\begin{thebibliography}{10} \bibitem{AHU74} A.~V. Aho, J.~E. Hopcroft, and J.~D. Ullman. \newblock {\em The Design and Analysis of Computer Algorithms}. \newblock Addison-Wesley Publishing Company, 1974. \bibitem{Aleksandrov1996} L.~Aleksandrov and H.~Djidjev. \newblock Linear algorithms for partitioning embedded graphs of bounded genus. \newblock {\em SIAM J. Discret. Math.}, 9(1):129--150, February 1996. \bibitem{Aleksandrov2007} Lyudmil Aleksandrov, Hristo Djidjev, Hua Guo, and Anil Maheshwari. \newblock Partitioning planar graphs with costs and weights. \newblock {\em J. Exp. Algorithmics}, 11, February 2007. \bibitem{alon2004probabilistic} N.~Alon and J.~H. Spencer. \newblock {\em The probabilistic method}. \newblock John Wiley \& Sons, 3rd edition, 2007. \bibitem{Indyk06} A.~Andoni and P.~Indyk. \newblock Near-optimal hashing algorithms for approximate nearest neighbor in high dimensions. \newblock In {\em Foundations of Computer Science, 2006. FOCS '06. 47th Annual IEEE Symposium on}, pages 459--468, 2006. \bibitem{Appel2002} Martin J.~B. Appel and Ralph~P. Russo. \newblock {The connectivity of a graph on uniform points on $[0,1]^d$}. \newblock {\em Statistics \& Probability Letters}, 60(4):351--357, 2002. \bibitem{balister2008percolation} P.~Balister, A.~Sarkar, and B.~Bollob{\'a}s. \newblock Percolation, connectivity, coverage and colouring of random geometric graphs. \newblock In {\em Handbook of Large-Scale Random Networks}, pages 117--142. Springer, 2008. \bibitem{BKMS15} Ahmad Biniaz, Evangelos Kranakis, Anil Maheshwari, and Michiel Smid. \newblock Plane and planarity thresholds for random geometric graphs. \newblock In {\em Proc.\ ALGOSENSORS 2015 (Patras, Greece)}, Lecture Notes in Computer Science, Berlin, Germany, 2015. Springer. \bibitem{bollobas2001random} B.~Bollob{\'a}s. \newblock {\em Random graphs}. \newblock Cambridge University Press, 2001. \bibitem{Bollobas1987} B{\'{e}}la Bollob{\'{a}}s and Andrew Thomason. \newblock Threshold functions. \newblock {\em Combinatorica}, 7(1):35--38, 1987. \bibitem{Bollobas1998} Béla Bollobás. \newblock {\em Modern Graph Theory}. \newblock Graduate texts in mathematics. Springer, Heidelberg, corrected edition, 1998. \bibitem{Bondy1976} J.~A. Bondy and U.~S.~R. Murty. \newblock {\em Graph Theory with Applications}. \newblock Elsevier, New York, 1976. \bibitem{boruuvka1926a} Otakar Borůvka. \newblock O jist{\'e}m probl{\'e}mu minim{\'a}ln{\'\i}m. \newblock {\em Práce mor. přírodověd. spol. v Brně III}, 3:37--58, 1926. \bibitem{boruuvka1926b} Otakar Borůvka. \newblock P{\v{r}}{\'\i}sp{\v{e}}vek k {\v{r}}e{\v{s}}en{\'\i} ot{\'a}zky ekonomick{\'e} stavby elektrovodn{\'\i}ch s{\'\i}t{\'\i}. \newblock {\em Elektrotechnick{\`y} obzor}, 15:153--154, 1926. \bibitem{Bou85} J.~Bourgain. \newblock On lipschitz embedding of finite metric spaces in hilbert space. \newblock {\em Israel Journal of Mathematics}, 52(1-2):46--52, 1985. \bibitem{Bourgain1998} Jean Bourgain and Gil Kalai. \newblock Threshold intervals under group symmetries. \newblock {\em Convex Geometric Analysis MSRI Publications Volume 34}, pages 59--63, 1998. \bibitem{Bradonjic2014} Milan Bradonji\'{c} and Will Perkins. \newblock On sharp thresholds in random geometric graphs. \newblock In {\em Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, {APPROX/RANDOM}}, pages 500--514, 2014. \bibitem{Broder98} Andrei~Z. Broder, Moses Charikar, Alan~M. Frieze, and Michael Mitzenmacher. \newblock Min-wise independent permutations. \newblock {\em Journal of Computer and System Sciences}, 60:327--336, 1998. \bibitem{Broder97} A.Z. Broder. \newblock On the resemblance and containment of documents. \newblock In {\em Compression and Complexity of Sequences 1997. Proceedings}, pages 21--29, 1997. \bibitem{chan1998} Timothy~M. Chan. \newblock Backwards analysis of the {K}arger-{K}lein-{T}arjan algorithm for minimum spanning. \newblock {\em Inf. Process. Lett.}, 67(6):303--304, 1998. \bibitem{CLRS} T.~H. Cormen, C.~E. Leiserson, R.~L. Rivest, and C.~Stein. \newblock {\em Introduction to Algorithms}. \newblock The MIT Press, 3rd edition, 2009. \bibitem{clrs2009} Thomas~H. Cormen, Charles~E. Leiserson, Ronald~L. Rivest, and Clifford Stein. \newblock {\em Introduction to Algorithms, Third Edition}. \newblock The MIT Press, 3rd edition, 2009. \bibitem{DG02} Sanjoy Dasgupta and Anupam Gupta. \newblock An elementary proof of a theorem of {J}ohnson and {L}indenstrauss. \newblock {\em Random Struct. Algorithms}, 22(1):60--65, 2003. \bibitem{Indyk04} Mayur Datar and Piotr Indyk. \newblock Locality-sensitive hashing scheme based on p-stable distributions. \newblock In {\em In SCG ’04: Proceedings of the twentieth annual symposium on Computational geometry}, pages 253--262. ACM Press, 2004. \bibitem{dePaula10} L.B. de~Paula, R.S. Villaca, and M.F. Magalhaes. \newblock A locality sensitive hashing approach for conceptual classification. \newblock In {\em Semantic Computing (ICSC), 2010 IEEE Fourth International Conference on}, pages 408--413, 2010. \bibitem{dijkstra1959} E.W. Dijkstra. \newblock A note on two problems in connexion with graphs. \newblock {\em Numerische Mathematik}, 1(1):269--271, 1959. \bibitem{dixon1992} B.~Dixon, M.~Rauch, and R.~Tarjan. \newblock Verification and sensitivity analysis of minimum spanning trees in linear time. \newblock {\em SIAM Journal on Computing}, 21(6):1184--1192, 1992. \bibitem{DP09} D.P. Dubhashi and A.~Panconesi. \newblock {\em Concentration of measure for the analysis of randomized algorithms}. \newblock Cambridge University Press, 2009. \bibitem{erd6s1960evolution} P.~Erd\"{o}s and A.~R{\'e}nyi. \newblock On the evolution of random graphs. \newblock {\em Publ. Math. Inst. Hungar. Acad. Sci}, 5:17--61, 1960. \bibitem{Even79} Shimon Even. \newblock {\em Graph Algorithms}. \newblock W. H. Freeman \& Co., New York, NY, USA, 1979. \bibitem{Frederickson1987} Greg~N. Frederickson. \newblock Fast algorithms for shortest paths in planar graphs, with applications. \newblock {\em SIAM J. Comput.}, 16(6):1004--1022, 1987. \bibitem{Frederickson1991} Greg~N. Frederickson. \newblock Optimal algorithms for tree partitioning. \newblock In {\em Proceedings of the Second Annual ACM-SIAM Symposium on Discrete Algorithms}, SODA '91, pages 168--177, Philadelphia, PA, USA, 1991. Society for Industrial and Applied Mathematics. \bibitem{friedgut1996every} E.~Friedgut and G.~Kalai. \newblock Every monotone graph property has a sharp threshold. \newblock {\em Proceedings of the American Mathematical Society}, 124(10):2993--3002, 1996. \bibitem{GS72} D.~Gale and L.~S. Shapley. \newblock College admissions and the stability of marriage. \newblock {\em The American Mathematical Monthly}, 69(1):9--15, 1972. \bibitem{gilbert1959random} E.~Gilbert. \newblock Random graphs. \newblock {\em The Annals of Mathematical Statistics}, pages 1141--1144, 1959. \bibitem{gilbert1961random} E.~Gilbert. \newblock Random plane networks. \newblock {\em Journal of the Society for Industrial \& Applied Mathematics}, 9(4):533--543, 1961. \bibitem{Indyk97} Aristides Gionis, Piotr Indyk, and Rajeev Motwani. \newblock Similarity search in high dimensions via hashing. \newblock pages 518--529, 1997. \bibitem{Godehardt1996} Erhard Godehardt and Jerzy Jaworski. \newblock On the connectivity of a random interval graph. \newblock {\em Random Struct. Algorithms}, 9(1-2):137--161, 1996. \bibitem{goel2004sharp} A.~Goel, S.~Rai, and B.~Krishnamachari. \newblock Sharp thresholds for monotone properties in random geometric graphs. \newblock In {\em Proceedings of STOC}, pages 580--586. ACM, 2004. \bibitem{Gorisse12} D.~Gorisse, M.~Cord, and F.~Precioso. \newblock Locality-sensitive hashing for chi2 distance. \newblock {\em Pattern Analysis and Machine Intelligence, IEEE Transactions on}, 34(2):402--409, 2012. \bibitem{Gupta1998} Piyush Gupta and P.~R. Kumar. \newblock Critical power for asymptotic connectivity in wireless networks. \newblock In {\em Stochastic Analysis, Control, Optimization and Applications}, pages 547--566, 1998. \bibitem{hagerup2010} Torben Hagerup. \newblock An even simpler linear-time algorithm for verifying minimum spanning trees. \newblock In Christophe Paul and Michel Habib, editors, {\em Graph-Theoretic Concepts in Computer Science}, volume 5911 of {\em Lecture Notes in Computer Science}, pages 178--189. Springer Berlin Heidelberg, 2010. \bibitem{HR88} Torben Hagerup and Christine R{\"u}b. \newblock A guided tour of {C}hernoff bounds. \newblock {\em Inf. Process. Lett.}, 33(6):305--308, 1990. \bibitem{hall1985coverage} P.~Hall. \newblock On the coverage of $ k $-dimensional space by $ k $-dimensional spheres. \newblock {\em The Annals of Probability}, 13(3):991--1002, 1985. \bibitem{H-Book69} Frank Harary. \newblock {\em Graph theory}. \newblock Addison-Wesley, 1991. \bibitem{harel1984} D.~Harel and R.~Tarjan. \newblock Fast algorithms for finding nearest common ancestors. \newblock {\em SIAM Journal on Computing}, 13(2):338--355, 1984. \bibitem{harel1980} Dov Harel. \newblock A linear time algorithm for the lowest common ancestors problem. \newblock In {\em Foundations of Computer Science, 1980., 21st Annual Symposium on}, pages 308--319, 1980. \bibitem{Hu05} S.~Hu. \newblock Efficient video retrieval by locality sensitive hashing. \newblock In {\em Acoustics, Speech, and Signal Processing, 2005. Proceedings. (ICASSP '05). IEEE International Conference on}, volume~2, pages 449--452, 2005. \bibitem{Indyk98} Piotr Indyk and Rajeev Motwani. \newblock Approximate nearest neighbors: Towards removing the curse of dimensionality. \newblock In {\em Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing}, STOC '98, pages 604--613, New York, NY, USA, 1998. ACM. \bibitem{Ir85} Robert~W. Irving. \newblock An efficient algorithm for the "stable roommates" problem. \newblock {\em J. Algorithms}, 6(4):577--595, 1985. \bibitem{janson1986random} S.~Janson. \newblock Random coverings in several dimensions. \newblock {\em Acta Mathematica}, 156(1):83--118, 1986. \bibitem{Jeon12} Woojay Jeon and Yan-Ming Cheng. \newblock Efficient speaker search over large populations using kernelized locality-sensitive hashing. \newblock In {\em Acoustics, Speech and Signal Processing (ICASSP), 2012 IEEE International Conference on}, pages 4261--4264, 2012. \bibitem{Jing08} Yushi Jing and S.~Baluja. \newblock Visualrank: Applying pagerank to large-scale image search. \newblock {\em Pattern Analysis and Machine Intelligence, IEEE Transactions on}, 30(11):1877--1890, 2008. \bibitem{Kala13} K.A. Kala and K.~Chitharanjan. \newblock Locality sensitive hashing based incremental clustering for creating affinity groups in hadoop; hdfs - an infrastructure extension. \newblock In {\em Circuits, Power and Computing Technologies (ICCPCT), 2013 International Conference on}, pages 1243--1249, 2013. \bibitem{Kang04} Zixiang Kang, Wei~Tsang Ooi, and Qibin Sun. \newblock Hierarchical, non-uniform locality sensitive hashing and its application to video identification. \newblock In {\em Multimedia and Expo, 2004. ICME '04. 2004 IEEE International Conference on}, volume~1, pages 743--746 Vol.1, 2004. \bibitem{karger1995} David~R. Karger, Philip~N. Klein, and Robert~E. Tarjan. \newblock A randomized linear-time algorithm to find minimum spanning trees. \newblock {\em J. ACM}, 42(2):321--328, March 1995. \bibitem{king1997} V.~King. \newblock A simpler minimum spanning tree verification algorithm. \newblock {\em Algorithmica}, 18(2):263--270, 1997. \bibitem{king1995} Valerie King. \newblock A simpler minimum spanning tree verification algorithm. \newblock In SelimG. Akl, Frank Dehne, Jörg-Rüdiger Sack, and Nicola Santoro, editors, {\em Algorithms and Data Structures}, volume 955 of {\em Lecture Notes in Computer Science}, pages 440--448. Springer Berlin Heidelberg, 1995. \bibitem{klein1994} Philip~N. Klein and Robert~E. Tarjan. \newblock A randomized linear-time algorithm for finding minimum spanning trees. \newblock In {\em Proceedings of the twenty-sixth annual ACM symposium on Theory of computing}, STOC '94, pages 9--15, New York, NY, USA, 1994. ACM. \bibitem{KT05} Jon Kleinberg and Eva Tardos. \newblock {\em Algorithm Design}. \newblock Addison-Wesley Longman Publishing Co., Inc., Boston, MA, USA, 2005. \bibitem{K1} Donald~E. Knuth. \newblock {\em The art of computer programming, volume 1-3}. \newblock Addison Wesley Longman Publishing Co., Inc., Redwood City, CA, USA, 1998. \bibitem{komlos1984} J.~Komlos. \newblock Linear verification for spanning trees. \newblock In {\em Foundations of Computer Science, 1984. 25th Annual Symposium on}, pages 201--206, 1984. \bibitem{komlos1985} J.~Komlós. \newblock Linear verification for spanning trees. \newblock {\em Combinatorica}, 5(1):57--65, 1985. \bibitem{Kozen} D.~Kozen. \newblock {\em The design and analysis of algorithms}. \newblock Springer, 1992. \bibitem{Krishnamachari2002} Bhaskar Krishnamachari, Stephen~B. Wicker, R\'{a}mon B\'{e}jar, and Marc Pearlman. \newblock Critical density thresholds in distributed wireless networks. \newblock In {\em Communications, information and network security}, 2002. \bibitem{Kulis12} B.~Kulis and K.~Grauman. \newblock Kernelized locality-sensitive hashing. \newblock {\em Pattern Analysis and Machine Intelligence, IEEE Transactions on}, 34(6):1092--1104, 2012. \bibitem{LT79} Richard~J. Lipton and Robert~E. Tarjan. \newblock A separator theorem for planar graphs. \newblock {\em SIAM Journal on Applied Mathematics}, 36(2):177--189, 1979. \bibitem{LP-Matching86} L.~Lov\'asz and M.D. Plummer. \newblock {\em Matching Theory}. \newblock Akad\'emiai Kiad\'o, Budapest, 1986. \newblock Also published as Vol. 121 of the North-Holland Mathematics Studies, North-Holland Publishing, Amsterdam. \bibitem{AM12} A.~Maheshwari and M.~Smid. \newblock {\em Introduction to Theory of Computation}. \newblock Free Online, 2012. \bibitem{Manber94} Udi Manber. \newblock Finding similar files in a large file system. \newblock In {\em in Proceedings of the USENIX Winter 1994 Technical Conference}, pages 1--10, 1994. \bibitem{Mat02} Jiri Matousek. \newblock {\em Lectures on Discrete Geometry}. \newblock Springer-Verlag New York, Inc., Secaucus, NJ, USA, 2002. \bibitem{Mat96} Jiří Matoušek. \newblock On the distortion required for embedding finite metric spaces into normed spaces. \newblock {\em Israel Journal of Mathematics}, 93(1):333--344, 1996. \bibitem{Mccolm2001} Gregory~L. Mccolm. \newblock Threshold functions for random graphs on a line segment. \newblock {\em Combinatorics, Probability and Computing}, 13:373--387, 2001. \bibitem{M70} P.L. Meyer. \newblock {\em Introductory probability and statistical applications}. \newblock Addison-Wesley, Boston, MA, USA, 1970. \bibitem{Miller1997} Gary~L. Miller, Shang-Hua Teng, William Thurston, and Stephen~A. Vavasis. \newblock Separators for sphere-packings and nearest neighbor graphs. \newblock {\em J. ACM}, 44(1):1--29, January 1997. \bibitem{MU05} Michael Mitzenmacher and Eli Upfal. \newblock {\em Probability and Computing: Randomized Algorithms and Probabilistic Analysis}. \newblock Cambridge University Press, New York, NY, USA, 2005. \bibitem{Mondal13} Tanmoy Mondal, Nicolas Ragot, Jean-Yves Ramel, and Umapada Pal. \newblock A fast word retrieval technique based on kernelized locality sensitive hashing. \newblock In {\em Document Analysis and Recognition (ICDAR), 2013 12th International Conference on}, pages 1195--1199, 2013. \bibitem{MR95} Rajeev Motwani and Prabhakar Raghavan. \newblock {\em Randomized algorithms}. \newblock Cambridge University Press, New York, NY, USA, 1995. \bibitem{motwani1995} Rajeev Motwani and Prabhakar Raghavan. \newblock {\em Randomized algorithms}. \newblock Cambridge University Press, New York, NY, USA, 1995. \bibitem{nesetril2001} Jaroslav Nešetřil, Eva Milková, and Helena Nešetřilová. \newblock Otakar borůvka on minimum spanning tree problem translation of both the 1926 papers, comments, history. \newblock {\em Discrete Mathematics}, 233(1–3):3 -- 36, 2001. \newblock Czech and Slovak 2. \bibitem{Panchapakesan2001} Padma Panchapakesan and D~Manjunath. \newblock On the transmission range in dense ad hoc radio networks. \newblock In {\em Proceedings of IEEE Signal Processing Communication (SPCOM)}, 2001. \bibitem{penrose1997longest} Mathew~D. Penrose. \newblock The longest edge of the random minimal spanning tree. \newblock {\em The annals of applied probability}, pages 340--361, 1997. \bibitem{Penrose1999} Mathew~D. Penrose. \newblock On $k$-connectivity for a geometric random graph. \newblock {\em Random Struct. Algorithms}, 15(2):145--164, 1999. \bibitem{penrose2003random} Mathew~D. Penrose. \newblock {\em Random geometric graphs}, volume~5. \newblock Oxford University Press Oxford, 2003. \bibitem{RLU12} Anand Rajaraman and Jeffrey~David Ullman. \newblock {\em Mining of Massive Datasets}. \newblock Cambridge University Press, New York, NY, USA, 2011. \bibitem{Rasheed12} Z.~Rasheed, H.~Rangwala, and D.~Barbara. \newblock Lsh-div: Species diversity estimation using locality sensitive hashing. \newblock In {\em Bioinformatics and Biomedicine (BIBM), 2012 IEEE International Conference on}, pages 1--6, 2012. \bibitem{Rosen} Kenneth~H. Rosen. \newblock {\em Discrete Mathematics and Its Applications}. \newblock McGraw-Hill Higher Education, 5th edition, 2002. \bibitem{schieber1988} Baruch Schieber and Uzi Vishkin. \newblock On finding lowest common ancestors: Simplification and parallelization. \newblock In JohnH. Reif, editor, {\em VLSI Algorithms and Architectures}, volume 319 of {\em Lecture Notes in Computer Science}, pages 111--123. Springer New York, 1988. \bibitem{spencer1987ten} J.~H. Spencer. \newblock {\em Ten lectures on the probabilistic method}, volume~52. \newblock SIAM, 1987. \bibitem{STRA69} V.~Strassen. \newblock Gaussian elimination is not optimal. \newblock {\em Numerische Mathematik}, 13:354--356, 1969. \bibitem{tarjan1983} Robert Tarjan. \newblock {\em Data Structures and Network Algorithms}, volume~44, chapter 6. Minimum Spanning Trees, pages 71--83. \newblock SIAM, 1983. \bibitem{tarjan1975} Robert~Endre Tarjan. \newblock Efficiency of a good but not linear set union algorithm. \newblock {\em J. ACM}, 22(2):215--225, April 1975. \bibitem{tarjan1979} Robert~Endre Tarjan. \newblock Applications of path compression on balanced trees. \newblock {\em J. ACM}, 26(4):690--715, October 1979. \bibitem{Wang12} Qiang Wang, Zhiyuan Guo, Gang Liu, and Jun Guo. \newblock Entropy based locality sensitive hashing. \newblock In {\em Acoustics, Speech and Signal Processing (ICASSP), 2012 IEEE International Conference on}, pages 1045--1048, 2012. \end{thebibliography}