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Presented by: GROUP 7 Gayathri Gandhamuneni & Yumeng Wang

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AGENDA Synonyms Definition Historical Background Scientific Fundamentals Key Applications Future Direction & References

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SYNONYMS CCNVD – None Voronoi Diagram - Voronoi tessellation, a Voronoi decomposition, a Voronoi partition, or a Dirichlet tessellation

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FORMAL DEFINITION Capacity Constrained Network Voronoi Diagram (CCNVD): Partitions graph into set of contiguous service areas that honor service center capacities and minimize the sum of distances from graph-nodes allotted service centers.

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SIGNIFICANCE & APPLICATIONS Critical Societal applications Examples: Assigning consumers to gas stations in the aftermath of a disaster Assigning evacuees to shelters Assigning patients to hospitals Assigning students to school districts CCNVD Finite Spaces Continuous Spaces

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PROBLEM STATEMENT Input: A transportation network G – (Nodes N, Edges E) Set of service centers Constraints on the service centers Real weights on the edges. Objective: Minimize the sum of distances from graph-nodes to their allotted service centers while satisfying the constraints of the network. Constraints: Nodes - Assumed to be contiguous The effective paths can be calculated (maximum coverage and shortest paths) Output: Capacity Constrained Network Voronoi Diagram (CCNVD)

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HISTORICAL BACKGROUND Voronoi Diagram: Way of dividing space into a number of regions A set of points (called seeds, sites, or generators) is specified beforehand For each seed, there will be a corresponding region consisting of all points closer to that seed than to any othercloser Regions are called Voronoi cells

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RELATED WORK Minimizing sum of distances between graph nodes and their allotted service centers Honoring service center capacity constraints Service Area Contiguity Min-cost flow approaches Network Voronoi Diagrams (NVD) CCNVD

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RELATED WORK ILLUSTRATION WITH DIAGRAMS Input NVD

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RELATED WORK ILLUSTRATION WITH DIAGRAMS Min-Cost Flow without SA contiguity (min-sum=30)(Output) CCNVD (min-sum=30) (Output) – Pressure Equalizer Approach

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CHALLENGES Large size of the transportation network Uneven distribution - Service centers & Customers Constraint: Service areas must be contiguous in graph to simplify communication of allotments NP Hard

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FUTURE DIRECTION More factors of the problem into account Factors related to capacity of service centers, Example: Number and distance of neighboring nodes Service quality of the service center. Factors related to weight of each node Number of consumers The level of importance

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REFERENCES [1] KwangSoo Yang, Apurv Hirsh Shekhar, Dev Oliver, Shashi Shekhar: Capacity-Constrained Network-Voronoi Diagram: A Summary of Results. SSTD 2013: [2] Advances in Spatial and Temporal Databases - 13th International Symposium, SSTD 2013 Munich, Germany, August 2013 Proceedings [3] [4] Ahuja, R., Magnanti, T., Orlin, J.: Network flows: theory, algorithms, and applications, Prentice Hall [5] Goldberg, A.V., Tarjan, R.E.: Finding minimum-cost circulations by successive approximation. Mathematics of Operations Research 15(3), 430–466 (1990) [6] Klein, M.: A primal method for minimal cost flows with applications to the assignment and transportation problems. Management Science 14(3), 205–220 (1967) [7] Erwig, M.: The graph voronoi diagram with applications. Networks 36(3), 156–163 (2000) [8] Okabe, A., Satoh, T., Furuta, T., Suzuki, A., Okano, K.: Generalized network voronoi diagrams: Concepts, computational methods, and applications. International Journal of Geographical Information Science 22(9), 965–994 (2008)

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