CAPACITY CONSTRAINED NETWORK VORONOI DIAGRAM

(CCNVD)

Presented by:GROUP 7

Gayathri Gandhamuneni &Yumeng Wang

AGENDA

SynonymsDefinitionHistorical BackgroundScientific FundamentalsKey ApplicationsFuture Direction & References

SYNONYMS

CCNVD – None

Voronoi Diagram - Voronoi tessellation, a Voronoi decomposition, a Voronoi partition, or a Dirichlet tessellation

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.

SIGNIFICANCE & APPLICATIONSCritical Societal applicationsExamples:

Assigning consumers to gas stations in the aftermath of a disaster

Assigning evacuees to shelters Assigning patients to hospitalsAssigning students to school districts

CCNVDFinite SpacesContinuous Spaces

PROBLEM STATEMENTInput:

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)

HISTORICAL BACKGROUNDVoronoi Diagram:

Way of dividing space into a number of regionsA 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 other

Regions are called “Voronoi cells”

RELATED WORKMinimizing 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

RELATED WORK ILLUSTRATION WITH DIAGRAMS

InputNVD

RELATED WORK ILLUSTRATION WITH DIAGRAMS

Min-Cost Flow without SA contiguity(min-sum=30)(Output)

CCNVD (min-sum=30) (Output) – Pressure Equalizer Approach

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

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 nodeNumber of consumersThe level of importance

REFERENCES[1] KwangSoo Yang, Apurv Hirsh Shekhar, Dev Oliver, Shashi Shekhar: Capacity-Constrained Network-Voronoi Diagram: A Summary of Results. SSTD 2013: 56-73 [2] Advances in Spatial and Temporal Databases - 13th International Symposium, SSTD 2013 Munich, Germany, August 2013 Proceedings [3] http://en.wikipedia.org/wiki/Voronoi_diagram [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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