Abstract
Computational complexity and approximation algorithms are reported for a problem of stabbing a set of straight line segments with the least cardinality set of disks of fixed radii \(r>0\) where the set of segments forms a straight line drawing \(G=(V,E)\) of a planar graph without edge crossings. Close geometric problems arise in network security applications. We give strong NP-hardness of the problem for edge sets of Delaunay triangulations, Gabriel graphs and other subgraphs (which are often used in network design) for \(r\in [d_{\min },\eta d_{\max }]\) and some constant \(\eta \) where \(d_{\max }\) and \(d_{\min }\) are Euclidean lengths of the longest and shortest graph edges respectively. Fast \(O(|E|\log |E|)\)-time O(1)-approximation algorithm is proposed within the class of straight line drawings of planar graphs for which the inequality \(r\ge \eta d_{\max }\) holds uniformly for some constant \(\eta >0,\) i.e. when lengths of edges of G are uniformly bounded from above by some linear function of r.
Keywords
This work was supported by Russian Science Foundation, project 14-11-00109.
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- 1.
A graph without loops and parallel edges.
- 2.
\({\mathrm {int}}\,N\) is the set of interior points of N.
- 3.
When U coincides with some prescribed finite set.
- 4.
\(\mathrm{{bd}}\,T\) denotes the set of boundary points of T.
References
Agarwal, P.K., Efrat, A., Ganjugunte, S.K., Hay, D., Sankararaman, S., Zussman, G.: The resilience of WDM networks to probabilistic geographical failures. IEEE/ACM Trans. Netw. 21(5), 1525–1538 (2013)
Arkin, E.M., Anta, A.F., Mitchell, J.S., Mosteiro, M.A.: Probabilistic bounds on the length of a longest edge in Delaunay graphs of random points in \(d\) dimensions. Comput. Geom. 48(2), 134–146 (2015)
Bhattacharya, B.K., Jadhav, S., Mukhopadhyay, A., Robert, J.M.: Optimal algorithms for some intersection radius problems. Computing 52(3), 269–279 (1994)
Bose, P., De Carufel, J.-L., Durocher, S., Taslakian, P.: Competitive online routing on delaunay triangulations. In: Ravi, R., Gørtz, I.L. (eds.) SWAT 2014. LNCS, vol. 8503, pp. 98–109. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08404-6_9
Bose, P., Kirkpatrick, D.G., Li, Z.: Worst-case-optimal algorithms for guarding planar graphs and polyhedral surfaces. Comput. Geom. 26(3), 209–219 (2003)
Chan, T.M., Grant, E.: Exact algorithms and APX-hardness results for geometric packing and covering problems. Comput. Geom. 47(2), 112–124 (2014)
Efrat, A., Katz, M.J., Nielsen, F., Sharir, M.: Dynamic data structures for fat objects and their applications. Comput. Geom. 15, 215–227 (2000)
Gonzalez, T.F.: Covering a set of points in multidimensional space. Inf. Process. Lett. 40(4), 181–188 (1991)
Hasegawa, T., Masuyama, S., Ibaraki, T.: Computational complexity of the \(m\)-center problems on the plane. Trans. Inst. Electron. Commun. Eng. Japan Sect. E 64(2), 57–64 (1981)
Marx, D.: Efficient approximation schemes for geometric problems? In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 448–459. Springer, Heidelberg (2005). https://doi.org/10.1007/11561071_41
Onoyama, T., Sibuya, M., Tanaka, H.: Limit distribution of the minimum distance between independent and identically distributed d-dimensional random variables. In: de Oliveira, J.T. (ed.) Statistical Extremes and Applications. NATO ASI Series (Series C: Mathematical and Physical Sciences), vol. 131, pp. 549–562. Springer, Dordrecht (1984). https://doi.org/10.1007/978-94-017-3069-3_42
O’Rourke, J.: Art Gallery Theorems and Algorithms. Oxford University Press, New York (1987)
Har-Peled, S., Quanrud, K.: Approximation algorithms for polynomial-expansion and low-density graphs. In: Bansal, N., Finocchi, I. (eds.) ESA 2015. LNCS, vol. 9294, pp. 717–728. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48350-3_60
Stojmenovic, I., Urrutia, J., Bose, P., Morin, P.: Routing with guaranteed delivery in ad hoc wireless networks. Wirel. Netw. 7(6), 609–616 (2001)
Tamassia, R., Tollis, I.G.: Planar grid embedding in linear time. IEEE Trans. Circ. Syst. 36, 1230–1234 (1989)
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A Proof of the Lemma 1
A Proof of the Lemma 1
Proof
Let \(u=(x,y),\,v=(x_1,y_1)\) and \(w=(x_2,y_2)\) be distinct points of X. Consider an arbitrary radius r circle (out of two circles) which passes through v and w, and denote its center by O. A lower bound is obtained below for the distance \(\pi =\pi (u;v,w)\) from that circle to the point \(u\notin C(v,w).\)
Let \(\varDelta =|v-w|_2,\) \(\lambda =\sqrt{r^2-\frac{ \varDelta ^2}{ 4}}\), \(a=(u-v,u-w)\) and \(b=(u-v,(v-w)^{\perp })\), where \((v-w)^{\perp }=\pm (y_1-y_2,-x_1+x_2)\). The distance \(\pi >0\) can be written in the form:
Without loss of generality it is assumed that u is in the 2r radius disk centered at O. Indeed, we get \(\pi \ge r\ge \frac{1}{r}\) otherwise. Let us bound denominator of the fraction \(\pi ,\) taking into account that \(\varDelta \le 2r,\) \(|u-v|_2\le |u-O|_2+|O-v|_2\le 3r\) and \(|b|/\varDelta \le 3r:\)
As points of X have integer coordinates, a and b are integers. For \(\varDelta ^2=4r^2\) we get \(\pi \ge \frac{ 1}{ 5r}\). When \(\varDelta ^2\le 4r^2-1,\) it is enough to prove the inequality
Indeed, again, combining this bound with the aforementioned upper bound for denominator of the fraction \(\pi ,\) we get \(\pi \ge \frac{1}{480r^5}\).
For integer \(\frac{2\lambda b}{\varDelta }\) the left-hand side of the inequality (1) is at least 1. Thus, it remains for us to prove the inequality (1) for the case where \(\frac{2\lambda b}{\varDelta }\notin \mathbb {Z}\). Suppose that \(q=\left\{ \left| \frac{2\lambda b}{\varDelta }\right| \right\} >0\) and \(k=\left[ \left| \frac{2\lambda b}{\varDelta }\right| \right] ,\) where \(\{\cdot \}\) and \([\cdot ]\) denote fractional and integer part of real number respectively. In fact, the term \(\min \{q,1-q\}\) can be bounded from below. Let us start estimating with q. First, it is assumed that \(\gamma =\frac{ 4r^2b^2}{ \varDelta ^2}\in ~\mathbb {Z}.\) We have \(k^2<\frac{ 4\lambda ^2b^2}{ \varDelta ^2}<(k+1)^2.\) As \(q>0,\) we get \(q\ge \left\{ \sqrt{k^2+1}\right\} .\) Due to concavity of the square root we have
Now the case is considered where \(\gamma \notin \mathbb {Z}\). As \(2kq+q^2\ge \{2kq+q^2\}=\{\gamma \}\), we have that
Let us get a lower bound for \(1-q\). Again, assume that \(\gamma \in \mathbb {Z}\). Arguing analogously, we arrive at the bound
Resolving the quadratic inequality with respect to \(1-q\), we get:
Now let \(\gamma \notin \mathbb {Z}\). Let us consider the subcase, where \(\{\gamma \}+2q(1-q)>1\). We get
Resolving the corresponding inequality with respect to \(1-q\), we arrive at the analogous lower bound \(1-q\ge \frac{1}{48r^4}\).
Now we are to address the case where \(\{\gamma \}+2q(1-q)<1\). Obviously,
For \(1-q<\frac{ 1}{ 4\varDelta ^2}\) we have \(1-\{\gamma \}-2q(1-q) \ge \frac{1}{\varDelta ^2}-\frac{1}{2\varDelta ^2} =\frac{ 1}{ 2\varDelta ^2}.\) Arguing analogously, we obtain the following bound \(1-q\ge \frac{ 1}{ 96r^4}\); otherwise, we get \(1-q\ge \frac{ 1}{ 4\varDelta ^2}\ge \frac{ 1}{ 16r^2}\).
For \(\{\gamma \}+2q(1-q)=1\) we have:
Finally, we arrive at the claimed bound:
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Kobylkin, K. (2018). Stabbing Line Segments with Disks: Complexity and Approximation Algorithms. In: van der Aalst, W., et al. Analysis of Images, Social Networks and Texts. AIST 2017. Lecture Notes in Computer Science(), vol 10716. Springer, Cham. https://doi.org/10.1007/978-3-319-73013-4_33
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