Minimum Power Dominating Sets of Random Cubic Graphs - - PowerPoint PPT Presentation
Minimum Power Dominating Sets of Random Cubic Graphs Liying Kang Dept. of
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 1 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
Liying Kang
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 2 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 3 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
Power Domination Problem
Let G be a connected graph and S a subset of its vertices. Denote by M(S) the set monitored by S, defined algorithmically as follows:
M(S) ← S ∪ N(S)
If there exist any v ∈ M(S) and w / ∈ M(S) such that N(v) \ M(S) = {w} choose any such w and set M(S) ← M(S) ∪ {w}.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 4 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
In other words, the set M(G) is obtained from S as follows. First put into M(S) the vertices from the closed neighbourhood of S. Then repeatedly add to M(S) any vertex w that has a neighbour v in M(S) such that all the neighbours of v, apart from w, are already in M(S). When such a vertex w no longer exists, the construction of M(S) is complete. The set S is called a power dominating set of G if M(S) = V (G), and the power domination number γP(G) is the minimum cardinality of a power dominating set.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 5 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
when the input graph is bipartite; they presented a linear-time algo- rithm to solve PDS optimally on trees.
rithm that finds an optimal solution for graphs that have bounded tree-width.
gorithm based on dynamic-programming for optimally solving PDS
O(ck2 · n), where c is a constant.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 6 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
sented a polynomial-time algorithm for solving PDS optimally on interval graphs.
nar graphs and showed that their methods cannot improve on this approxi- mation guarantee.
the size of an optimal power dominating set, for n × m grids.
dominating set is at most n/3 for any connected graph G of order n ≥ 3, and γP(G) ≤ n/4 for any connected claw-free cubic graph G of order n.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 7 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
We consider random cubic graphs that are generated uniformly at random (u.a.r.). We say that a property B = Bn of a random graph holds asymptoti- cally almost surely (a.a.s.) if the probability that B holds tends to 1 as n tends to infinity. When discussing any cubic graph on n vertices, we assume n to be even to avoid parity problems. In this paper, we present two heuristics for finding a small power dominating set
vertex cubic graphs using differential equations and obtain two upper bounds on the size of the power dominating set P returned by these algorithms. We show that, for the second heuristic, P asymptotically almost surely satisfies |P| ≤ 0.067801n. On the other hand, we give the lower bound |P| ≥ 1/29.7n ≈ 0.03367n a.a.s.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 8 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
Define the boundary ∂S to be the set of edges with one end in S and the other end in V (G) \ S.
i(G) = min
|S|≤V (G)|/2
|∂S| |S| .
bw(G) = min
|S|=|V (G)|/2 |∂S|.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 9 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
Proposition 1 If G is cubic, then γP(G) ≥ bw(G) − 1 3 . Kostochka and Melnikov showed that a random cubic graph G a.a.s. satisfies i(G) ≥ n/4.95n + o(n). This immediately implies bw(G) ≥ n/9.9 + o(n), which then gives the following. Corollary 2 A random cubic graph G on n vertices a.a.s. satisfies γP(G) ≥ n/29.7 + o(n). Note that 1/29.7 > 0.03367, which is quite close to half of the upper bound featuring as the main result of this paper.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 10 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
3.1. Generating Random Cubic Graphs
The model used to generate a cubic graph u.a.r., originally proposed by Bol- lobas.
bucket
pair as an edge.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 11 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
3.2. Analysis Using Differential Equations
One method of analyzing the performance of a randomized algorithm is to use a system of differential equations to express the expected changes in variables de- scribing the state of the algorithm during its execution. N. Wormald (1995) gave an exposition of this method; it has been applied to many other combinatorial
The algorithm will be applied to the random pseudogaph generated by the pair-
choose the pairs sequentially. The first point, pi, of a pair may be chosen by any rule, but in order to ensure that the cubic graph is generated u.a.r., the second point, pj, of that pair must be selected u.a.r. from all the remaining free points. The freedom of choice of pi enables us to select it u.a.r. from the vertices of given degree in the evolving graph. Using B(pk) to denote the bucket that the point pk belongs to, we say that the edge (B(pi), B(pj)) is exposed when the second point of the pair has been selected.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 12 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
In order to analyse our algorithm using a system of differential equation, we in- corporate the algorithm into the pairing process that generates the random graph. In this way, we may generate the pairs of the random pairing in precisely the or- der that the corresponding edges are examined by the algorithm. Throughout the execution of the generation process, vertices will increase in degree until the generation is complete and all vertices have degree 3. We refer to the (part of) the graph which has been generated at any particular time as the evolving graph.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 13 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
3.3. Nature of the algorithms
The algorithms we use to find a power dominating set P of cubic graphs are greedy algorithms based on selecting vertices of given degree. We say that our algorithms proceeds as a series of operations. We denote the set of vertices of degree i of the evolving graph at time t by Vi = Vi(t) and let Yi = Yi(t) denote |Vi|. It will turn out that, in order to analyse our algorithms, the only knowledge we require of the state of the evolving graph at any point during the execution is determined by Y0, Y1, and Y2.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 14 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
A Type 1 operation consists of selecting a vertex u u.a.r. from V1 and then doing the following. Expose the edges incident with u, to vertices v1, v2 say). If, as in Figure 1(a) or (b), either of v1 or v2 is now in V2, the operation finishes. So we are left with considering v1, v2 ∈ V1. In this case, select v u.a.r. from {v1, v2}, expose all edges incident with v (to vertices w1, w2 say) and do the following. If w1, w2 ∈ V1, set P ← P ∪ {w1}, and expose all edges incident with w1; see Figure 1(c).
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 15 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
✒✑ ✓✏ ❝ s
u
s ✒✑ ✓✏ s s ❝
v1
✒✑ ✓✏ s s ❝ v2 ✡ ✡ ✡ ❏ ❏ ❏
(1a)
✒✑ ✓✏ ❝ s
u
s ✒✑ ✓✏ s s s
v1
✒✑ ✓✏ s s ❝ v2 ✡ ✡ ✡ ❏ ❏ ❏
(1b)
✒✑ ✓✏ ❝ s s
u
✒✑ ✓✏ s s s
v1
✒✑ ✓✏ s s s v2 ✡ ✡ ✡ ❏ ❏ ❏ ✒✑ ✓✏ s s s w2 ✒✑ ✓✏ s s s
w1
❏ ❏ ❏ ✡ ✡ ✡
(1c)
✒✑ ✓✏ ❝ s s
u
✒✑ ✓✏ s s s
v1
✒✑ ✓✏ s s s v2 ✡ ✡ ✡ ❏ ❏ ❏ ✒✑ ✓✏ s s s w2 ✒✑ ✓✏ s s ❝
w1
❏ ❏ ❏ ✡ ✡ ✡ ✒✑ ✓✏ s s ❝
w (1d)
✒✑ ✓✏ ❝ s s
u
✒✑ ✓✏ s s s
v1
✒✑ ✓✏ s s s v2 ✡ ✡ ✡ ❏ ❏ ❏ ✒✑ ✓✏ s s s w2 ✒✑ ✓✏ s s ❝
w1
✒✑ ✓✏ s s s
w
❏ ❏ ❏ ✡ ✡ ✡ ✒✑ ✓✏ s
z (1e)
✒✑ ✓✏ s s s
v1
✒✑ ✓✏ s s s v2 ✒✑ ✓✏ ❝ s s
u
✡ ✡ ✡ ❏ ❏ ❏ ✒✑ ✓✏ s s ❝
w1
✒✑ ✓✏ s s ❝ w2 ✡ ✡ ✡ ❏ ❏ ❏ ✒✑ ✓✏ s s ❝
w (1f)
✒✑ ✓✏ ❝ s s
u
✒✑ ✓✏ s s s
v1
✒✑ ✓✏ s s s v2 ✡ ✡ ✡ ❏ ❏ ❏ ✒✑ ✓✏ s ❝ s w2 ✒✑ ✓✏ s s ❝
w1
❏ ❏ ❏ ✡ ✡ ✡ ✒✑ ✓✏ s s s w ✒✑ ✓✏ s z
(1g)
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 16 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
If w1 ∈ V2, w2 ∈ V1, expose an edge incident with w1, to a vertex w (say). If w ∈ V2, do nothing, see Figure 1(d). On the other hand, if w ∈ V1, expose an edge incident with w to a vertex z (say), set P ← P ∪ {z}, and expose all edges incident with z. If w1 ∈ V1, w2 ∈ V2, do the same thing with w1 and w2 swapped. If w1, w2 ∈ V2, expose an edge incident with w2, to a vertex w (say). If w ∈ V2, do nothing. On the other hand, if w ∈ V1, expose an edge incident with w to a vertex z (say), set P ← P ∪ {z}, and expose all edges incident with z.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 17 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
A Type 2 operation consists of selecting a vertex u u.a.r. from V2 and exposing an edge incident with u. Since u was in S and had two neighbours in S, it power dominates the new vertex reached. See Figure 2.
✖✕ ✗✔ t ❞ ❞
u
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 18 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
Algorithm PD1 select u u.a.r. from V0 P ← {u} expose all edges incident with u while (Y1 + Y2 > 0) do if (Y2 > 0) perform the Type 2 operation else perform the Type 1 operation end while
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 19 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
4.1. An Upper Bound
Theorem 3 A random cubic graph on n vertices asymptotically almost surely has a minimum power dominating set with less than 0.116603n vertices. 4.1.1. Preliminary analysis In order to analyse our randomized algorithms for finding a power dominating set P of cubic graphs, we calculate the expected change in this state over one unit of time in relation to the expected change in the size of P.
the history of the process.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 20 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
We then regard E∆Yi/E∆P as the derivative dYi/dP, which gives a system of differential equations. The solutions to these equations describe functions which represent the behavior
expected size of the power dominating set may be deduced from these results.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 21 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
Let s = s(t) denote the number of free points available in all buckets at a given time t. Note that s =
2
(3 − i)Yi. The expected change in Yi due to changing the degree of an increase from i to i + 1 by exposing one of its incident edges (at time t) is ρi + o(1), where ρi = ρi(t) = (i − 3)Yi + (4 − i)Yi−1 s , 0 ≤ i ≤ 2.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 22 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
4.1.2. Preliminary Equations Operations of Type 2 involve the selection of a vertex v from V2. The expected changes in Yi for an operation of Type 2 given in Fig. 1 is αi + o(1), where αi = −δi2 + ρi. (1) The expected increase in P is just E∆P = 0. (2)
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 23 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
We now consider operation of Type 1. The expected change in Yi for operation 1h given in Fig. 2 (at time t) is βhi + o(1), where βai = 2δi2 − 3δi1, 0 ≤ i ≤ 2, βbi = δi2 − δi1 − δi0, 0 ≤ i ≤ 2, βci = δi1 − 4δi0 + 2ρi, 0 ≤ i ≤ 2, βdi = δi2 − δi1 − 3δi0, 0 ≤ i ≤ 2, βei = δi2 − 4δi0 + 3Y0 s (2ρi − δi0) + 2Y1 s (ρi − δi1), 0 ≤ i ≤ 2, βfi = 2δi2 − 3δi1 − 2δi0, 0 ≤ i ≤ 2, δgi = 2δi2 − 2δi2 − 3δi0 + 3Y0 s (2ρi − δi0) + 2Y1 s (ρi − δi1), 0 ≤ i ≤ 2.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 24 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
Therefore, the probabilities that operations of type 1a, 1b, 1c, 1d, 1e, 1f, 1g are given by P(1a) = 4Y 2
1
s2 + o(1), P(1b) = 12Y0Y1 s2 + o(1), P(1c) = 81Y 4 s4 + o(1), P(1d) = 216Y 3
0 Y 2 1
s5 + o(1), P(1e) = 324Y 4
0 Y 1 1
s5 + o(1), P(1f) = 72Y 2
0 Y 3 1
s5 + o(1)P(1g) = 108Y 3
0 Y 2 1
s5 + o(1). respectively.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 25 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
So the expected change in Yi for an Operation Type 1 is βi + o(1), where βi = 4Y 2
1
s2 βai + 12Y0Y1 s2 βbi + 81Y 4 s4 βci + 216Y 3
0 Y 2 1
s5 βdi + 324Y 4
0 Y1
s5 βei +72Y 2
0 Y 3 1
s5 βfi + 108Y 3
0 Y 2 1
s5 βgi, 0 ≤ i ≤ 2. The expected increase in P is just E∆P = (3Y0 s )2(1 − 2 × (2Y1 2 )2 × 3Y0 s − (2Y1 s )3) = 9Y 2 s2 − 216Y 2
1 Y 3
s5 − 72Y 2
0 Y 3 1
s5 .
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 26 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
We define a birth to be the generation of a vertex in V2 by processing a vertex of V1 or V2. The expected number of births from processing a vertex from V1 (at time t) is ν1 + o(1), where ν1 = 8Y 2
1
s2 + 12Y0Y1 s2 + 324Y 4
0 Y1
s5 + 216Y 3
0 Y 2 1
s5 + 324Y 4
0 Y1
s5 · 6Y0 + 2Y1 s · 2Y1 s +144Y 2
0 Y 3 1
s5 + 108Y 3
0 Y 2 1
s5 · (1 + 6Y0 + 2Y1 s · 2Y1 s ).
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 27 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
The expected number of birth from a Type 2 operation (at time t) is ν2 + o(1), where ν2 = ν2(t) = 2Y1 s .
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 28 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
We define a clutch to be a series of operations from an operation of Type 1 up to not in including the next operation of Type 1. Consider the Type 1 operation at the start of the clutch to be the first generation of a birth-death process in which the individual are the vertices in V2, each give birth to a number of children (essentially independent of the others) with expected number ν2. Then, the expected number in the jth generation is ν1νj−1
2
and the expected total number of births in the clutch is ν1 1 − ν2 .
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 29 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
The equation giving the expected change in Yi for a clutch is given by E∆Yi = βi + ν1αi 1 − ν2 + o(1). (3) The equation giving the expected increase in D for a clutch is given by E∆P = 9Y 2 s2 − 216Y 2
1 Y 3
s5 − 72Y 2
0 Y 3 1
s5 (4)
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 30 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
4.1.3. The Differential Equations Write Yi(t) = nzi(t/n), s(t) = nξ(t/n), ρi(t) = nψi(t/n), αi(t) = nτi(t/n), βi(t) = nϕi(t/n), νi(t) = nωi(t/n). Using equation (3), representing E∆Yi for processing a clutch, suggests the differential equation z′
i = ϕi +
ω1 1 − ω2 τi + o(1). (5)
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 31 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
Using equation (4), representing the increasing in the size of P after processing a clutch, and write P(t) = nz(t/n) suggests the differential equation for z as z′ = 9z2 ξ2 − 216z3
0z2 1
ξ5 − 72z2
0z3 1
ξ5 . (6) The solution to these system of differential equations represents the cardinalities
with initial conditions z0(0) = 1, z1(0) = 0, z2(0) = 0 and z(0) = 0.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 32 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
We use a result from N. Wormald (1999) to show that, the function representing the solutions to the differential equations almost surely approximate the variable Yi and P with error o(1). Theorem 4 Let ˆ W = ˆ W(n) ⊆ Ra+1. For 1 ≤ l ≤ a, where a is fixed, let yl : S(n)+ → R and fl : Ra+1 → R, such that for some constant C0 and all l, |yl(hl)| < c0n for all hl ∈ S(n)+ for all n. Let Yl(t) denote the random counterpart of yl(hl). Assume the following three conditions hold, where in (ii) and (iii) W is some bounded connected open set containing the closure of {(0, z1, . . . , za) : P(Yl(0) = zln, 1 ≤ l ≤ a) = 0 for some n}. (i) For some functions β = β(n) ≥ 1 and γ = γ(n), the probability that max1≤l≤a|Yl(t + 1) − Yl(t)| ≤ β. condition upon Hl is at least 1 − γ for t < min{TW, T ˆ
W}.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 33 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
(ii) For some function λ1 = λ1(n) = 0(1), for all l ≤ a, |E(Yl(t + 1) − Yl(t)|Ht) − fl(t/n, Y1(t)/n, . . . , Ya(t)/n| ≤ λ1 for t < min{TW, T ˆ
W}.
(iii) Each function fl is continuous, and satisfies a Lipschitz condition, on W ∩ {(t, z1, . . . , za) : t ≥ 0}, with the same Lipschitz constant for each l.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 34 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
Then the following are true: (a) For (0, z1, . . . , za) ∈ W the system of differential equations dzl dx = fl(x, z1, . . . , za), l = 1, . . . , a has a unique solution in W for zl : R → R passing through zl(0) = zl, 1 ≤ l ≤ a, and which extends to points arbitrarily close to the boundary of W; (b) Let λ > λ1 + C0nγ with λ = o(1). For a sufficiently large constant C, with probability 1 − O(nγ + β
λexp(−nλ3 β3 )),
Yl(t) = nzl(t/n) + O(λn) uniformly for 0 ≤ t ≤ min{σn, T ˆ
W} and for each l, where zl(x) is the solution
in (a) with zl = 1
nYl(0), and σ = σ(n) is the supremum of those x to which the
solution can be extended before reaching within l∞-distance Cλ of the boundary
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 35 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
By Theorem 4, the random variables Yi/n and P/n a.a.s. remain within o(1)
until a point arbitrarily close to where it leaves the set W, or until t = T ˆ
W if that
We compute the ratio dzi/dz, and we have dzi dz = ϕi +
ω1 1−ω2τi 9z2 ξ2 − 216z3
0z2 1
ξ5
− 72z2
0z3 1
ξ5
, i ∈ {0, 1, 2} where, differentiation is with respect to z and all function can be taken as func- tions of z. By solving this we find that the solution hits a boundary of W at ξ = ε. The differential equations were solved using a Runge-Kutta method, giving ξ = ε at z < 0.116603. This corresponds to the size of the power dominating set (scaled by 1
n) when all vertices are used up, thus proving the theorem.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 36 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
In this section we present Algorithm PD2 in order to improve the bound obtained using Algorithm PD1.In Algorithm PD2,
will be dominated from below.
next below is in V2 or will be dominated from below. A Type 1 operation consists of selecting a vertex u u.a.r. from V1 and then doing the following. Expose the edges incident with u, to vertices v1, v2 (say). If, as in Fig 3(a) or (b), either of v1 or v2 is now in V2, the operation finishes. If v1, v2 ∈ V1, label v1 as H, H ← H ∪ {v1}. See Figure 3(c).
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 37 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
✚✙ ✛✘ ❡ ✉
u
✉ ✚✙ ✛✘ ✉ ❡ ❡
v1
✚✙ ✛✘ ✉ ✉ ❡ v2 ✡ ✡ ✡ ✡ ❏ ❏ ❏ ❏
3(a)
✚✙ ✛✘ ❡ ✉
u
✉ ✚✙ ✛✘ ✉ ✉ ✉
v1
✚✙ ✛✘ ✉ ✉ ❡ v2 ✡ ✡ ✡ ✡ ❏ ❏ ❏ ❏
3(b)
✚✙ ✛✘ ❡ ✉ ✉
u
✚✙ ✛✘ ✉ ✉ ✉
v1 H
✚✙ ✛✘ ✉ ✉ ✉ v2 ✡ ✡ ✡ ✡ ❏ ❏ ❏ ❏
3(c)
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 38 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
✚✙ ✛✘ ✉ ❡ ❡
u
A Type 2 operation consists of selecting a vertex u u.a.r. from V2 and expose an edge incident with u. See Figure 4. A Type 3 operation consists of selecting a vertex u u.a.r. from F and exposing an edge incident with u, to vertices v (say). If v ∈ V2, the operation finishes. If v ∈ V1, label v as H, H ← H ∪ {v}. See Figure 5.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 39 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
✚✙ ✛✘ ❡ ✉
u
❡
F
✚✙ ✛✘ ✉ ✉ ❡
v 5(a)
✚✙ ✛✘ ❡ ✉
u
❡
F
✚✙ ✛✘ ✉ ✉ ✉
v H 5(b) Figure 5
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 40 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
A Type 4 operation consists of selecting a vertex u u.a.r. from H and then doing the following. Expose the edges with u, to vertices v1, v2 (say). If, as in Fig 6(a) or (b), v1 ∈ V2 ∧ v2 ∈ V2, label v2 as F, F ← F ∪ {v2}. If v1 ∈ V2∧v2 ∈ V1, label v1 as F, F ← F ∪{v1}. So we are left with considering v1, v2 ∈ V1. In this case, select v u.a.r. from {v1, v2}, expose all edges incident with v to vertices w1, w2 (say) and do the following. If w1 ∈ V2 ∧ w2 ∈ V2, label w1 as F, F ← F ∪ {w1}. See Figure 6(c) If w1 ∈ V2 ∧ w2 ∈ V1, P ← P ∪ {w2}, expose edges incident with w2. See Figure 6(d). If w1 ∈ V1 ∧ w2 ∈ V1, label w1 as H, H ← H ∪ {w1} P ← P ∪ {w2}, expose edges incident with w2. See Figure 6(e).
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 41 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
✒✑ ✓✏ ❝ s
u
s
H
✒✑ ✓✏ s s ❝
v1
✒✑ ✓✏ s s ❝ v2
F
✡ ✡ ✡ ❏ ❏ ❏
6(a)
✒✑ ✓✏ ❝ s
u
s
H
✒✑ ✓✏ s s s
v1
✒✑ ✓✏ s s ❝ v2
F
✡ ✡ ✡ ❏ ❏ ❏
6(b)
✒✑ ✓✏ ❝ s s
u H
✒✑ ✓✏ s s s
v1
✒✑ ✓✏ s s s v2 ✡ ✡ ✡ ❏ ❏ ❏ ✒✑ ✓✏ s s ❝ w2 ✒✑ ✓✏ s s ❝
w1 F
❏ ❏ ❏ ✡ ✡ ✡
6(c)
✒✑ ✓✏ s s s
v1
✒✑ ✓✏ s s s v2 ✒✑ ✓✏ ❝ s s
u H
✡ ✡ ✡ ❏ ❏ ❏ ✒✑ ✓✏ s s ❝
w1
✒✑ ✓✏ s s s w2 ✡ ✡ ✡ ❏ ❏ ❏
6(d)
✒✑ ✓✏ s s s
v1
✒✑ ✓✏ s s s v2 ✒✑ ✓✏ ❝ s s
u H
✡ ✡ ✡ ❏ ❏ ❏ ✒✑ ✓✏ s s s
w1 H
✒✑ ✓✏ s s s w2 ✡ ✡ ✡ ❏ ❏ ❏
6(e) Figure 6
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 42 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
5.1. Algorithm PD2
select u u.a.r. from V0 P ← {u} expose all edges incident with u while (Y1 + |F| + |H| > 0) do if (|F| > 0) perform the Type 3 operation. else if |H| > 0, perform the Type 4 operation. else if Y1 > 0, perform the Type 1 operation. else perform the Type 2 operation. end while
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 43 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
5.2. The upper Bound
We analyse the combined algorithm and pairing process using differential equa- tions and in this way prove the following theorem. Theorem 5 A random cubic graph on n vertices asymptotically almost surely has a minimum power dominating set with less than 0.067801n vertices.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 44 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
5.2.1. Preliminary Equations Operations of Type 1 involve the selection of a v from V1. The expected changes in Yi for an operation of Type 1 given in Fig.1 is βi + o(1), where βi = 4Y 2
1
s2 · (2δi2 − 3δi1) + 12Y0Y1 s2 · (δi2 − δi1 − δi0) + 9Y 2 s2 · (δi1 − 2δi0). The expected increase in P is just E∆P = 0.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 45 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
Operations of Type 2 involve the selection of a v from V2. The expected changes in Yi for an operation of Type 2 given in Fig.1 is αi + o(1), where αi = −δi2 + ρi. The expected increase in P is just E∆P = 0.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 46 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
The expected change in Yi for an operation of Type 3 is γi + o(1), where γi = 2Y1 s (−δi1) + 3Y0 s (−δi2 + δi1 − δi0). The expected increase in P E∆P = 0.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 47 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
The expected change in Yi for an operation of Type 4 is θi + o(1), where θi = 4Y 2
1
s2 · (2δi2 − 3δi1) + 12Y1Y0 s2 · (δi2 − δi1 − δi0) + 36Y 2
1 Y 2
s4 · (2δi2 − 2δi1 − 2δi0) +108Y1Y 3 s4 · (δi2 − δi1 − 3δi0 + 2ρi) + 81Y 4 s4 · (δi1 − 4δi0 + 2ρi). The expected increased in P is E∆P = 108Y1Y 3 s4 + 81Y 4 s4 .
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 48 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
The expected number of vertices F, H, V2 from processing a vertex from V1 is ν1f + o(1), ν1h + o(1), ν12 + o(1) respectively, where ν1f = 0, ν1h = 3Y0 s 2 = 9Y 2 s2 , ν12 =
s 2 + 2 · 3Y0 s · 2Y1 s
1
s2 + 12Y1Y0 s2 . The expected number of vertices F, H, V2 from processing a vertex from V2 is ν2f + o(1), ν2h + o(1), ν22 + o(1), where ν2f = ν2h = 0, ν22 = 2Y1 s .
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 49 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
The expected number of vertices F, H, V2 from processing a vertex from F is νff + o(1), νfh + o(1), νf2 + o(1) respectively, where νff = 0, νfh = 3Y0 s , νf2 = 2Y1 s . The expected number of vertices F, H, V2 from processing a vertex from H is νhf + o(1), νhh + o(1), νh2 + o(1) respectively, where νhf = 2Y1 s 2 + 2 · 2Y1 s
3Y0 s
2Y1 s 2 · 3Y0 s 3 = 4Y 2
1
s2 + 12Y1Y0 s2 + 36Y 2
1 Y 2
s4 , νhh = 3Y0 s 4 = 81Y 4 s4 , νh2 = 2Y1 s 2 + 3Y0 s 2 · 2Y1 s 2 + 2 · 3Y0 s 3 · 2Y1 s
s
3Y0 s 4 · 4Y1 s
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 50 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
We define a clutch to be a series of operations from an operation of Type 1 up to not in including the next operation of Type 1. Let the expected total number of vertices V2, F, H in the clutch is a, b, c, respectively. Let matrix D =
νhf νhh
and det = (1 − νhh) · (1 − νff) − νhf · νfh. Then (I − D)−1 = 1 det
νfh νhf 1 − νff
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 51 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
(b, c) = (ν1f, ν1h) + (ν1f, ν1h) · D + (ν1f, ν1h) · D2 + . . . = (ν1f, ν1h)(I + D + D2 + . . .) = (ν1f, ν1h)(I − D)−1 =
s2
= 9Y 2 s2 · det · νhf, 9Y 2 s2 · det
So we have b = 9Y 2 s2 · det · νhf, c = 9Y 2 s2 · det.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 52 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
a = (ν12 + ν12 · ν22 + ν12 · ν2
22 + . . .) + b · (νf2 + νf2 · ν22 + νf2 · ν2 22 + . . .)
+c · (νh2 + νh2 · ν22 + νh2 · ν2
22 + . . .)
= (ν12 + b · νf2 + c · νh2)(1 + ν22 + ν2
22 + . . .)
= ν12 + b · νf2 + c · νh2 1 − ν22 . The equation giving the expected change in Yi for a clutch is given by E∆Yi = βi + a · αi + b · γi + c · θi. (7) The equation giving the expected increasing in P for a clutch is given by E∆P = c · 108Y 3
0 Y1
s4 + 81Y 4 s4
(8)
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 53 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
5.2.2. The Differential Equations Using equation (7), representing E∆Yi for processing a clutch,suggests the dif- ferential equation z′
i = ϕi + A · τi + B · ζi + C · ηi.
(9) Using equation (8), representing the increasing in the size of P after processing a clutch, and write P(t) = nz(t/n) suggests the differential equation for z as z′ = C · 108z3
0z1
ξ4 + 81z4 ξ4
(10)
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 54 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ
The solution to these system of differential equations represents the cardinalities
with initial conditions z0(0) = 1, z1(0) = 0, z2(0) = 0 and z(0) = 0. Using the similar discussion as in Section 3, we compute the ratio dzi/dz, and we have dzi dz = ϕi + A · τi + B · ζi + C · ηi C · 108z3
0z1
ξ4
+ 81z4
ξ4
i ∈ {0, 1, 2} where, differentiation is with respect to z and all function can be taken as func- tions of z. By solving this we find that the solution hits a boundary of W at ξ = ε. The differential equations were solved using a Runge-Kutta method, giving ξ = ε at z < 0.067801. This corresponds to the size of the power dominating set (scaled by 1
n) when all vertices are used up, thus proving the theorem.
✒➭✵☛ ✝✳➚➀➥❑ ➀✝✳➚➀➥❑ ä❦✗➦✛➚➀➥❑
↔ ✔ ➘ ➄ ■ ❑ ➄ ◭◭ ◮◮ ◭ ◮ ✶ 55 ➄ 55 ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ