How to Use Spanning Trees to Navigate in Graphs

Abstract

In this paper, we investigate three strategies of how to use a spanning tree T of a graph G to navigate in G, i.e., to move from a current vertex x towards a destination vertex y via a path that is close to optimal. In each strategy, each vertex v has full knowledge of its neighborhood N _G [v] in G (or, k-neighborhood D _k (v,G), where k is a small integer) and uses a small piece of global information from spanning tree T (e.g., distance or ancestry information in T), available locally at v, to navigate in G. We investigate advantages and limitations of these strategies on particular families of graphs such as graphs with locally connected spanning trees, graphs with bounded length of largest induced cycle, graphs with bounded tree-length, graphs with bounded hyperbolicity. For most of these families of graphs, the ancestry information from a Breadth-First-Search-tree guarantees short enough routing paths. In many cases, the obtained results are optimal up to a constant factor.

Appendix A: Experimental Results

In this appendix, we empirically compare the performance of TDGR, IGR and IGRF, and their corresponding k-localized versions, on Unit Disk Graphs (UDGs), which often model the wireless ad hoc networks. We use three kinds of spanning trees, breadth-first-search tree (BFST, for short), minimum-spanning tree (MST, for short), and depth-first-search tree (DFST, for short) for TDGR, IGR, and IGRF. Since the IGR scheme is the same as the IGRF scheme if the spanning tree is BFST, we only need to report on one of them. Therefore, the routing scheme pairs (“strategy type”–“tree type”), we report on, are BFST-IGR, BFST-TDGR, MST-IGR, MST-IGRF, MST-TDGR, DFST-IGR, DFST-IGRF, and DFST-TDGR. All methods are implemented in C++.

To generate a Unit Disk Graph, we first fix an area S, a radius R, and the number of vertices N. Then, in S we randomly generate N vertices/points. Two vertices/points are connected by an edge if and only if their Euclidean distance is at most R.

In the experiments, we care about the maximum multiplicative-stretch factor and average multiplicative-stretch factor of each routing scheme using different spanning trees. The multiplicative-stretch factor of two vertices u and v is defined as $\frac{g_{G,T}(u,v)}{d_{G}(u,v)}$, which is a good indication of how close the routing path is to the shortest path. Here, g _G,T (u,v) is the length of the route produced by an appropriate strategy from u to v on G using tree T, and d _G (u,v) is the distance in G between u and v. The maximum stretch factor of a graph G=(V,E) is defined as $\max_{u,v\in V}\{\frac{g_{G,T}(u,v)}{d_{G}(u,v)}\}$, and the average stretch factor is defined as $\frac{1}{n^{2}}\sum_{u,v\in V}\frac{g_{G,T}(u,v)}{d_{G}(u,v)}$.

1.1 A.1 Performance Under Various Densities

In this set of experiments, we report on the performance of these routing schemes on randomly generated UDGs with different densities, i.e., |E|/|V|. However, it is difficult to “randomly” generate a UDG with a fixed density. Instead, we vary densities by choosing the radius R to be 150, 170, 190, 210, 230, 250, 270 and 290, with |V| fixed to be 100. For each radius R, we randomly generate 10 UDGs. The average density of 10 UDGs corresponding to each R is listed in Table 3. In the following figures, each value is an average result on the 10 randomly generated UDGs.

Table 3 Average densities for different radiuses

The maximum multiplicative-stretch factors achieved by routing strategies under different radiuses are shown in Fig. 14. We see that DFST-IGRF and MST-IGRF have the worst maximum multiplicative-stretch factors and their performances are not stable when the radius changes. Other routing schemes have quite low maximum multiplicative-stretch factors which decrease gradually when the radius increases. Among them, BFST-IGR, BFST-TDGR, MST-IGR, and MST-TDGR have the lowest maximum multiplicative-stretch factors.

Fig. 14

Maximum multiplicative-stretch factors by varying densities (Color figure online)

Figure 15 shows average multiplicative-stretch factors achieved by routing strategies under different radiuses. Again, DFST-IGRF and MST-IGRF have the worst performances and BFST-IGR, BFST-TDGR, MST-IGR, and MST-TDGR have the best performances.

Fig. 15

Average multiplicative-stretch factors by varying densities (Color figure online)

1.2 A.2 Performance Under Various Localities

In this appendix, we show how the k-localized version of these routing strategies performs. The experimental settings are similar to those from Appendix A.1 except that |V| is fixed at 120 and the radius is fixed at 130. We randomly generate 10 UDGs. The average diameter of these UDGs is 18 (ranging from 16 to 20). For each UDG, we range the locality k from 1 to 8 to see how each routing strategy performs. Each value in the following figures is an average result on the 10 randomly generated UDGs.

Figure 16 shows the maximum multiplicative-stretch factors achieved by each routing scheme with different localities, and Fig. 17 shows the average multiplicative-stretch factors achieved by each routing scheme with different localities. In both figures, we observe that the multiplicative-stretch factor of each k-localized routing scheme converges to 1 when locality increases from 1 to 8. Increase in locality allows to obtain better routing paths, however, it also increases the computational and communication costs. A good tradeoff between stretch factor and locality is needed.

Fig. 16

Maximum multiplicative-stretch factors by varying localities (Color figure online)

Fig. 17

Average multiplicative-stretch factors by varying localities (Color figure online)

Finally, when locality is more than 4, except for DFST-IGRF and MST-IGRF, all the other routing schemes have quite small maximum and average multiplicative-stretch factors. This is consistent with the observations from the previous subsection.