Connected Placement of Disaster Shelters in Modern Cities

Transcription

1 Connected Placement of Diater Shelter in Modern Citie Huanyang Zheng and Jie Wu Department of Computer and Information Science Temple Univerity, USA {huanyang.zheng, ABSTRACT Thi paper i motivated by the fact that modern citie are urpriingly vulnerable to large-cale emergencie, uch a the recent terrorit attack on Pari that reulted in the death of 130 people. Diater helter are one of the mot e ective method to handle large-cale emergencie. Hence, thi paper etablihe diater helter with bounded cot. The objective i to minimize the total etablihment cot of diater helter under three contraint. The firt contraint i a ditance contraint, which require that people mut be aigned to diater helter within a certain range. The econd contraint i a capacity contraint, which require that diater helter mut have the capacity to hold incoming people. The third contraint i a connection contraint, which require that diater helter hould be connected to avoid being iolated. Two bounded algorithm are propoed to e ciently etablih diater helter. Real data-driven experiment are conducted to demontrate the e ciency and e ectivene of the propoed algorithm. CCS Concept Network! Cyber-phyical network; Keyword Diater helter; mart city; connected placement; greedy algorithm; approximation ratio; 1. INTRODUCTION Recently, France ha witneed unprecedented attack on Pari that reulted in the death of 130 people [1]. While e ective policie are neceary to reolve thi crii, we are motivated by the fact that modern citie are vulnerable to large-cale emergencie, uch a terrorit attack and refugee crie. Meanwhile, diater helter are one of the mot effective method to handle large-cale emergencie. They are defenive building with food torage and medical facilitie. Permiion to make digital or hard copie of all or part of thi work for peronal or claroom ue i granted without fee provided that copie are not made or ditributed for profit or commercial advantage and that copie bear thi notice and the full citation on the firt page. Copyright for component of thi work owned by other than ACM mut be honored. Abtracting with credit i permitted. To copy otherwie, or republih, to pot on erver or to reditribute to lit, require prior pecific permiion and/or a fee. Requet permiion from permiion@acm.org. CHANTS 16, October 03-07, 2016, New York City, NY, USA c 2016 ACM. ISBN /16/10... $15.00 DOI: Figure 1: Scenario of the diater helter placement. Hawaii ha etablihed underground diater helter to protect people from hurricane [2]. While diater helter can e ectively ave live in emergencie, few reearch e ort were made. Etablihment cot of diater helter are currently unbounded in exiting literature [3]. In contrat, thi paper explore the bounded placement of diater helter to minimize their etablihment cot. Our cenario i hown in Fig. 1, which i a three-dimenional Euclidean urface repreenting the ground of the city. People are ditributed in the Euclidean urface (known a priori). Diater helter of di erent capacitie are etablihed with di erent etablihment cot. A large-capacity diater helter ha a higher cot than a mall-capacity one. The objective of thi paper i to minimize the total etablihment cot of diater helter under three contraint. The number, location, and capacitie of diater helter erve a variable that need to be determined. The firt contraint (ditance contraint) i that people mut be aigned to diater helter within a certain range. Otherwie, people may not have enough time to reach diater helter in large-cale emergencie. The econd contraint (capacity contraint) i that diater helter mut have the capacity to hold incoming people. A crowded diater helter can lead to epidemic preading, inu cient amount of food, and evacuation pace problem. The third contraint i that diater helter hould be connected to avoid iolation (connection contraint, formally defined later in the problem formulation). Thi contraint aim to maintain the communication between di erent diater helter, uch that earche and recue can be facilitated. For example, family member eparated in di erent diater helter can be found through thoe communication. Recue team can alo be upported by nearby diater helter. The remainder of thi paper i organized a follow. Section II urvey related work. Section III decribe the model and then formulate the problem. Section IV analyze the problem. Section V propoe bounded olution. Section VI include the experiment. Section VII conclude the paper.

2 2. RELATED WORK Few reearch e ort have been made with repect to the placement of the diater helter. Park et al. [3] propoed a combinational optimization method to determine location of tunami vertical evacuation helter. Genetic algorithm were ued to olve the placement problem without a bounded performance. Gama et al. [4] tudied the helter placement for mitigating urban flood diater. A coverage model wa propoed to maximize a gradual demand coverage function, which repreent a trade-o between a full coverage objective and a ditance objective. Heßler and Hamacher [5] explored the ink location problem in evacuation planning, uch that diater helter are reachable for people. Their reult were evaluated by numerical tet including random data a well a real world data from the city of Kaierlautern, Germany. The above work are conducted by the tranportation reearch community. Few bounded reult were preented. Due to the recent development of Delay Tolerant Network (DTN), the earch and recue after diater were tudied by the wirele network community. Mae and Okada [6] deigned Unmanned Aerial Vehicle (UAV) for meage deliverie in large-cale diater environment. Meage are ent and received through wirele link between device and the UAV. DTN can be built baed on the UAV. Yang et al. [7] focued on the capacity contrained Voronoi diagram, which partition the network to minimize imbalanced tra c load among diater helter. We additionally conider the communication between diater helter. Our diater helter placement problem extend the claic et cover problem in term of the connection contraint. Given ome element and a collection of et of element, the claic et cover problem aim to elect minimum et to cover all given element [8]. Element in a et are covered if thi et i elected. In our problem, diater helter are mapped to et and people are mapped to element. Geometric et cover technique [9] are ued, while the connection contraint i poed. 3. PROBLEM FORMULATION Our cenario i baed on a three-dimenional Euclidean urface that repreent the ground of a city. Let p denote a peron that i planned for diater helter. P i the et of people, i.e., P = {p}. Location of people are known a priori. Let denote a diater helter. S i the et of the diater helter, i.e., S = {}. Then, the location, capacity and etablihment cot of are denoted by l, c, and e, repectively. We conider that the capacity of a diater helter i bounded, i.e., c min apple c apple c max. The etablihment cot of a diater helter depend on it capacity. Clearly, a large-capacity diater helter ha a higher etablihment cot than a mall-capacity one. We ue e = F (c ) to denote the pre-known etablihment cot function. Let x p, denote a boolean deciion variable. x p, = 1 mean that the peron, p, i aigned to the diater helter,. Finally, D( ) i a pre-known ditance function. Our objective i to minimize the total etablihment cot, P e, of the diater helter. The number, location, and capacitie of the diater helter (i.e., S, l, and c ) are variable. We determine the aignment plan of x p,. Three contraint are poed. The firt contraint (ditance contraint) i that people mut be aigned to diater helter within a certain range of r, i.e., P xp, = 1 and D(p, ) x p, apple r. Thi contraint guarantee that people have enough time to reach diater helter in large-cale e- mergencie. The econd contraint (capacity contraint) i that diater helter mut have the capacity to hold incoming people. We have P p xp, apple c to enure u cient reource for people in diater helter. The third contraint i that diater helter hould be connected to avoid iolation. We ay two diater helter are neighboring, if the ditance between them i no larger than a threhold that i denoted by R. The third contraint (connection contraint) mean that the induced neighboring graph of the diater helter include all diater helter and i connected. Thi aim to maintain temporary communication between di erent diater helter, ince diater may detroy wired and wirele communication infratructure. Vehicle equipped with wirele device can move around diater helter to provide temporary communication baed on DTN. A a trade-o, we need to balance the number and capacitie of the diater helter. For the ame cot, we can either etablih many mall-capacity diater helter or a few large-capacity diater helter, depending on the ditribution of the people. Thi trade-o become even more complex, when the location problem i involved. Since we require that diater helter hould be connected, the etablihment of a diater helter can be ued to hold people or erve a an intermediate relay to connect other diater helter. Hence, our problem face unique challenge. 4. PROBLEM ANALYSIS We tart with the problem hardne: Theorem 1. Our placement problem i NP-hard. Proof: The proof i done through two pecial aumption, under which our problem become equivalent to the geometry et cover problem [10]. Given ome element and a collection of et of element, the et cover problem aim to elect a minimum number of et to cover all given element. The geometry et cover problem i a pecial cae, in which element are ditributed in the Euclidean pace and et are geometric ball. An element i covered by a et, if it i geometrically included by the ball of that et. Let u conider the diater helter placement problem in the Euclidean pace. The firt aumption i that the etablihment cot of a diater helter barely depend on it capacity. A a reult, the capacity of each helter can be large enough to hold all incoming people, i.e., the econd contraint (capacity contraint) i relaxed. The econd aumption i that R i large enough, uch that we can relax the third contraint (connection contraint). When R goe to infinity, all diater helter are neighboring and connected. Then, let u reduce the geometry et cover problem to our diater helter placement problem. Thi reduction i done by mapping element and et to people and diater helter, repectively. The geometric ball of each et ha a radiu of r, i.e., the diater helter will hold all people within the range of r. Our problem can alo reduce to the geometry et cover problem in the ame manner (they become equivalent to each other). Since the geometry et cover problem i NP-hard, our problem i alo NP-hard. We apply a dicrete framework to olve the diater helter placement problem. The three-dimenional Euclidean urface, which model the ground of the city, i horizontally dicretized by a grid, i.e., the vertical projection of the dicretized urface would be a grid at the horizontal plane. An

3 dicrete tep three-dimenional Euclidean urface grid interection dicrete tep Figure 2: An illutration of the dicrete framework. example of the dicretization i hown in Fig. 2. Then, each quare in the grid ha a ize of, in which i the dicrete tep meauring the ditance between neighboring grid interection. Let L denote the et of location at grid interection on the dicretized three-dimenional Euclidean urface (the ground of the city). We aume that diater helter are only etablihed at location that belong to L. Since the lope of the ground i limited, uch a dicretization bring a limited information lo. bring a trade-o between the accuracy and the time complexity. A maller bring a higher accuracy at the cot of a higher time complexity, ince the et L become larger. In addition, the etablihment cot function, e = F (c ), i alo dicretized. Thi i becaue the helter capacity i dicretized, i.e., the number of people that it can hold i an integer. We have: Definition 1. The coverage radiu,, ofadiaterhel- ter,, ithemaximumpoibleditancethatatifie: (i) apple r, and(ii) ha a u cient capacity to hold all people within.peoplewithinthecoverageradiuof are covered by, andtheetofcoveredpeopleidenotedbyp. In Definition 1, r come from the ditance contraint, in which people mut be aigned to diater helter within a range of r. A diater helter will not cover people without the range of r. In addition, the coverage radiu i retricted by the capacity contraint, in which diater helter mut have the capacity to hold incoming people. Note that the area within the coverage radiu of a helter i not necearily a dik, ince the ground of the city may not be flat. For a diater helter at a given location, it coverage radiu can be determined once it capacity i known. 5. BOUNDED SOLUTIONS Two bounded algorithm are propoed for our problem. The firt and econd algorithm independently and cooperatively conider the connection contraint, repectively. 5.1 Two-Stage Algorithm We propoe a Two-Stage Algorithm () to place diater helter. The firt tage reolve the ditance and capacity contraint, and then, the econd tage place additional diater helter to atify the connection contraint. i preented a Algorithm 1. Line 1 to 4 correpond to the firt tage. In line 1 to 3, diater helter at all poible location with each poible capacity are mapped to a collection of et. People are mapped element, and people that are covered by a diater helter are mapped to element in the correponding et. In line 4, an exiting approximation algorithm i ued to olve the geometric et cover problem. Algorithm 1 Two-Stage Algorithm () Input: A et of people, P, and their location, An etablihment cot function, F ( ), A ditance function on Euclidean urface, D( ), A et of dicrete location, L, A range, r, and a connection threhold, R. Output: S, their location, and capacitie. 1: for each location, l 2 L do 2: for each dicrete capacity, c min apple c apple c max do 3: Calculate the et of covered people, P, for the diater helter,, withl = l and c = c. Map to a et and P to element in. Ue the etablihment cot, e = F (c ), a the et weight. 4: For the mapped et and element, ue an approximation algorithm to olve the geometric et cover problem. The reultant collection of elected et are mapped back to the diater helter, including location and capacitie. 5: Map location in L a node in a graph (node are connected if their ditance i no larger than R). Ue an approximation algorithm to olve a Steiner tree problem, which elect node to connect the terminal (diater helter in line 4). Selected node are recorded a new diater helter with minimum capacitie. 6: return the reultant diater helter in line 4 and 5. The et, which are elected by the approximation algorithm, are mapped back to diater helter at given location with given capacitie. The reultant diater helter atify the ditance and capacity contraint, but may not atify the connection contraint. To atify the connection contraint, we introduce the econd tage, which include line 5. The reultant diater helter of the firt tage are regarded a terminal in a Steiner tree problem. Through another exiting approximation algorithm, they can be connected by placing additional diater helter with minimum capacitie. i bounded: Theorem 2. Let a 1 and a 2 denote the approximation ratio of exiting algorithm for the geometric et cover problem and the Steiner tree problem, repectively. Then, the approximation ratio of i a 1 + a 2 (1 + b2r/rc a 1). Proof: Let 1 and 2 denote the cot of the diater helter placed by the firt and econd tage of, repectively. Let OPT 1 and OPT 2 denote the optimal cot of the diater helter for the firt and econd tage of T- SA, repectively. Let OPT denote the optimal cot of the diater helter in our problem. Baed on the definition of the approximation ratio, we have the following inequalitie: OPT 1 apple 1 apple a 1 OPT 1 (1) OPT 2 apple 2 apple a 2 OPT 2 (2) Since the optimal placement in our problem i alo a olution to the geometric et cover problem in the firt tage, we have: 1 apple a 1 OPT 1 apple a 1 OPT (3) Let u conider a pecial olution to the Steiner tree problem in the econd tage, baed on the optimal placement in our problem. Since people mut be aigned to diater helter within a certain range of r, the ditance between a diater helter in 1 and it cloet diater helter in OPT mut

4 be no larger than 2r. Conequently, we can ue b2r/rc diater helter with minimum capacitie (and thu minimum etablihment cot) to connect an arbitrary diater helter in 1 to diater helter in OPT. Hence, we can ue a total cot of OPT + b2r/rc OPT 1 a a pecial olution to the Steiner tree problem. The cot of thi pecial olution i no maller than OPT 2 by the definition. We have: 2 apple a 2 OPT 2 apple a 2 (OPT + b 2r c 1) (4) R If we combine Eq 3 and 4, we have: = 1 + a 2 (OPT + b 2r R c 1) apple a 1 + a 2 (1 + b 2r R c a1) OPT (5) The proof complete. Let P denote the number of people (et cardinality of P ). Since there exit approximation algorithm [10, 11] with a 1 =log P and a 2 = 3, we have the following corollary: 2 Corollary 1. The approximation ratio of can be 3 +(1+ 3 b 2r c) log P. When2r <R,iti 3 +log P. 2 2 R 2 The bound of depend on b2r/rc, which repreent the tightne of the connection contraint. When R i large enough, the connection contraint can be relaxed, and then, we can focu on the ditance and capacity contraint. The time complexity of depend on the time complexitie of approximation algorithm for the geometric et cover problem and the Steiner tree problem. olve the ditance and capacity contraint by the geometric et cover problem, and then, olve the connection contraint by the Steiner tree problem. olve contraint independently. 5.2 Tree Growth Algorithm Thi ubection preent a Tree Growth Algorithm () to place diater helter. The ditance, capacity, and connection contraint are cooperatively conidered. The idea of i to iteratively place a diater helter that can connect to an exiting diater helter, until all people are covered. We tart with the following definition: Definition 2. The adjacent location i defined baed on aetofthediaterhelter,s. l i adjacent to S, ifiti(i) in the range of R of at leat one diater helter in S, and (ii) not in the coverage radiu of a diater helter in S. The coverage radiu i in Definition 1, and R i the threhold for the connection contraint. If we place a diater helter at an adjacent location of S, then it i connected to S, and therefore atifying the connection contraint. Meanwhile, ince people in adjacent location of S are not covered by S, the placement of a diater helter at an adjacent location can alo provide e cient coverage to people (the ditance and capacity contraint are tified). To quantify the connection contraint, we lightly abue the notation, and ue D(l, S) to denote the ditance between l and it cloet diater helter in S. If l i adjacent to S, thenwe have min apple D(l, S) apple R. i preented in Algorithm 2. In line 1, it initialize S = ;, i.e., no diater helter ha been placed. In line 2, a- mong all the location, it place the firt diater helter that minimize F (c). Here, e = F (c) i the etablihment cot Algorithm 2 Tree Growth Algorithm () Input: A et of people, P, and their location, An etablihment cot function, F ( ), A ditance function on Euclidean urface, D( ), A et of dicrete location, L, A range, r, and a connection threhold, R. Output: S, their location, and capacitie. 1: Initialize S = ;. 2: Among location l 2 L, place a diater helter,, with a capacity, c that minimize F (c). 3: Add to S, and then, remove P from P. 4: while P 6= ; do 5: Find the et of location, L 0, that are adjacent to S. 6: Among location l 2 L 0, place a diater helter,, with a capacity, c, that minimize F (c) D(l,S). 7: Add to S, and then, remove P from P. 8: return S, their location, and capacitie. of a diater helter with a capacity, c, and i the number of people covered by the diater helter,. Therefore, F (c ) repreent the cot-to-benefit ratio, which hould be minimized. Note that F (c) involve the ditance and capacity contraint, but not the connection contraint. Thi i becaue line 2 only place the firt diater helter. Line 3 update the diater helter placement, and remove covered people. Line 4 to 7 iteratively place a diater helter, until all people are covered. Line 5 calculate the et of adjacent location of the exiting diater helter, S. Among all the adjacent location of S, line 6 place a diater helter that minimize F (c) D(l,S), which repreent an improved cotto-benefit ratio. Here, D(l, S) i the ditance between l and it cloet diater helter in S. It additionally incorporate the connection contraint into F (c). Line 7 update the diater helter placement, and remove covered people. The iteration terminate when all people are covered. In, the ditance and capacity contraint are atified by the definition of the coverage radiu. Meanwhile, the connection contraint i atified by the definition of the adjacent location. Let min denote the minimum coverage radiu in. Then, i bounded a follow: Theorem 3. In term of minimizing the total etablihment cot, ha an approximation ratio of R min log P to the optimal diater helter placement algorithm. Proof: A new concept of the peron weight i introduced to prove thi theorem. i an iterative algorithm, and in each iteration, a diater helter i etablihed to minimize F (c) D(l,S). Note that people, who have already been covered in previou iteration, are not included in P. The weight of each peron in P i defined a F (c) D(l,S), and i denoted by w p. To unify line 2, the weight of each peron covered by the firt diater helter i pecially defined a F (c ) R. Then, the total weight of people in i: p2p w p = min F (c ) D(l, S) w p = P p2p p2p F (c ) (6)

5 Thi i becaue P 1 p2p = 1. We have D(l, S) min baed on the definition of adjacent location. Let denote a diater helter in the optimal olution, S. We have: w p = w p (7) p2p p2p We claim that, for each diater helter in the optimal olution, the following inequality i atified: w p apple log P F (c ) R (8) p2p To prove our claim, let u conider how cover people in P. Let n k be the number of uncovered people in P, after the k-th iteration. For a people, p, who i covered by in and by in the optimal olution, we have: w p = F (c) D(l, S) apple F (c ) D(l, S ) P apple F (c ) R n k (9) The firt inequality reult from the greedine, in which minimize F (c) D(l,S) among all poible placement. The econd inequality reult from P n k and D(l, S ) apple R. A a reult, we have [11]: p2p w p apple k (n k n k+1 ) F (c ) R n k apple log P F (c ) R (10) The proof of Eq. 8 complete. Combining Eq. 6, 7, and 8, the following bound can be obtained: min F (c ) apple w p = w p p2p p2p apple log P F (c ) R (11) Eq. 11 can be rewritten a: R F (c ) apple min log P F (c ) (12) Note that P F (c) and P F (c ) are the total etablihment cot of and the optimal olution, repectively. Therefore, the proof complete. bound depend on it minimum coverage radiu, which in turn depend on the ditribution of the people and the etablihment cot function, F ( ). When F ( ) i a contant function, the etablihment cot of a diater helter i a contant, and i not related to the capacity. We have min = r, ince diater helter can have u cient and free capacitie to hold all people within r. Hence, we have: Corollary 2. The approximation ratio of i R log P, r when F ( ) i a contant etablihment cot function. The time complexity of i O( L 2 + L P c max c min ). L i the number of location, P i the number of people, and c max c min i the number of di erent capacitie. Thi time complexity can be obtained by pre-computation. O( L 2 ) come from pre-computation of the ditance between all pair of location in L, in order to determine the adjacent location. O( L P c max c min ) come from precomputation of the coverage radiu for all poible diater helter with all poible capacitie. 6. EPERIMENTS 6.1 Setting Our dataet i baed on the city of Cannon Beach, Oregon, United State [12]. The dataet information ha been illutrated in Park work [12]. It include a long and narrow area (6.1km by 1.5km). The location of 1,382 houe are collected. For implicity, we aume each houe correpond to a ingle peron in our experiment. Algorithm 1 and 2 are denoted a and, repectively. Two baeline algorithm are ued a comparion. (i) tand for a bounded clutering method in [13]. To guarantee that each peron i within in a range, r, of the diater helter, the radiu of each cluter i r. Diater helter are etablihed at the center of each cluter to atify the ditance and capacity contraint. To atify the connection contraint, a panning tree i additionally built. (ii) i a divide and conquer method. iteratively divide people into two cluter through a k-mean algorithm [14]. The center of thee two cluter are connected through placing diater helter with minimum capacitie. The iteration terminate when the ditance and capacity contraint are atified. We et = 100m to dicretize the location. Note that the dataet include a long and narrow area of 6.1km by 1.5km. Therefore, uch a dicretization could maintain the accuracy. To reveal the capacity contraint, we et c min =10 and c max = 100. Two etablihment cot function are ued: F (c p )= c and F (c )= c.coe cientof2.4 and are ued, ince F (c min) and F (c max) can be the ame for the firt and econd etablihment cot function. Therefore, fair comparion can be obtained. We do not ue hyper-linear etablihment cot function, ince the etablihment of multiple mall-capacity diater helter can be alway better than thoe of a few largecapacity diater helter. Moreover, hyper-linear etablihment cot function are alo empirically impractical. Parameter, r and R, are tuned to repreent the impact of the ditance and connection contraint, repectively. While r range from m to m, we et R to be m, 600m, and 800m. In thi etting, we have R r to guarantee that the connection contraint i not ignored or relaxed. 6.2 Evaluation Reult The evaluation reult under F (c )= p c are hown in Fig. 3. Three ubfigure of Fig. 3(a), Fig. 3(b), and Fig. 3(c) correpond to R =m,r = 600m, and R = 800m for the connection contraint, repectively. and outperform and under all the etting. outperform, ince it cooperatively conider the connection contraint. P For all algorithm, the total etablihment cot, e, decreae with repect to the, r. The ditance contraint require that people mut be aigned to diater helter within a certain range of r. A larger r mean that the ditance contraint i more relaxed, and thu, the total etablihment cot i maller. When r goe to infinity, the ditance contraint i completely relaxed. Another notable point i that, when the connection threhold, R, become larger, the total etablihment cot alo lightly decreae. Similarly, a larger R mean that the connection contraint can be more relaxed, and thu, the total etablihment cot i maller. When R goe to infinity, the connection contraint can be ignored. A hown in Fig. 3, the performance gap between and i mall when

6 total etablihment cot total etablihment cot total etablihment cot (a) R =m. (b) R =600m. (c) R =800m. Figure 3: Evaluation reult under F (c p )= c. total etablihment cot (a) R =m. (b) R =600m. (c) R =800m. Figure 4: Evaluation reult under F (c )= c. total etablihment cot total etablihment cot R = m, but i large when R = 600m and R =800m. Thi i becaue and independently and cooperatively conider the connection contraint, repectively. The evaluation reult under F (c )= c are hown in Fig. 4. Three ubfigure of Fig. 4(a), Fig. 4(b), and Fig. 4(c) correpond to R =m,r = 600m, and R = 800m for the connection contraint, repectively. The reult in Fig. 4 are imilar to reult in Fig. 3. The only di erence i that the total etablihment cot of and decreae more lowly in Fig. 4. Under F (c p )= c, we ignificantly favor the etablihment of large-capacity diater helter, ince they are cot-e cient, epecially when r i large (a diater helter can cover more people due to the ditance contraint). On the other hand, when F (c )= c, we only lightly favor the etablihment of large-capacity diater helter, ince the etablihment cot function cale linearly with repect to the diater helter capacity. 7. CONCLUSION Thi paper minimize the total etablihment cot of the diater helter under three contraint: the ditance contraint, the capacity contraint, and the connection contraint. The ditance contraint require that people mut be aigned to diater helter within a certain range. The capacity contraint require that diater helter mut have the capacity to hold incoming people. The connection contraint require that diater helter hould be connected to avoid iolation. Two bounded algorithm are propoed to etablih diater helter. Experiment demontrate the e ciency and e ectivene of the propoed algorithm. ACKNOWLEDGEMENT Thi reearch wa upported in part by NSF grant CNS , CNS , CNS , CNS , CNS , and ECCS REFERENCES [1] A. Sazonov, Pari attack and europe aŕ, repone to the refugee crii, Hu pot Politic, [2] K. Kim, P. Pant, and E. Yamahita, Evacuation planning for plauible wort cae inundation cenario in Honolulu, Hawaii. Journal of Emergency Management, [3] S. Park, J. W. Van de Lindt, R. Gupta, and D. Cox, Method to determine the location of tunami vertical evacuation helter, Natural Hazard, [4] M. Gama, M. P. Scaparra, and B. Santo, Optimal location of helter for mitigating urban flood, in European Working Group on Tranportation, [5] P. Heßler and H. W. Hamacher, Sink location to find optimal helter in evacuation planning, EURO Journal on Computational Optimization, [6] K. Mae and H. Okada, Meage communication ytem uing unmanned aerial vehicle under large-cale diater environment, in IEEE PIMRC, [7] K. Yang, A. H. Shekhar, D. Oliver, and S. Shekhar, Capacity-contrained network-voronoi diagram, IEEE Tranaction on Knowledge and Data Engineering, [8] A. Dehpande, L. Hellertein, and D. Kletenik, Approximation algorithm for tochatic boolean function evaluation and tochatic ubmodular et cover, in ACM-SIAM SODA, [9] S. Stergiou and K. Tioutioulikli, Set cover at web cale, in ACM SIGKDD, [10] K. L. Clarkon and K. Varadarajan, Improved approximation algorithm for geometric et cover, Dicrete & Computational Geometry, 7. [11] Z. Zhang,. Gao, and W. Wu, Algorithm for connected et cover problem and fault-tolerant connected et cover problem, Theoretical Computer Science, 9. [12] S. Park, Undertanding and mitigating tunami rik for coatal tructure and communitie, PhD Thei, Colorado State Univerity, 7. [13] T. F. Gonzalez, Clutering to minimize the maximum intercluter ditance, Theoretical Computer Science, [14] J. Wang, J. Wang, Q. Ke, G. Zeng, and S. Li, Fat approximate k-mean via cluter cloure, in Multimedia Data Mining and Analytic, 2015.