# Sum-of-Local-Effects Data Structures for Separable Graphs

## Document Type

Conference Proceeding

## Publication Date

2024

## Department

Department of Computer Science

## Abstract

It is not difficult to think of applications that can be modelled as graph problems in which placing some facility or commodity at a vertex has some positive or negative effect on the values of all the vertices out to some distance, and we want to be able to calculate quickly the cumulative effect on any vertex’s value at any time or the list of the most beneficial or most detrimential effects on a vertex. In this paper we show how, given an edge-weighted graph with constant-size separators, we can support the following operations in time polylogarithmic in the number of vertices and the number of facilities placed on the vertices, where distances between vertices are measured with respect to edge weights: Add (v,f,w,d) places a facility of weight w and with effect radius d onto vertex v.Remove (v,f) removes a facility f previously placed on v using ADD from v.Sum (v) or Sum (v,d) returns the total weight of all facilities affecting v or, with a distance parameter d, the total weight of all facilities whose effect region intersects the “circle” with radius d around v.Top (v,k) or Top (v,k,d) returns the k facilities of greatest weight that affect v or, with a distance parameter d, whose effect region intersects the “circle” with radius d around v. The weights of the facilities and the operation that SUM uses to “sum” them must form a semigroup. For TOP queries, the weights must be drawn from a total order.

## Publication Title

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

## ISBN

9783031491894

## Recommended Citation

Lyu, X.,
Gagie, T.,
He, M.,
Nekrich, Y.,
&
Zeh, N.
(2024).
Sum-of-Local-Effects Data Structures for Separable Graphs.
*
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics),
14422 LNCS*, 195-206.
http://doi.org/10.1007/978-3-031-49190-0_14

Retrieved from: https://digitalcommons.mtu.edu/michigantech-p2/448