# Slides

```Presented by:
GROUP 7
Gayathri Gandhamuneni &amp;
Yumeng Wang
AGENDA
 Synonyms
 Definition
 Historical Background
 Scientific Fundamentals
 Key Applications
 Future Direction &amp; 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 &amp; 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
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)
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 other
 Regions are called “Voronoi cells”
RELATED WORK
Minimizing sum of distances between graph
nodes and their allotted service centers
Honoring service center
capacity constraints
Min-cost flow
approaches
Service Area Contiguity
CCNVD
Network Voronoi
Diagrams (NVD)
RELATED WORK ILLUSTRATION WITH DIAGRAMS
Input
NVD
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 &amp; 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 node
 Number of consumers
 The 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)
QUESTIONS?
THANK YOU
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