Yu Zhang
Abstract
Partitioning the Maximal Redundant Rigid Components (MRRC) and Maximal Global Rigid Components (MGRC) in generic 2D graphs are critical problem for network structure analysis, network localizability detection, and localization algorithm design. This article presents efficient algorithms to partition MRRCs and MGRCs and develops an open-sourced toolbox, GPART, for these algorithms to be conveniently used by the society. We firstly propose conditions and an efficient algorithm to merge the over-constrained regions to form the maximal redundant rigid components (MRRC). The detected MRRCs are proved to be maximal and all the MRRCs are guaranteed to be detected. The time to merge the over-constrained regions is linear to the number of nodes in the over-constrained components. To detect MGRCs, the critical problem is to decompose 3-connected components in each MRRC. We exploit SPQR-tree based method and design a local optimization algorithm, called MGRC_acce to prune the unnecessary decomposition operations so that the SPQR-tree functions can be called much less number of times. We prove the MGRCs can be detected inside MRRCs using at most O(mn) time. Then a GPART toolbox is developed and extensively tested in graphs of different densities. We show the proposed MRRC and MGRC detection algorithms are valid and MGRC_acce greatly outperforms the direct SPQR-tree based decomposition algorithm. GPART is outsourced at https://github.com/inlab-group/gpart.
- [1] . 2013. Generic global rigidity of body–bar frameworks. Journal of Combinatorial Theory, Series B 103, 6 (2013), 689–705.Google Scholar
Digital Library
- [2] . 2010. Characterizing generic global rigidity. American Journal of Mathematics 132, 4 (2010), 897–939.Google ScholarCross Ref
- [3] . 2001. A linear time implementation of SPQR-Trees. In Graph Drawing, , , , and (Eds.). Vol. 1984. Springer Heidelberg, Heidelberg, 77–90.
DOI: .Series Title: Lecture Notes in Computer Science. Google ScholarCross Ref - [4] . 1992. Conditions for unique graph realizations. SIAM Journal on Computing 21, 1 (1992), 65–84.Google ScholarDigital Library
- [5] . 1972. Finding the triconnected components of a graph. Technical Report TR 140, Dept. of Computer Science, Cornell Univ.Google Scholar
- [6] . 1973. Dividing a graph into triconnected components. SIAM J. Comput. 2, 3 (1973), 135–158.
DOI: .arXiv:https://doi.org/10.1137/0202012 Google ScholarDigital Library - [7] . 2007. Notes on the rigidity of graphs. In Levico Conference Notes, Vol. 4.Google Scholar
- [8] . 2005. Connected rigidity matroids and unique realizations of graphs. Journal of Combinatorial Theory, Series B 94, 1 (
May 2005), 1–29.DOI: Google ScholarDigital Library - [9] . 1997. An algorithm for two-dimensional rigidity percolation: The Pebble game. J. Comput. Phys. 137, 2 (1997), 346–365.Google ScholarDigital Library
- [10] . 1970. On graphs and rigidity of plane skeletal structures. Journal of Engineering Mathematics 4, 4 (
Oct. 1970), 331–340.DOI: Google ScholarCross Ref - [11] . 2010. Location, localization, and localizability. Journal of Computer Science and Technology 25, 2 (2010), 274–297.Google ScholarCross Ref
- [12] . 2004. Robust distributed network localization with noisy range measurements. In SenSys’04.
DOI: Google ScholarDigital Library - [13] . 2020. HGO: Hierarchical graph optimization for accurate, efficient, and robust network localization. In Proceedings of the 29th International Conference on Computer Communications and Networks, ICCCN. 1–9.
DOI: Google ScholarCross Ref - [14] Haodi Ping, Yongcai Wang, Xingfa Shen, Deying Li, and Wenping Chen. 2022. On node localizability identification in barycentric linear localization. ACM Trans. Sen. Netw. 19, 1, Article 19 (Dec 2022), 26 pages. Google ScholarDigital Library
- [15] H. Ping, Y. Wang, D. Li, and T. Sun. 2022. Flipping free conditions and their application in sparse network localization. IEEE Transactions on Mobile Computing 21, 3 (2022), 986–1003.Google Scholar
- [16] . 2013. Localizability of wireless sensor networks: Beyond wheel extension. In Stabilization, Safety, and Security of Distributed Systems
, , , , , and (Eds.). Lecture Notes in Computer Science, Springer International Publishing, Cham, 326–340. DOI: Google ScholarCross Ref - [17] . 2010. Rigidity, global rigidity, and graph decomposition. European Journal of Combinatorics 31, 4 (2010), 1121–1135.Google ScholarDigital Library
- [18] Ileana Streinu and Audrey Lee. 2008. Pebble game algorithms and (k, l)-sparse graphs. Discrete Mathematics & Theoretical Computer Science 308, 8 (2008), 1425–1437.
DOI: Google ScholarDigital Library - [19] . 2018. WCS: Weighted component stitching for sparse network localization. IEEE/ACM Transactions on Networking 26, 5 (2018), 2242–2253.Google ScholarDigital Library
- [20] . 1971. Depth-first search and linear graph algorithms. In Proceedings of the 12th Annual Symposium on Switching and Automata Theory (swat 1971). 114–121.
DOI: Google ScholarDigital Library - [21] . 2018. Formation tracking in sparse airborne networks. IEEE Journal on Selected Areas in Communications 36, 9 (2018), 2000–2014.Google ScholarDigital Library
- [22] . 2017. Triangle extension: Efficient localizability detection in wireless sensor networks. IEEE Transactions on Wireless Communications 16, 11 (
Nov. 2017), 7419–7431.DOI: Google ScholarCross Ref - [23] . 2010. Understanding node localizability of wireless Ad-hoc networks. In Proceedings of the 2010 IEEE INFOCOM. 1–9.
DOI: Google ScholarCross Ref - [24] . 2009. Beyond trilateration: On the localizability of wireless ad-hoc networks. In Proceedings of the IEEE INFOCOM 2009. 2392–2400.
DOI: Google ScholarCross Ref - [25] . 2019. Regular topology formation based on artificial forces for distributed mobile robotic networks. IEEE Transactions on Mobile Computing 18, 10 (2019), 2415–2429.Google ScholarCross Ref
Index Terms
- GPART: Partitioning Maximal Redundant Rigid and Maximal Global Rigid Components in Generic Distance Graphs
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