the equilateral triangle placement, points which are Towards Precise . . . closest to a given detector forms a hexagonal area: Optimal Placement of . . . h ✛ ✲ Estimating the Effect . . . r r r r ❆ ✁ ❆ ✁ ❆ ✁ ❆ Precise Formulation of . . . ❑ ❆ ❆ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ❆ ✟ ❍ ✟ ❍ ❆ h ❆ ✁ ✟ ❆ ✁ ❍ ❆ ✁ ❆ ✟ ❍ ❆ Solution ❆ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ❆ ❯ ❆ ❆ ✁ ❆ ✁ ❆ ✁ Towards More . . . r r r ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ Fuzzy Techniques May . . . ❍ ✟ ❍ ✟ ✁ ❆ ✁ ❍ ✟ ❆ ✁ ❆ ✁ ✟ ❍ Home Page ✁ ❆ ✁ ❆ ✁ ❆ ✁ ✁ ❆ ✁ ❆ ✁ ❆ ✁ r r r r Title Page This hexagonal area consists of 6 equilateral triangles: ◭◭ ◮◮ h ✛ ✲ ◭ ◮ r r r r ❆ ✁ ❆ ✁ ❆ ✁ ❆ ❆ ❆ ❑ ❆ ✁ ❆ ✁ ❆ ✁ ❆ Page 5 of 14 ✟ ❆ ❍ ✟ ❍ ❆ h ❆ ✁ ✟ ❆ ✁ ❍ ❆ ✁ ❆ ✟ ❍ ❆ ❆❆ ❩ ✚ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ❩ ✚✚ ❯ ❆ Go Back ❆ ✁ ❩ ❆ ✁ ❆ ✁ ✚ ❩❩ r r r ✁ ❆ ✁ ❆ ✁ ❆ ✚ ✚ ❩ ✁ ❆ ✁ ❆ ✁ ❆ ✁ Full Screen ✟ ❍ ❍ ✟ ✁ ❆ ✁ ❍ ✟ ❆ ✁ ❆ ✁ ❍ ✟ ✁ ❆ ✁ ❆ ✁ ❆ ✁ Close ✁ ❆ ✁ ❆ ✁ ❆ ✁ r r r r Quit
Formulation of the . . . 4. Optimal Placement of Sensors (cont-d) Towards Precise . . . Optimal Placement of . . . • In each △ , the height h/ 2 is related to the side s by √ √ Estimating the Effect . . . the formula h 3 3 2 = s · cos(60 ◦ ) = s · 2 , hence s = h · 3 . Precise Formulation of . . . Solution • Thus, the area A t of each triangle is equal to Towards More . . . √ √ A t = 1 2 · s · h 2 = 1 3 · 1 3 3 2 · h 2 = Fuzzy Techniques May . . . 12 · h 2 . 2 · Home Page • So, the area A s of the whole set is equal to 6 times the Title Page √ 3 2 · h 2 . triangle area: A s = 6 · A t = ◭◭ ◮◮ ◭ ◮ • In a region of area A , there are A · ρ d ( x ) sensors, they cover area ( A · ρ d ( x )) · A s . Page 6 of 14 √ 3 Go Back 2 · h 2 • The condition A = ( A · ρ d ( x )) · A s = ( A · ρ d ( x )) · � 2 Full Screen c 0 def implies that h = � , with c 0 = √ 3 . Close ρ d ( x ) Quit
Formulation of the . . . 5. Estimating the Effect of Sensor Placement Towards Precise . . . Optimal Placement of . . . • The adversary places the bio-weapon at a location which Estimating the Effect . . . is the farthest away from the detectors. Precise Formulation of . . . • This way, it will take the longest time to be detected. Solution Towards More . . . • For the grid placement, this location is at one of the vertices of the hexagonal zone. Fuzzy Techniques May . . . Home Page • At these vertices, the distance from each neighboring √ Title Page 3 detector is equal to s = h · 3 . ◭◭ ◮◮ c 0 c 1 • By know that h = � , so s = � , with ◭ ◮ ρ d ( x ) ρ d ( x ) Page 7 of 14 √ √ √ 4 3 3 · 2 c 1 = 3 · c 0 = . Go Back 3 Full Screen • Once the bio-weapon is placed, it starts spreading until Close it reaches the distance d 0 from the detector. Quit
Formulation of the . . . 6. Effect of Sensor Placement (cont-d) Towards Precise . . . Optimal Placement of . . . c 1 • The bio-weapon is placed at a distance s = � Estimating the Effect . . . ρ d ( x ) Precise Formulation of . . . from the nearest sensor. Solution • Once the bio-weapon is placed, it starts spreading until Towards More . . . it reaches the distance d 0 from the detector. Fuzzy Techniques May . . . • In other words, it spreads for the distance s − d 0 . Home Page • During this spread, the disease covers the circle of ra- Title Page dius s − d 0 and area π · ( s − d 0 ) 2 . ◭◭ ◮◮ • The number of affected people n ( x ) is equal to: ◭ ◮ � � 2 Page 8 of 14 c 1 n ( x ) = π · ( s − d 0 ) 2 · ρ ( x ) = π · � − d 0 · ρ ( x ) . Go Back ρ d ( x ) Full Screen Close Quit
Formulation of the . . . 7. Precise Formulation of the Problem Towards Precise . . . Optimal Placement of . . . • For each location x , the number of affected people n ( x ) Estimating the Effect . . . is equal to: Precise Formulation of . . . � � 2 c 1 Solution n ( x ) = π · � − d 0 · ρ ( x ) . ρ d ( x ) Towards More . . . Fuzzy Techniques May . . . • The adversary will select a location x for which this Home Page number n ( x ) is the largest possible: Title Page � � 2 c 1 π · . ◭◭ ◮◮ n = max � − d 0 · ρ ( x ) x ρ d ( x ) ◭ ◮ • Resulting problem: Page 9 of 14 – given population density ρ ( x ), detection distance d 0 , Go Back and number of sensors N , Full Screen – find a function ρ d ( x ) that minimizes the above ex- � Close pression n under the constraint ρ d ( x ) dx = N . Quit
Formulation of the . . . 8. Main Lemma Towards Precise . . . Optimal Placement of . . . • Reminder: we want to minimize the worst-case damage Estimating the Effect . . . n = max n ( x ). x Precise Formulation of . . . • Lemma: for the optimal sensor selection, n ( x ) = const. Solution Towards More . . . • Proof by contradiction: let n ( x ) < n for some x ; then: Fuzzy Techniques May . . . – we can slightly increase the detector density at the Home Page locations where n ( x ) = n , Title Page – at the expense of slightly decreasing the location ◭◭ ◮◮ density at locations where n ( x ) < n ; ◭ ◮ – as a result, the overall maximum n = max n ( x ) will x Page 10 of 14 decrease; – but we assumed that n is the smallest possible. Go Back Full Screen • Thus: n ( x ) = const; let us denote this constant by n 0 . Close Quit
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