3 SYNONYMSCCNVD – NoneVoronoi Diagram - Voronoi tessellation, a Voronoi decomposition, a Voronoi partition, or a Dirichlet tessellation
4 FORMAL DEFINITIONCapacity 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.
5 SIGNIFICANCE & APPLICATIONS Critical Societal applicationsExamples:Assigning consumers to gas stations in the aftermath of a disasterAssigning evacuees to sheltersAssigning patients to hospitalsAssigning students to school districtsCCNVDFinite SpacesContinuous SpacesReferences for CCNVD in Finite Spaces and Continuous Spaces – Not discussing due to time constraint
6 PROBLEM STATEMENT Input: Objective: Constraints: Output: A transportation network G – (Nodes N, Edges E)Set of service centersConstraints on the service centersReal 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 contiguousThe effective paths can be calculated (maximum coverage and shortest paths)Output:Capacity Constrained Network Voronoi Diagram (CCNVD)
7 HISTORICAL BACKGROUND Voronoi Diagram:Way of dividing space into a number of regionsA set of points (called seeds, sites, or generators) is specified beforehandFor each seed, there will be a corresponding region consisting of all points closer to that seed than to any otherRegions are called “Voronoi cells”One of the early applications of Voronoi diagrams was implemented by John Snow to study the 1854 Broad Street cholera outbreak in Soho, England. He showed the correlation between areas on the map of London using a particular water pump, and the areas with most deaths due to the outbreak.
8 RELATED WORKMinimizing sum of distances between graph nodes and their allotted service centersHonoring service center capacity constraintsService Area ContiguityMin-cost flow approachesCCNVDNetwork Voronoi Diagrams (NVD)
9 RELATED WORK ILLUSTRATION WITH DIAGRAMS Every edge is associated with a distance (e.g., travel time), Every graph-node has unit demand.Every service center has a capacity to serve 5 graph units.NVD allots every graph-node to the nearest service center – No account for load balance, which may lead to congestion and delay at service centersInputNVD
10 RELATED WORK ILLUSTRATION WITH DIAGRAMS Min-cost flow: minimizes the sum of distances from graph-nodes to their allotted service centers. Although it achieves its goal (min-sum), service areas for Y and Z violate SA contiguity.CCNVD: load is balanced, service areas are contiguous for all service centers.Min-Cost Flow without SA contiguity(min-sum=30)(Output)CCNVD (min-sum=30) (Output) – Pressure Equalizer Approach
11 CHALLENGES Large size of the transportation network Uneven distribution - Service centers & CustomersConstraint:Service areas must be contiguous in graph to simplify communication of allotmentsNP Hard
12 FUTURE DIRECTION More factors of the problem into account Factors related to capacity of service centers,Example:Number and distance of neighboring nodesService quality of the service center.Factors related to weight of each nodeNumber of consumersThe level of importance
13 REFERENCES KwangSoo Yang, Apurv Hirsh Shekhar, Dev Oliver, Shashi Shekhar: Capacity-Constrained Network-Voronoi Diagram: A Summary of Results. SSTD 2013:  Advances in Spatial and Temporal Databases - 13th International Symposium, SSTD 2013 Munich, Germany, August 2013 Proceedings   Ahuja, R., Magnanti, T., Orlin, J.: Network flows: theory, algorithms, and applications, Prentice Hall  Goldberg, A.V., Tarjan, R.E.: Finding minimum-cost circulations by successive approximation. Mathematics of Operations Research 15(3), 430–466 (1990)  Klein, M.: A primal method for minimal cost flows with applications to the assignment and transportation problems. Management Science 14(3), 205–220 (1967)  Erwig, M.: The graph voronoi diagram with applications. Networks 36(3), 156–163 (2000)  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)