We now outline the idea of the proof. In the following, we show that every subsequent pass during which the single ants move from subroot r1 to subroot g0 at least doubles the u-value of subroot g0. Thus, the total number of edge traversals grows exponentially in the number of down passes that move the single ants from subroot r1 to subroot g0. The down pass is over when the single ants switch direction and move from subroot g0 back to subroot r1.
To ensure that this number is large, we need to ensure that the number of down passes is large and thus that many up passes end before the goal vertex is reached. Assume that the second pass during which the single ants visit subroot gi is pass p.
This pass must be an up pass since pass 0 is a down pass and ends at subroot g0. Consequently, there are a large number of up and thus also down passes. Subroot ri has been visited at least once after pass 0 and thus its u-value is at least two. According to our tie breaking rules, the next subroot that the ants move to is subroot gi. Thus, the up pass ends and the single ants move towards subroot g0 again. The purpose of the particular topology of the graph then is to ensure that the single ants move from subroot r1 to subroot g0 during each down pass and do not stop the down pass earlier.
In the following, we provide a sketch of the proof that this is indeed the case. Lemma 9. Theorem Each down pass ends at subroot g0. In the following, we need two inequalities. Theorem 11 follows. We report only results for small m because the number of edge traversals grows quickly.
How the number of edge traversals and its lower bound relate is shown in Figure Visual inspection of the graphs suggests that the lower bound is approximately at a constant distance from the number of edge traversals in the log plot, suggesting that the lower bound underestimates the number of edge traversals by roughly a constant factor.
Theorem 11 can easily be extended to groups of k ants for any constant k. In this case, we replicate our example graph k times, see Figure 9 right. We introduce a new start vertex, with undirected edges between it and all old start vertices. If k ants that use Node Counting are started in the new start vertex, each ant can move to a di erent one of the old start vertices and then exhibit the behavior described above in the context of single ants.
Corollary Sample Eulerian Graph and its Conversion can be exponential in the square root of the number of vertices even for planar undirected trees. The cover time of a given number of ants that use Node Counting on strongly connected undirected graphs can be exponential in the square root of the number of vertices even for planar undirected trees.
Cover Time of Node Counting for Special Cases: So far, we have shown that the cover time of ants that use Node Counting is not guaranteed to be polynomial in the number of vertices for either directed or undirected graphs, including planar undirected trees. We are also able to describe a graph property that guarantees a polynomial cover time of single ants that use Node Counting.
A Eulerian graph is a graph that contains a Eulerian tour. This implies for directed Eulerian graphs that each vertex has an equal number of incoming and outgoing directed edges. Now consider an arbitrary strongly connected directed Eulerian graph whose number of edges is at most polynomial in the number of its vertices for example, because no two edges connect the same vertices in the same direction , and a graph that is derived from the directed Eulerian graph by replacing each of the directed edges with two directed edges that are connected with a unique intermediate vertex.
Figure 12 shows an example. It is currently unknown whether grids and other graph topologies that represent realistic terrains guarantee ants that use Node Counting a polynomial cover time. Discussion of the Theoretical Results So far, we have shown that the cover time of single ants that use Node Counting can be exponential in the square root of the number of vertices even for planar undirected trees. While we realized early on that the cover time of single ants that use Node Counting is exponential on directed graphs, we tried to prove for a long time that it was polynomial on undirected graphs.
There also exist other real-time search methods whose cover time on directed or undirected graphs is polynomial in the number of vertices perhaps with the re- striction that the graphs have no edges that leave the current vertex unchanged. Table 4 shows some of these real-time search methods.
Their cover times were analyzed by their authors and shown to be polynomial in the number of vertices for single ants. We have shown that the cover time of Node Counting can be exponential in the number of vertices for single ants. We do this in form of a guideline for how to design real-time search methods that result in a polynomial cover time. The results can then be generalized to groups of ants, which we do elsewhere.
Condition 1: At every point in time, the u-values u s are integers. The proof is in three parts. Suppose that the induction hypothesis holds at time step t. The only u-value that changes at time step t is u st for the non-goal vertex st. Consider an arbitrary time step t. The only u-value that changes is u st for the non-goal vertex st. Finally, we prove the main theorem. The main theorem follows.
Real-time search methods that satisfy Conditions 1, 2, and 3 of Theorem 14 cover strongly connected graphs repeatedly since the proof in Section 3 applies to them. This is so because all three conditions of the theorem hold for these three real-time search P methods for a functionPf n that is polynomial in n.
Consequently, the cover time is polynomial in n. On the other hand, all three conditions of the theorem hold for Node Counting only for a function f n that can be exponential in n. The ants could move to each of the four neighboring cells of their current cell provided that the destination cell was traversable white. Cells were untraversable if they contained either walls black or furniture grey.
The ants executed the real-time search method independently even if several ants shared a cell. Each ant moved once during each time step in a given sequential order.
This is important for one-time vacuum cleaning since each cell has to be vacuumed at least once. Figure 14 left shows the results. The trend was the same for all real-time search methods. The cover times improved as more ants were added although the rate of improvement decreased. In gen- eral, the cover times of all real-time search methods were similar.
To di erentiate better between the real-time search methods, Figure 14 right shows the total number of moves made by all ants, that is, roughly the number of ants times the cover time. It appears that the di erence in the total number of moves of the di erent real-time search methods was independent of the number of ants. The cover time of single ants was between 3, and 3, with a standard deviation between and Thus, the di erence in performance was dominated by the standard deviation.
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