Approximate counting, uniform generation and rapidly mixing Markov chains
Information and Computation
The electrical resistance of a graph captures its commute and cover times
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
A technique for lower bounding the cover time
STOC '90 Proceedings of the twenty-second annual ACM symposium on Theory of computing
Randomized algorithms
Convergence rates for Markov chains
SIAM Review
The Markov chain Monte Carlo method: an approach to approximate counting and integration
Approximation algorithms for NP-hard problems
Random Structures & Algorithms
A tight lower bound on the cover time for random walks on graphs
Random Structures & Algorithms
Next century challenges: scalable coordination in sensor networks
MobiCom '99 Proceedings of the 5th annual ACM/IEEE international conference on Mobile computing and networking
Wireless integrated network sensors
Communications of the ACM
WSNA '02 Proceedings of the 1st ACM international workshop on Wireless sensor networks and applications
Rumor routing algorthim for sensor networks
WSNA '02 Proceedings of the 1st ACM international workshop on Wireless sensor networks and applications
The cover time of sparse random graphs.
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Random Walk for Self-Stabilizing Group Communication in Ad-Hoc Networks
SRDS '02 Proceedings of the 21st IEEE Symposium on Reliable Distributed Systems
Gossip-Based Computation of Aggregate Information
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
On the Cover Time for Random Walks on Random Graphs
Combinatorics, Probability and Computing
Efficient and robust query processing in dynamic environments using random walk techniques
Proceedings of the 3rd international symposium on Information processing in sensor networks
Sharp thresholds For monotone properties in random geometric graphs
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
The bin-covering technique for thresholding random geometric graph properties
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Opportunistic forwarding in wireless networks with duty cycling
Proceedings of the third ACM workshop on Challenged networks
The cover time of random geometric graphs
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
A packet-centric approach to distributed rateless coding in wireless sensor networks
SECON'09 Proceedings of the 6th Annual IEEE communications society conference on Sensor, Mesh and Ad Hoc Communications and Networks
Error scaling laws for linear optimal estimation from relative measurements
IEEE Transactions on Information Theory
Probabilistic quorum systems in wireless Ad Hoc networks
ACM Transactions on Computer Systems (TOCS)
Rateless packet approach for data gathering wireless sensor networks
IEEE Journal on Selected Areas in Communications - Special issue on simple wireless sensor networking solutions
Radio cover time in hyper-graphs
Proceedings of the 6th International Workshop on Foundations of Mobile Computing
Efficient broadcast on random geometric graphs
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Location-aided fast distributed consensus in wireless networks
IEEE Transactions on Information Theory
EUROCRYPT'11 Proceedings of the 30th Annual international conference on Theory and applications of cryptographic techniques: advances in cryptology
Analysis of latency of stateless opportunistic forwarding in intermittently connected networks
IEEE/ACM Transactions on Networking (TON)
Mobile geometric graphs: detection, coverage and percolation
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Fast decentralized averaging via multi-scale gossip
DCOSS'10 Proceedings of the 6th IEEE international conference on Distributed Computing in Sensor Systems
Diameter and broadcast time of random geometric graphs in arbitrary dimensions
ISAAC'11 Proceedings of the 22nd international conference on Algorithms and Computation
Radio cover time in hyper-graphs
Ad Hoc Networks
| Hi-index | 5.35 |
The cover time and mixing time of graphs has much relevance to algorithmic applications and has been extensively investigated. Recently, with the advent of ad hoc and sensor networks, an interesting class of random graphs, namely random geometric graphs, has gained new relevance and its properties have been the subject of much study. A random geometric graph G(n,r) is obtained by placing n points uniformly at random on the unit square and connecting two points iff their Euclidean distance is at most r. The phase transition behavior with respect to the radius r of such graphs has been of special interest. We show that there exists a critical radius r"o"p"t such that for any r=r"o"p"tG(n,r) has optimal cover time of @Q(nlogn) with high probability, and, importantly, r"o"p"t=@Q(r"c"o"n) where r"c"o"n denotes the critical radius guaranteeing asymptotic connectivity. Moreover, since a disconnected graph has infinite cover time, there is a phase transition and the corresponding threshold width is O(r"c"o"n). On the other hand, the radius required for rapid mixing r"r"a"p"i"d=@w(r"c"o"n), and, in particular, r"r"a"p"i"d=@Q(1/poly(logn)). We are able to draw our results by giving a tight bound on the electrical resistance and conductance of G(n,r) via certain constructed flows.