Virtual Edge Weights

The most critical piece of solving for mutually-visible deployments is designing the edge weights that give rise to the cost of the shortest paths and Steiner trees. These

weights give us the opportunity to define what we consider to be desirable config- urations of robots. We will focus on three separate considerations: keeping a tight deployment, making use of larger visibility cells whenever possible, and using as few robots as possible.

To this end, we define our virtual edge weight as follows, with each of the three costs to be discussed next:

Wv(i, j) = ca(i, j) +cb(i, j) +cc(i, j) (19)

Distance Cost

For keeping a tight deployment, we would like to prefer that given two pairs of visible cells, the pair that are closer together are given preference. To this end, we will define the first cost term as the distance between the centroids of the visibility cells involved:

ca(i, j) = kakcent(Ci)−cent(Cj)k2 (20)

where ka is a coefficient used to weight this first cost term versus the others.

Area Cost

Since we would like to prefer pairs of mutually visible cells where one or both of the cells are larger, we need a cost term that scales inversely with cell area. Two costs that achieve this goal are a simple inverse area and a negative exponential:

cb(i, j) =kb 1 area(Ci) + 1 area(Cj) (21)

cb(i, j) = kb(exp(−area(Ci)) +exp(−area(Cj))) (22)

where kb is again a coefficient to weight this cost term relative to the others.

To compare these two, Fig. 25 presents a histogram of the inverse-area cost, while Fig. 26 presents a histogram of the negative-exponential cost, both taken over all mutually-visible cell pairs in the aspect decomposition of Fig. 21.

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 0 2000 4000 6000 8000 10000 12000 0 2 4 6 8 10 12 14 16 18 20 0 50 100 150 200 250 300 350 400 450 500

Figure 25: Histogram of the inverse area measure for the MVG of the decomposed environ- ment seen in Fig. 21. The view on the right is a truncated version since the distribution has a heavy tail. The mean is 191.02 and the standard deviation is 1.383e3.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 50 100 150 200 250 300 350 400 450 500

Figure 26: Histogram of the negative exponential measure (de) for the MVG of the decom-

posed environment seen in Fig. 21. The mean is 1.25 and the standard deviation is 0.46. Note that by definition the measure cannot exceed 2.

The inverse area cost results in very explicit separation between visibility pairs with nominal areas and those with very small areas, as evidenced by the heavy tail on the distribution. The negative exponential cost, by comparison, is very smooth and has a hard limit of 2 by definition.

We will make use of the negative exponential cost. The relative weighting between the negative exponential and centroid distance costs will be discussed shortly. Minimum Deployment

The preference to minimize the robot count is realized by simply adding a large penalty term to each edge. The need for this is seen in Fig. 27. Without a penalty term, many small hops are preferred to a few large hops because the closer centroid distances result in a shorter path. Though including costs related to the area can mitigate this to some extent, we want the preference to be strong. To this end, we define the deployment cost as:

cc(i, j) = Z, (23)

Z > sup

i,j (c

a(i, j) +cb(i, j)) (24)

By ensuring that is Z is larger than the other costs combined, we are guaranteed

that at no point will it be preferable to use two visibility hops where one will suffice. Combining Costs

Consider the histograms of the centroid distance and negative exponential area costs in Fig. 28. We need to combine the two to make the edge weights prefer both tight deployments and the use of larger visibility cells. However, the relative scaling between the two is important since weighting one too heavily will make the other

meaningless. Set ka = 1 and consider just the changing of kb. The combined cost

Wv(i, j) =kcent(Ci)−cent(Cj)k2+kb(exp(−area(Ci))+exp(−area(Cj))) for various

Figure 27: Using only centroid distances as the edge weighting, the resulting shortest path can involve taking many small hops instead of fewer large hops. By adding a large penalty term to each edge, in this case the length of the diagonal of the entire area, it will always be preferable to use the smallest number of hops.

0 2 4 6 8 10 12 14 16 18 20 0 100 200 300 400 500 600 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 50 100 150 200 250 300 350 400 450 500

Figure 28: The statistics for the centroid distance (left) and negative exponential area (right) costs from the MVG for the decomposition of Fig. 21. We need to mix the two to obtain a cost that stresses both tight deployments and the use of larger visibility cells.

that of the centroid distance alone to that of the negative-exponential area alone. Expressing the preferences of both requires some in-between histogram.

0 2 4 6 8 10 12 14 16 18 20 0 100 200 300 400 500 600 0 5 10 15 20 25 0 100 200 300 400 500 600 0 5 10 15 20 25 0 100 200 300 400 500 600 700 800 0 5 10 15 20 25 30 35 0 100 200 300 400 500 600 700 0 10 20 30 40 50 60 70 80 0 100 200 300 400 500 600 0 50 100 150 200 250 0 50 100 150 200 250 300 350 400 450 500

Figure 29: The statistics using the measure Wv(i, j) = kcent(Ci) − cent(Cj)k2 +

kb(exp(−area(Ci)) +exp(−area(Cj))) for a range of kb values. Across from top left to

bottom right, the values are kb = 0, 1, 5, 10, 30, and 100. Note how the distributions go

from that of the left histogram of Fig. 28 to the right histogram as kb becomes larger and

the negative-exponential-area measure is more represented (albeit scaled).

The ultimate goal is to manage the relative weighting so that both costs are of the same magnitude on average. One way to do this is to divide each cost by its maximum value to normalize it, and then they can be averaged together on equal scaling. This approach results in the histogram of Fig. 30. Note that the result looks similar to a Gaussian.

The normalized approach results in the following virtual edge weight:

Wv(i, j) =1

2

kcent(C

i)−cent(Cj)k2

maxi,jkcent(Ci)−cent(Cj)k2

. . . (25)

+ exp(−area(Ci)) +exp(−area(Cj))

maxi,j(exp(−area(Ci)) +exp(−area(Cj)))

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 100 200 300 400 500 600 700

Figure 30: Each cost (ca, cb) was normalized by dividing by its max and then the total

measure was taken as the average of the two, resulting in this almost Gaussian distribution.

combining all of the desired preferences into a weight that balances between tighter deployments while preferring larger cells and minimizing the number of deployed robots.