Global Synchronization in Sensornets Jeremy Elson, Richard Karp, Christos Papadimitriou, Scott Shenker

latin, 8/4/42 Clocks in networks Difficult problem, largely ignored in practice

latin, 8/4/43 …but in sensornets: synchronization is crucial transmission delays are negligible, and so very tight synchronization can (in principle) be achieved

latin, 8/4/44 each vertex i has a local offset T i T 0 = 0 a synchronization signal is sent at (unknown) time U k received by i at time U k +T i + e ki Gaussian error with mean zero and known variance V ki

latin, 8/4/45 Need to estimate T i - T j Estimate must be globally consistent: (T i - T j ) + (T j - T k ) + (T k - T i ) = 0 Must minimize total variance (Or maximize likelihood?)

latin, 8/4/46 Variance minimization by flows The unbiased estimators of T i – T j are flows from i to j in the network The minimum variance flow corresponds to the min-cost flow with costs V ij x 2 But this is the same as the power in electric networks!

latin, 8/4/47 effective variance  effective resistance! min variance estimate  current

latin, 8/4/48 Computation? By analog computer… By approximation of min-cost flow Idea: cost = x 2 capacities = 

latin, 8/4/49 Maximum likelihood? error is Gaussian  likelihood is exp(sum of squares)  optimum is by least squares Surprise: Same estimates as with min variance! Random walk method gives same expectation (but huge variance)

latin, 8/4/410 Synchronization design Suppose that we have a network How do you design an optimum synchronization protocol which minimizes –variance of the estimates (assume is bounded) –total synchronization activity (messages)?

latin, 8/4/411 Resistive network design You are given a graph Allocate metal to the edges (resistance inversely proportional to weight) So as to achieve small effective resistance between all nodes (or small weighted sum) With the minimum amount of copper

latin, 8/4/412 Theorem: Optimum can be found in polynomial time Idea: minimize Σ x i subject to x  K K: all weight allocations x that achieve effective resistance  b Surprise: K is convex, and has a polynomial-time separation oracle

latin, 8/4/413 Also: Clocks with different drifts t i = a i t + b i can be also synchronized by a reduction to the equal drift case (as it turns out, by setting T i = log a i ) Soon-to-be-deployed distributed clock synchronization protocol based on these ideas