# Load Balancing in Network Voronoi Diagrams Under Overload Penalties

## Abstract

Input to the problem of Load Balanced Network Voronoi Diagram (LBNVD) consists of the following: (a) a road network represented as a directed graph; (b) locations of service centers (e.g., schools in a city) as vertices in the graph and; (c) locations of demand (e.g., school children) also as vertices in the graph. In addition, each service center is also associated with a notion of *capacity* and an *overload penalty* which is “charged” if the service center gets overloaded. Given the input, the goal of the LBNVD problem is to determine an *assignment* where each of the demand vertices is allotted to a service center. The objective here is to generate an assignment which minimizes the sum of the following two terms: (i) total distance between demand vertices and their allotted service centers and, (ii) total penalties incurred while overloading the service centers. The problem of LBNVD finds its application in the domain of urban planning. Research literature relevant to this problem either assume infinite capacity or do not consider the concept of “overload penalty.” These assumptions are relaxed in our LBNVD problem. We develop a novel algorithm for the LBNVD problem and provide a theoretical upper bound on its worst-case performance (in terms of solution quality). We also present the time complexity of our algorithm and compare against the related work experimentally using real datasets.

## Notes

### Acknowledgement

We would like to thank Prof Sarnath Ramnath, St. Cloud State University and the reviewers of DEXA 2018 for giving their valuable feedback towards improving this paper. This paper was in part supported by the IIT Ropar, Infosys Center for AI at IIIT Delhi and DST SERB (ECR/2016/001053).

## References

- 1.Wikipedia: Catchment area. http://en.wikipedia.org/w/index.php?title=Catchment%20area
- 2.Okabe, A., et al.: Generalized network voronoi diagrams: concepts, computational methods, and applications. Int. J. GIS
**22**(9), 965–994 (2008)Google Scholar - 3.Demiryurek, U., Shahabi, C.: Indexing network voronoi diagrams. In: Lee, S., Peng, Z., Zhou, X., Moon, Y.-S., Unland, R., Yoo, J. (eds.) DASFAA 2012. LNCS, vol. 7238, pp. 526–543. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29038-1_38CrossRefGoogle Scholar
- 4.Yang, K., et al.: Capacity-constrained network-voronoi diagram. IEEE Trans. Knowl. Data Eng.
**27**(11), 2919–2932 (2015)CrossRefGoogle Scholar - 5.U, L.H., et al.: Optimal matching between spatial datasets under capacity constraints. ACM Trans. Database Syst.
**35**(2), 9:1–9: 44 (2010)Google Scholar - 6.U, L.H., et al.: Capacity constrained assignment in spatial databases. In: Proceeding of the International Conference on Management of Data (SIGMOD), pp. 15–28 (2008)Google Scholar
- 7.Aurenhammer, F.: Power diagrams: properties, algorithms and applications. SIAM J. Comput.
**16**(1), 78–96 (1987)MathSciNetCrossRefGoogle Scholar - 8.Yao, B., et al.: Dynamic monitoring of optimal locations in road network databases. VLDB J.
**23**(5), 697–720 (2014)CrossRefGoogle Scholar - 9.Xiao, X., Yao, B., Li, F.: Optimal location queries in road network databases. In: Proceedings of the 27th International Conference on Data Engineering (ICDE), pp. 804–815 (2011)Google Scholar
- 10.Diabat, A.: A capacitated facility location and inventory management problem with single sourcing. Optim. Lett.
**10**(7), 1577–1592 (2016)MathSciNetCrossRefGoogle Scholar - 11.Cormen, T.H., Stein, C., Rivest, R.L., Leiserson, C.E.: Introduction to Algorithms, 2nd edn. McGraw-Hill Higher Education, New York (2001)MATHGoogle Scholar
- 12.Delling, D., et al.: PHAST: hardware-accelerated shortest path trees. J. Parallel Distrib. Comput.
**73**(7), 940–952 (2013)CrossRefGoogle Scholar - 13.Bast, H., et al.: Fast routing in road networks with transit nodes. Science
**316**(5824), 566 (2007)MathSciNetCrossRefGoogle Scholar - 14.Jing, N., et al.: Hierarchical encoded path views for path query processing: an optimal model and its performance evaluation. IEEE Trans. KDE
**10**(3), 409–432 (1998)Google Scholar - 15.Bortnikov, E., Khuller, S., Li, J., Mansour, Y., Naor, J.S.: The load-distance balancing problem. Networks
**59**(1), 22–29 (2012)MathSciNetCrossRefGoogle Scholar - 16.Kleinberg, J., Tardos, E.: Algorithm Design. Pearson Education, London (2009)Google Scholar