In this section we consider provable properties of the routing methods just introduced. Experimental results follow in the next section. Routing Rule 1 directly implies the following lemma for tree vertices.
Lemma 5.4.1 Routing Rule 1 guarantees message delivery ifs and t are in the same tree. Otherwise, the message will be passed from s to its root vertex if s is a tree vertex. If t is a tree vertex, the message will be passed from its root to t.
Theorem 5.4.2 BR, GBR, AGBR and GBFR guarantee message delivery.
This theorem implies that the application of Rule1guarantees message deliv-ery on 3-connected graphs, which is also proved in [96]. Anyhow, we present a simpler proof relying only on (5.2).
Proof: Consider Fig.5.3. The message has to be passed from source vertexs to target vertex t. Assume that both vertices are not tree vertices, otherwise Lemma 5.4.1 could first be applied. The (infinite) line −→
st divides the plane into two half-planes. Because either s is in the barycenter of its neighbors orsis a vertex on the convex polygonC, there exists a neighborv ofsin the closed right hand half-plane with minimum deviation angle. If the direction angle from s to v does not increase, the distance to t decreases. Hence, if the message is not delivered to tthe message path finally leads to a vertexw with a direction angle of ϕ ≥ 2π. To prove Theorem 5.4.2 it is sufficient to show thatwhas to be within the star-shaped (open) domainD, delimited by the message path and the (dashed) line−→
ts. Assume the contrary, as depicted in Fig. 5.3. Let GD be the subgraph induced by all vertices within D. All the edges connecting a vertex from the message path to a vertex in GD have negative deviation angle. All the other edges connecting GD to GGG\GD
have to pass the dashed line between s and the upper intersection point with the edge to w. Since D is star-shaped with t as center there exists a sufficiently small angle, such that the rotation of GD by this angle reduces the lengths of all edges leaving GD, in contradiction to (5.2). This proves message delivery for BR.
Since the greedy mode of GBR (GBFR) will only be restarted after BR mode reaches a vertex strictly closer to t, GBR (GBFR) also guarantees delivery. The case when AGBR reaches a vertex v, such that Rule 2 would lead to a vertex w with d(w, t)b > Rd(s, t) remains to be shown. First noteb that the Theorem of Whitney [123] implies the existence of a homeomorphism g: R2 −→ R2 that sends the original graph GGG (with non-straight-line
virtual edges that do not cross) to its barycentric embedding. Hence, v is a vertex on the outer face fR of the graph GR formed by all vertices u with d(u, t)b ≤Rd(s, t). Ifb twas not surrounded by the walk alongfR, thentis not reachable within GR and R has to be increased. If t was surrounded then t is reachable. There is a BR-path from v to t that has to intersect fR. If not reachingt earlier, Rule4leads to the BR-path from the last intersection
point of the BR-path to t.
s
t
v w
D
Figure 5.3: BR and GBR guarantee message delivery. The message cannot be passed to a vertex w outside the domain delimited by the message path and the (dashed) line from t, passings
Figure 5.4: Example, where routing suffers from finite precision arithmetic Note that Theorem 5.4.2 actually requires the exact coordinates of the barycentric embedding pC that may not be representable with finite preci-sion arithmetic. Consider Fig. 5.4, where a message from the uppermost vertex has to be sent to the center vertex, and assume that perimeter ver-tices w1, w2 at 3 and 9 o’clock have virtual coordinates (30,1) and (-30,-1),
respectively. The exacty-coordinates of the inner vertices adjacent to w1, w2 are 2/3,−2/3. Thus, for finite precision arithmetic, the deviation angles atw1 andw2 remain negative after each application of Jacobi iteration (5.1), even in the (finite precision) limit. Rounding the exact coordinates (that may have been obtained somehow) to finite precision yields the same nega-tive result. A heuristical workaround that works well in practice is to modify Rule 1 from ψt(v, w) ≥ 0 to ψt(v, w) ≥ −ε for some small ε > 0. Another workaround is a rational representation of coordinates as follows.
Theorem 5.4.3 Let the virtual coordinates of vertices in C be small integer numbers, representable by q ≤ 2 log|C| bits (|x(v)| ≤ 2q for all v ∈ C).
Then the coordinates of pC are rational numbers with at most η:=d2.416ne bits for denominators and η +q bits for numerators. Furthermore, using finite precision arithmetic (2(η+q) bits) the rational coordinates of pC can be reconstructed after finitely many steps of Jacobi iteration (5.1).
Proof: Let xI and xC again denote x-positions of inner and perimeter ver-tices, respectively (this notation will also be used for other variables, e.g., VI, VC, for which we wrote V, ∂V◦ in Chap. 2). Then xI = L−1I (−eL)xC, where LI is the submatrix of L induced by inner vertices and Le is im-plicitly defined by Lx = 0. Hence, by Laplace’s formula for matrix inver-sion (2.3), denominators (defined as positive integers) of xI are not greater than det(LI) ≤ τ(GGG) ≤ (16/3)n, where the first inequality is Kirchhoff’s matrix tree theorem, see Theorem 2.1.9, together with the fact that merging the vertices of (the circle) C does not increase the number of spanning trees, and the last inequality is from [107]. Note that this yields a bound very much better than the Hadamard bound nn/2(degmax)n, see, e.g., [115]. The size of the numerators then follows from |x(v)| ≤2q for all v ∈ V. Since we use 2(η+q) bits, Jacobi iteration (5.1) will approximate the exact coordi-nates of each vertex up to a deviation smaller than 2−2η−1, which is sufficient to reconstruct the finite continued fraction of the exact rational coordinates simply by the Euclidean algorithm applied to the approximate coordinates, see, e.g., [117], for a similar application, or Sect. A.5.
A strictly convex polygon C with integer coordinates of bit length at most 2 log|C|may be obtained for vertices j = 0, . . . , c:=d|C|/8e by coordinates (Pj
k=1c−k+ 2,(c+ 5)c/2−j) and defining the remaining positions sym-metrically. Actually, a strictly convex polygon can also be defined within a square of width in O(|C|3/2), see [1], instead of O(|C|2). Note that arith-metic with 2(η+q) ∈Θ(n) bits maybe replaced by arithmetic with a fixed number of bits and stepwise storing highest bits that already remain fixed
by Jacobi iteration (5.1), see also, e.g., [117]. Also note that reconstruction of the exact coordinates should be attempted repeatedly from time to time.
A reconstruction may be possible very early and it can easily be checked, whether the exact coordinates are found since only these set each vertex to the barycenter of its neighbors. Bounds when the reconstruction is possible for sure, can be obtained as follows. Let J denote the mapping on inner vertices implicitly defined by Jacobi iteration (5.1). Furthermore, let ˆxI (and ˆyI) denote some initial positions of the inner vertices within C, e.g., (scaled) original positions, and let xI (and yI) denote the exact rational co-ordinates. We first consider a special case. Let d = maxv∈GGGdist(v, VC), where dist(v, VC) denotes the minimum hop-distance fromv to a vertex inC.
Let GGG be a quadratic grid with n = 4(d+ 1)2 vertices (d >0), where the 4(√
n−1) perimeter vertices are set to the integer positions of the rectangle with diagonal (0,0) to (√
n−1,√
n−1) (thus C is not strictly convex, but, anyhow, pC is planar). The bound for denominators for the grid given by (16/3)n ≥ det(LI)≥ 2n−1/n (see [84]) can be replaced by 1 since the exact coordinates are integers. Since the matrix D−1I AI is 1/4 times the (sym-metric) adjacency matrix of a grid with 4d2 vertices, its spectral radius ρ is cos(π/(d+ 1)), see, e.g., [31], or use the sum formula for the sine. The initial positions ˆxI of the inner vertices are in the worst-case (for| · | convergence), such that ˆxI−xI is an eigenvector corresponding to ρ, say, with maximum-norm | · |∞ equal to 1 (such that ˆxI is within the perimeter vertices). Hence,
|Jt(ˆxI)−xI|∞ =ρt
(neglecting the fact that we actually may have a rounding error smaller than 2−2η−q for each iteration), such that
t= 1
2 log sec(π/(d+ 1)) = ln 2
π2 ·n+ Θ(1) = 0.070. . .·n+ Θ(1) , where the constant in Θ(1) is negative, is the sharp bound, when the re-construction is possible for sure in this case. Note that worse initial co-ordinates, such that |ˆxI −xI|∞ = c, we still have t ≤ n(1 + logc) ln 2/π2 and did not lose very much. Also note that if we had the worst-case de-nominators as large as 22.416n (removal of a few inner edges of the grid increases denominators, while R remains nearly constant), we still have t ≤n(1 + 4.831n) ln 2/π2 ≤n2/3. The bound ofO(n2) may also be used for well-behaving networks in practice. Sufficient conditions may be given as fol-lows. Let degj(v) denote the number of neighborswofvwith dist(w, VC) =j.
Theorem 5.4.4 Let d,|VC| ∈ O(√
n). For each v ∈ VI with dist(v, VC) =j let degj−1(v) ≥ degj+1(v) and let degj(v) ≤ c1 be bounded by a constant.
Then the exact coordinates can be reconstructed after t ∈ O(n2) iterations.
Proof: Since |VC| ∈ O(√
n), any initial positions ˆxI that are within C fulfill
|ˆxI−xI|∞≤c2n for a constant c2. Thus
|(ˆxI−xI)v| ≤c2ndsinπ
2 ·dist(v, VC) d
. Hence by Jacobi iteration (5.1) and the sum formula for the sine
|Jt(ˆxI)−xI|∞ ≤c2nd
cos
π 2(c1+ 1)d
t
,
such that t= (4.831n+ 1 + logc2nd)·0.562(c1+ 1)2d2 ∈ O(n2) iterations are sufficient for reconstruction of the exact coordinates with worst-case
denom-inators and worst-case initial positions.
Figure 5.5: Example for slow convergence to the barycentric embedding However, for the general case one can also construct instances, where conver-gence is worse. Consider the graph of Fig. 5.5, suitable for our application, which needs the worst-case number of iterationst∈Θ(n3) becaused ∈Θ(n).
LetGGGbe a rectangular grid with dimension 7×(d+2), where some vertices and edges are removed according to Fig. 5.5. Let the left lower and upper perimeter vertices have coordinates (0,0) and (0,6), respectively (again, pC is planar althoughC is not strictly convex). Furthermore, let ˆxI have integer coordinates according to Fig. 5.5. Then analogous to the last proof
|Jt(ˆxI)−xI|∞≥
cos π 2d
t
.
The determinant of LI is the number of spanning trees of a 3×(d+ 1) grid (see also [84]), and the exact coordinates of the uppermost inner vertex is given for d = 1,2 by 4/5,19/24 and for d ≥ 3 by ad/bd, both sequences ad, bdwith characteristic polynomial T2−5T+ 1. Allad, bdare coprime since 1 = ad−1bd−bd−1ad, such that the denominators grow exponentially with respect to d (and n). Hence, t ∈ Ω(n3). The following theorem states that on the other hand O(n3) is already the worst-case.
Theorem 5.4.5 Exact coordinates can be reconstructed after O(n3) itera-tions.
Proof: Let GGG be any graph as considered above. In general, D−1I AI is not symmetric, so we define the matrix norm | · |DI =|DI1/2 ·D−1/2I |2 (where| · |2 is the matrix norm that corresponds to the Euclidean vector norm) that coincides with the spectral radius ρon DI−1AI. First,
ρ(D−1I AI) = 1− 1
ρ(L−1I DI) ≤1− 1
18n2 , (5.4)
where the equation holds because the spectrum of DI−1AI is in ]0,1[ (LI is essentially diagonally dominant). Before we discuss the last inequality note that L−1I is known as an M-matrix (see, e.g., [72]), and here, in particular, diagonal elements of L−1I are not smaller than absolute values of the ele-ments in the corresponding row. Together with (2.10) and interlacing of eigenvalues of L and LI (see, e.g., [62]) this already yields a good bound on ρ(D−1I AI). A slightly better bound comes from the lower bound for the smallest eigenvalue λ1 of DI−1LI given by
λ1 ≥ 1 2 min
W⊆VI
κ(W)
|W|deg 2
,
see [54] and (2.9), where κ(C) is the number of edges with vertices both in W and V \W, and |W|deg denotes the sum of degrees of vertices in W. Since GGG is planar and 2-connected, the minimum is greater than 2/(6n).
Any initial positions ˆxI within C fulfill|ˆxI−xI|∞ ≤n2/4. Hence,
|Jt(ˆxI)−xI|∞ ≤n3· |D−1I AI|tD
I =n3·ρ(D−1I AI)t.
This yields a bound of t = (4.831n + 1 + 3 logn)· 12.477n2 ∈ O(n3) for worst-case denominators and worst-case initial positions.
The last theorems together have shown that both memory and time con-sumption to compute exact coordinates may exceed memory and time needed for complete routing tables, which is obviously in O(nlogn) andO(n2) (e.g., n-times application of BFS), respectively. Hence, for practical purposes, clearly, GBFR is the better choice.
Now we state results about the lengths of the message paths that are computed by our routing methods.
Theorem 5.4.6 There is a series of unit disk graphs with a designated target vertex t, such that the expected quotient of the length of the BR-path (and GBR-path) between a random source vertex s and t divided by the length of a shortest st-path is unbounded.
Proof: Consider Fig. 5.6. The left hand side displays a restricted Gabriel GraphGGGof a series of unit disk graphs, where the numbern−1 of vertices on the half circle can be varied. Let the lowermost vertex be the target vertex t. The right hand side shows G in barycentric embedding, which is trivial in this case, because there are no inner vertices. For all source verticessdifferent fromt and its neighbors the length of a shortest path is 2, whereas the BR-path (and GBR-path) has an expected length of aboutn/4.
Figure 5.6: From left to right: Restricted Gabriel Graph of a unit disk graph G satisfying the conditions of Theorem 5.4.6. Barycentric embedding of G Theorem 5.4.7 The length of an AGBR-path is in O(`2), where ` is the length of a shortest st-path.
Proof: The proof of Theorem 5.4.7 is analogous to the corresponding for GOAFR+ [85] and is thus only sketched here. The backbone graph GBG
is a dominating set for the vertices of G. The length of a shortest st-path using only intermediate vertices ofGBG is only a constant factor longer than a shortest st-path in G. Furthermore, GBG has bounded degree, such that the maximum number of vertices in a circle of radius r is in O(r2). This together with the fact that the number of vertices visited during application of Rule 4 in the circle of radius r = Rd(s, t)b ≥ ` is a constant times the number of vertices in the circle concludes the proof.
Note that analogous to the extension from GBR to AGBR we can also extend the rules of GBFR (together with GOAFR+ instead of Face Routing) to obtain an algorithm that has the same upper bound of O(`2).
Theorem 5.4.8 There is a series of unit disk graphs with a designated target vertex t, such that the expected length of an AGBR-path between a random source vertex s and t is in Ω(`2), where ` is the length of a shortest st-path.
Proof: In the following the distance of the vertices on the AGBR-path to t will never increase, such that we can restrict ourselves to GBR. The lhs of Fig.5.7 shows an example of a series of unit disk graphs that are constructed as follows. Choose the number of windings of the double spiral, here it is three. Draw the inner windings as in Fig. 5.7. For each further quarter of a winding choose the number of vertices to be used for that quarter, such that the outmost vertices are still connected. Finally, add the outer circle and the vertices of the cross-hair. The resulting unit disk graph is a planar, 3-connected graph, and it is equal to its restricted Gabriel Graph. The rhs of Fig. 5.7 shows pC, where C is the outer circle. Let t be the center vertex that by symmetry will remain in the center of the circle. Now all except the two upper and lowermost edges on the crosshair directed to t have negative deviation angle. Hence, each BR message path, starting from a source s, will avoid these edges and instead be a full spiral path to t. Thus, the length of the message path is roughly proportional to the area of the circle around t with radius determined by the original position of s, whereas the length of a shortest st-path is only proportional to the radius. The very same example also holds for GBR if s is not a vertex of the crosshair. Greedy mode will be either stopped on the circle or after moving to roughly the center of a quarter of a winding. The following BR phase will as before avoid the edges of the crosshair and will only stop at some vertex of the next quarter of the
winding.
Figure 5.7: Left-right: A unit disk graph G satisfying the conditions of Theorem 5.4.8. Barycentric embedding ofG
5.5 Experimental Results
The mean algorithm cost is defined here as the quotient of the length of the message path divided by the length of a shortest path. Figure 5.11 shows results of tests on random unit disk graphs with 1000 vertices. For densities from 1.0 to 20.0 in steps of 0.25 we created 100 networks, and routed messages between 1000 st-pairs. Figure 5.8 shows the improvement of greedy success rate, i.e., the ratio of messages delivered only in greedy mode. Moreover, the number of tree vertices and inserted virtual edges are shown. Figures 5.9 and5.10finally show mean algorithm costs and greedy success rates of GBFR with various maximum numbers of applications of Jacobi iteration (5.1).
5.6 Conclusion
We presented four routing methods for networks, where geographic positions are available. Actually, it is even often sufficient if a planar subgraph and perimeter vertices in C would be known. All methods use a barycentric embedding as virtual coordinates obtained from a one-time-precomputation phase. This allows simple routing rules with guaranteed delivery, and very short routes in practice, outperforming algorithms running only on geometric coordinates. Attempts to reduce the consuming precomputation phase are made with GBFR. Nevertheless, our methods clearly apply to static, long-time-living networks, where the precomputation is affordable and short routes are of great importance. Considering GBFR as a heuristic on the other hand might also make it a candidate for dynamic networks, where some few iterations (even only local) are used to update the barycentric embeddingpC.
60
50
40
30
20
10
2 4 6 8 10 12 14 16 18 20
Number of tree nodes/inserted edges
Node Density [nodes per unit disk]
Tree nodes Inserted edges (2-con.) Inserted edges (3-con+)
0 0.2 0.4 0.6 0.8 1
2 4 6 8 10 12 14 16 18 20
Greedy Success Rate
Node Density [nodes per unit disk]
Greedy Barycentric Routing GOAFR+
Figure 5.8: Number of tree vertices and number of virtual edges that were inserted is displayed. Furthermore, success rates of greedy routing for GBR and GOAFR+
3
2.5
2
1.5
1
0.5
2 4 6 8 10 12 14 16 18
Mean Algorithm Cost
Node Density [nodes per unit disk]
GBFR 2 iterations GBFR 5 iterations GBFR 10 iterations GBFR 20 iterations GBFR 50 iterations
0 0.2 0.4 0.6 0.8 1
2 4 6 8 10 12 14 16 18
Greedy Success Rate
Node Density [nodes per unit disk]
GBFR 2 iterations GBFR 5 iterations GBFR 10 iterations GBFR 20 iterations GBFR 50 iterations
Figure 5.9: Mean algorithm costs and greedy success rates of GBFR (and call of Face Routing if pure GBR failed) at maximum number of iterations 2,5,10,20,50,100
0 0.2 0.4 0.6 0.8 1
2 4 6 8 10 12 14 16 18
Delivery Ratio
Node Density [nodes per unit disk]
GBR 2 iterations GBR 5 iterations GBR 10 iterations GBR 20 iterations GBR 50 iterations
Figure 5.10: The delivery ratio of pure GBR (without call of Face Routing)
6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5
2 4 6 8 10 12 14 16 18 20
Mean Algorithm Cost
Node Density [nodes per unit disk]
Barycentric Routing Greedy Barycentric Routing GOAFR+
Figure 5.11: The mean algorithm costs of BR, GBR and GOAFR+ are compared on random unit disk graphs with 1000 vertices
Color Assignment of Vertices
Recently, Dillencourt, Eppstein, and Goodrich introduced a graph coloring variant, where the main goal is to maximize the perceived difference of the colors assigned to adjacent vertices [38]. Although this is achieved by em-bedding the graph in a color space, the criteria are very different from those usually considered in graph drawing because, in particular, edges should be long. We present refined approaches for computing color space embeddings of weighted graphs and give two specific applications.
6.1 Introduction
In graph theory, graph coloring usually refers to a set of combinatorial opti-mization problems, e.g., given an undirected graph G= (V, E), what is the minimum number of colors for its vertices, such that each two vertices joined by an edge are colored differently. Here, color is a nominal attribute, and only adjacent vertices are considered.
In visualization, color assignment is a different task because here it is important that colors support visual distinction. While the selection of colors and properties of color spaces are well-studied issues (see, e.g., [69,112,120]), adjacency seems to have entered the picture only recently [38].
Slightly reformulated, the problem introduced in [38] is as follows. Given an undirected, possibly weighted graph, color its vertices, such that
• adjacent vertices are colored as different as possible, and
• all used colors can be well distinguished.
Note that the graph may just be a model for the adjacencies of any kind of ge-ometric objects, where edge weights represent the required color differences.
In [38], the problem is solved by refining an initially random embedding of the vertices in a color space according to a specific quality measure.
To arrive at even better color assignments we propose to substitute both steps, initialization and refinement, by graph drawing methods adapted to the unusual criteria, and also discuss the assignment of edge weights. Since we require that the final visualization uses only displayable colors from a space such as RGB on a monitor or a CMYK on a printer, a modified clipping
To arrive at even better color assignments we propose to substitute both steps, initialization and refinement, by graph drawing methods adapted to the unusual criteria, and also discuss the assignment of edge weights. Since we require that the final visualization uses only displayable colors from a space such as RGB on a monitor or a CMYK on a printer, a modified clipping