Sensor Network Localization

snl solves the sensor network problem with partial distance (squared) matrix D, and anchor positions anchors, in dimension d.

snl(D, d, anchors = NULL)

Arguments

D	The partial distance matrix specifying the known distances between nodes. If anchors is specified (and is a pxr matrix), the p final columns and p final rows specify the distances between the anchors specified in anchors.
d	the dimension for the resulting completion
anchors	a pxr matrix specifying the d dimensional locations of the p anchors. If the anchorless problem is to be solved, anchors = NULL

Value

X the d-dimensional positions of the localized sensors. Note that it may be the case that not all sensors could be localized, in which case X contains the positions of only the localized sensors.

Details

Set anchors=NULL to solve the anchorless (Euclidean distance matrix completion) problem in dimension d.

NOTE: When anchors is specified, the distances between the anchors must be in the bottom right corner of the matrix D, and anchors must have d columns.

References

Nathan Krislock and Henry Wolkowicz. Explicit sensor network localization using semidefinite representations and facial reductions. SIAM Journal on Optimization, 20(5):2679-2708, 2010.

Examples

D <- matrix(c(0,NA,.1987,NA,.0595,NA,.0159,.2251,.0036,.0875,
              NA,0,.0481,NA,NA,.0515,NA,.2079,.2230,NA,
              .1987,.0481,0,NA,NA,.1158,NA,NA,.1553,NA,
              NA,NA,NA,0,NA,NA,NA,.2319,NA,NA,
              .0595,NA,NA,NA,0,NA,.1087,.0894,.0589,.0159,
              NA,.0515,.1158,NA,NA,0,NA,NA,NA,NA,
              .0159,NA,NA,NA,.1087,NA,0,.3497,.0311,.1139,
              .2251,.2079,NA,.2319,.0894,NA,.3497,0,.1918,.1607,
              .0036,.2230,.1553,NA,.0589,NA,.0311,.1918,0,.1012,
              .0875,NA,NA,NA,.0159,NA,.1139,.1607,.1012,0),nrow=10, byrow=TRUE)
              
anchors <- matrix(c(.5131,.9326,
                    .3183,.3742,
                    .5392,.7524,
                    .2213,.7631), nrow=4,byrow=TRUE)
d <- 2

#Anchorless Problem
edmc(D, method="snl", d=2, anchors=NULL)
#> $X
#>              [,1]         [,2]
#> [1,]  0.016934663  0.048956965
#> [2,] -0.004757603 -0.030583106
#> [3,] -0.007601688  0.156167720
#> [4,]  0.007282167 -0.193730966
#> [5,]  0.041458581  0.027923602
#> [6,] -0.053316120 -0.008734216
#> 

#Anchored Problem
edmc(D, method="snl", d=2, anchors=anchors)
#> $X
#>           [,1]      [,2]
#> [1,] 0.4359152 0.7483225
#> [2,] 0.3870077 0.6819505
#>