Reliable Broadcast in a Wireless Grid Network with ... - Semantic Scholar
Reliable Broadcast in a Wireless Grid Network with Probabilistic Failures Technical Report (October 12, 2005)
Vartika Bhandari
Nitin H. Vaidya
Dept. of Electrical and Computer Eng., and Coordinated Science Laboratory University of Illinois at Urbana-Champaign [email protected]
Dept. of Computer Science, and Coordinated Science Laboratory University of Illinois at Urbana-Champaign [email protected]
Abstract We consider a wireless grid network in which nodes are prone to failure. In the considered failure mode, each node has an independent probability of failure p, and failures may be either Byzantine or crash-stop in nature. All nodes are assumed to have a common transmission range r, and a resultant common degree d. We establish necessary and sufficient conditions for the degree of each node as a function of the total network size n and the failure probability p, so as to ensure that reliable broadcast succeeds with probability 1, as n → ∞. Our results 1 indicate that reliable broadcast ¶ is asymptotically achievable with Byzantine failures if p < 2 , and the degree of µ ln n 1 1 +ln ln 2p 2(1−p)
. These results exhibit similarity of form to results obtained for crash-stop failures µ ¶ that indicate a required degree of Θ ln n1 for p < 1.
each node is Θ
ln p
I. I NTRODUCTION We consider the problem of reliable broadcast in a wireless grid network prone to probabilistic failures. The node failures are assumed i.i.d. with probability p. Two separate failure types are considered, viz., Byzantine and crash-stop. We show that when nodes failures, reliable broadcast requires µ exhibit Byzantine ¶ ln n that p < 21 , and the node degree must be Θ ln 1 +ln for asymptotic achievability of reliable 1 2p 2(1−p) µ ¶ broadcast. This may alternatively be stated as Θ D(Qln1n||P) where Q 1 denotes a distribution with failure 1 2,
2
2
probability P denotes the actual distribution with failure probability p, and D(Q||P) denotes the relative entropy (or Kullback-Leibler distance) between distributions Q and P. For crash-stop failures, the problem µ ¶
ln n of reliable broadcast is equivalent to connectivity. For this case, we show that node degree must be Θ ln 1 p ³ ´ ln n for p < 1, or alternatively stated, Θ D(Q1 ||P) , where Q1 is the distribution with failure probability 1. This report comprises two independent parts. We consider the case of Byzantine failures in the first part. In the second part, we address the issue of crash-stop failures as a connectivity question. Along with connectivity, we also obtain conditions for coverage that point toward the same expression, except for the constants involved.
Minor Corrections made in Feb 2006. This report supercedes an earlier report ”Connectivity and Coverage in Failure-Prone Wireless Grid Networks”, dated September 2005. This research is supported in part by the National Science Foundation, and a Vodafone Graduate Fellowship.
II. S OME U SEFUL M ATHEMATICAL R ESULTS We state some mathematical results that have been used in our proofs: 1 ≥x FACT 1. ∀x ∈ [0, 1] : ln 1−x
1
FACT 2. If f (n) ≤ n 2 −ε (0 < ε < 12 ): µ ¶ f (n) n ( lim f (n)) lim 1 + = e n→∞ n→∞ n 1
Proof: Let f (n) ≤ n 2 −ε , where 0 < ε < 12 . Let g(n) = (1 +
f (n) n n ) .
Then:
µ ¶ f (n) 1 f (n) 2 1 f (n) 3 f (n) )=n − ( ) + ( ) − .... [1] ln g = n ln (1 + n n 2 n 3 n ∞ ∞ 1 f (n) k 1 f (n)k = n ∑ (−1)k−1 ( ) = f + ∑ (−1)k−1 ( k−1 ) k n k n k=1 k=2 ∞ ∞ 1 f (n) k−1 1 ≤ f (n) + f (n) ∑ ( ) < f (n) + f (n) ∑ ( √ )k−1 n n k=2 k ! ! Ã k=2 Ã ∞ 1 1 k = f (n) 1 + ∑ ( √ ) = f (n) 1 + n 1 − √1 k=1 n
µ
∴ 1+
f (n) n
¶n
≤ 2 f for n ≥ 4
≤ e2 f (n) for n ≥ 4
µ ¶ ∞ 1 f (n) k f (n) f (n) 1 f (n) 2 1 f (n) 3 )=n − ( ) + ( ) − .... [1] = n ∑ (−1)k−1 ( ) ln g = n ln (1 + n n 2 n 3 n k n k=1
∞ 1 f (n)k = f (n) + ∑ (−1)k−1 ( k−1 ) k n k=2
∞ 1 f (n)k lim ln g = lim f (n) + ∑ (−1)k−1 ( k−1 ) = lim f (n) n→∞ n→∞ n→∞ k n k=2 ( lim f (n))
∴ lim g(n) = e n→∞ n→∞
FACT 3. If c > 0 is a positive constant independent of n, and b ≥ 1 is another positive constant independent of n, then ∃no ∈ N such that: 1 − (ln1n)b ≤ 1nc for n > no n
Proof: ∵ ∴ 1−
1
1
1 − (ln1n)b
≥ e (ln n)b (from Fact 1 )
− 1 b 1 (ln n) = ≤ e (ln n)b
1 e
1 (ln n)b
1
=
ln n
e (ln n)(b+1)
1 c for large n nn n 1 c ∵ ∃no ∈ N s.t. , ∀n > no ≥ (ln n)(b+1) n =
1
1 (ln n)(b+1)
≤
LEMMA 1. (Jogdeo & Samuels [2]) Given X = Y1 + Y2 + ..., +Yn where ∀i,Yi = Bernoulli(pi ), and ∑ pi = np, the median m of the distribution is either bnpcordnpe, i.e., Pr[X ≤ m] ≥ 21 and Pr[X ≥ m] ≥ 21 . Corollary 1. Given X = Y1 + Y2 + ..., +Yn where ∀i,Yi = Bernoulli(p), the median m of the distribution is either bnpcordnpe, i.e., Pr[X ≤ m] ≥ 12 and Pr[X ≥ m] ≥ 12 . Proof: The proof proceeds by setting p1 = p2 = ... = pn = p and applying the above-stated Lemma. Corollary 2. Given X = Y1 + Y2 + ..., +Yn where n is even, and ∀i,Yi = Bernoulli(p) where p ≥ 12 , the median m of the distribution satisfies m ≥ n2 . Proof: We know that m is either bnpcordnpe. When p = 12 , m = m ≥ bnpc ≥ b n2 c = n2 .
n 2
(as n is even). For p > 12 ,
n
LEMMA 2. (Chernoff Bound) If X = ∑ Xi , where each Xi is Bernoulli(p), then for 0 ≤ β ≤ 1: i=1
Pr[X ≤ (1 − β)E[X]] ≤ exp(−
β2 E[X]) 2
(1) n
LEMMA 3. (Relative Entropy Form of Chernoff-Hoeffding Bound[3]) If X = ∑ Xi , where each Xi is Bernoulli(p), then for 0 ≤ β ≤ 1:
i=1
β
1−β
Pr[X ≥ βn] ≤ e−n(β ln p +(1−β) ln 1−p )
(2)
LEMMA 4. [4] If X1 , X2 ,..., Xn are drawn i.i.d. from alphabet χ according to Q(x), then probability of x is given by: Q(n) (x) = e−n(H(Px )+D(Px ||Q)) (3) where H and P denote the entropy and relative entropy functions (here considered w.r.t base e). Also, for any distributions P and Q, the size of type class T (P) satisfies: 1 enH(P) ≤ |T (p)| ≤ enH(P) |χ| (n + 1)
(4)
and, the probability of the type class T (P) under Q is governed by: 1 e−n(D(P||Q)) ≤ Q(n) (T (p)) ≤ e−n(D(P||Q)) (n + 1)|χ|
(5)
LEMMA 5. Suppose S1 and S2 are sets of Bernoulli random variables, such that S1 = {I1 , I1 , ..., Im } and S2 = {Ik+1 , ..., Ik+m }, where ∀i, Ii = Bernoulli(p). If N1 = ∑ I j and N2 = ∑ I j then: I j ∈S1
I j ∈S2
(6)
Pr[N2 < a|N1 < a] ≥ Pr[N2 < a] Proof: We know that S1 ∩ S2 = {Ik+1 , ..., Im }. Let M1 =
M1 = N1 − b where b = Pr[M1 (9) 2 Consider any set S ∈ P (nbd(u)). Then Pr[ at least half nodes in S faulty] ≥ 21 (from Lemma 1). Hence for any subset S, the probability of obtaining an erroneous value from S is at least 12 . Iteratively sampling over many such subsets is also not useful, as on sampling a sequence of m sets S 1 , S2 , ..., Sm , the probability that at least half the Si ’s had half or more faults, is at least 12 . An alternative way to view this is that corresponding to each fault configuration C1 with t ≥ d2 in nbd(u), there is another configuration C2 with t faults, such that all non-faulty nodes in C1 are faulty in C2 , while the non-faulty nodes in C2 were all faulty in C1 . Then, the faulty nodes can modulate their behavior so that u is unable to distinguish between the case where the correct broadcast value was v 1 and configuration was C1 and the case when the correct value was v2 and the configuration was C2 . THEOREM 2. When failure probability p ≥ 21 , and dn → ∞ (this happens when d = o(n)), lim Pr[ reliable broadcast fails] > η > 0 (for some positive constant η ≤ 1). In particular, if n(1−p) → ∞, d
n→∞
or if p ≥ 1 − o( n1 ), then: lim Pr[ reliable broadcast fails] = 1. n→∞
Proof: Suppose failure probability p ≥ 21 .
Fig. 1.
Division of network into disjoint neighborhoods
a) 12 ≤ p ≤ 1 − γ(0 < γ < 12 ): Note that in this case, γ can be an arbitrarily small constant, but must be independent of n. Consider a particular node j in the network. Then, if j is non-faulty, but more than half of its neighbors are faulty, reliable broadcast fails with probability at least half, as this node cannot get a correct view. Given that there are d neighbors, and each may fail independently with probability p, let Y j denote the number of failed neighbors of j. Then, Y takes values from 0, 1, ..., d, and E[Y ] ≥ d2 . Thus bE[Y ]c ≥ b d2 c = d2 (since d = 4r2 + 4r is always even). Thus, Pr[Y ≥ d2 ] ≥ Pr[Y ≥ bE[Y ]c] ≥ 21 (from lemma 1). Let us call this probability q. When p ≤ 1 − γ (for arbitrarily small constant γ), we have 1 − p ≥ γ > 0. Thus: γ Pr[ j alive; at least half nbd( j) faulty ] ≥ (1 − p)q ≥ > 0 (10) 2 Let us mark out a subset of nodes j such that the neighborhoods of these nodes are all disjoint, as in Fig. 1. Then the number of such nodes that we may obtain is approximately k = b dn c ≥ dn − 1 (a √ more precise number would be (b rn c)2 (where d = 4r2 + 4r), but the loss of precision is negligible for large n). Let I j be an indicator variable that takes value 1 if j is non-faulty but has at least half faulty neighbors. Then Pr[I j = 1] ≥ 2γ > 0, and all I j ’s are independent. Consider the case where dn → ∞. Let X be a random variable indicating ¢ the number of non-faulty nodes with at least half faulty neighbors. Then ¡ E[X] = ∑ Pr[I j = 1] ≥ 2γ dn − 1 → ∞. Thus from the Chernoff Bound in Lemma 2: Pr[X ≤ βE[X]] ≤ e−
(1−β)2 E[X] 2
lim Pr[X > βE[X]] > lim 1 − e−
n→∞
n→∞
(0 < β < 1)
(1−β)2 E[X] 2
= 1(∵ E[X] → ∞)
(11)
Thus, as n → ∞, the number of non-faulty nodes isolated by half or more faulty neighbors will also tend to infinity with probability 1. Given that each of these fails to receive broadcast with probability at least half, application of Chernoff bound leads to the conclusion that the probability of broadcast failure tends to 1.
b) 1 − γ < p ≤ 1 − ω( dn ): This is relevant if d is an increasing function of n and/or p. Once again, consider a particular node j in the network. Then, if j is non-faulty, but more than half of its neighbors are faulty, reliable broadcast fails. Given that there are d neighbors, and each may fail independently with probability p, let Y j denote the number of failed neighbors of j. Then, Y takes values from 0, 1, ..., d, and 1 E[Y ] = pd > (1 − γ)d > 12 d. We set β = 1 − 2(1−γ) and apply the Chernoff bound in Lemma 2. This yields: ³ ´2 1 d 1 − 2(1−γ) Pr[Y j ≤ ) ≤ exp − (1 − γ)d 2 2 ´ ³ Ã ! 1 1 − γ 1 4(1−γ) d < exp − 4 d if 0 < γ < ≤ exp − 2 2 16 1 (as d ≥ 8) e d 1 ∴ Pr[Y j ≥ ] > 1 − 2 e ≤
(12)
(13) (14) (15)
Since 1 − γ < p ≤ 1 − ω( dn ), we have dn (1 − p) → ∞. Also: 1 Pr[ j fault-free; at least half nbd( j) faulty ] ≥ (1 − p)(1 − ) e
(16)
Let us again mark out a subset of nodes j such that the neighborhoods of these nodes are all disjoint, as in Fig. 1. Then the number of such nodes obtained is approximately k = b dn c ≥ dn − 1. Let I j be an indicator variable that takes value 1 if j is non-faulty but has at least half faulty neighbors. Then Pr[I j = 1] ≥ (1 − p)(1 − 1e ), and all I j ’s are independent. Let X = ∑ I j be a random variable denoting ¡ ¢ n(1−p)(1− 1e ) → ∞ if dn → ∞. number of alive but isolated nodes. Then E[X] ≥ (1 − p)(1 − 1e ) dn − 1 ≈ d Thus from the Chernoff Bound in Lemma 2, for 0 < β < 1 (e.g. β = 31 ) : Pr[X ≤ βE[X]] ≤ e−
(1−β)2 E[X] 2
lim Pr[X > βE[X]] > lim 1 − e−
n→∞
n→∞
(1−β)2 E[X] 2
= 1(∵ E[X] → ∞)
(17)
Thus, as n → ∞, the number of non-faulty nodes isolated by half or more faulty neighbors will also tend to infinity with probability 1. c) 1 − ω( dn ) < p ≤ 1 − ω( 1n ): Note that n(1 − p) → ∞. Thus, it is easily seen there will still be a large number of fault-free nodes in the network (and this number will also tend to infinity as n increases). The cases of interest are those in which at least two non-neighboring nodes in the entire network are alive (else the broadcast issue is either trivialized or moot), and as n(1 − p) → ∞ in this range, there will indeed be at least two such alive nodes with probability q → 1 (as may be verified by application of the Chernoff bound from Lemma 2). Then, consider each of these alive nodes, say A and B. The probability that half or more of A’s neighbors are faulty can be no less that that in the previous case, i.e., Pr[ A has half or more faulty neighbors] ≥ 1 − 1e . Similarly, Pr[ B has half or more faulty neighbors] ≥ 1− 1e . Then Pr[at least A or B has half or more faulty neighbors] ≥ 1− 1e > 0. From Theorem 1, we know that having half faulty neighbors leads to failure with probability at least half. Hence reliable broadcast fails with a significant positive probability.
d) p = 1 − Θ( 1n ): Pr[All nodes faulty;broadcast issue moot] = pn ¶ µ α 1 n ≥ 1 − Θ( ) = (1 − g(n))n where g(n) = n n
(18) (19)
lim Pr[All nodes faulty; broadcast issue moot] ³ α ´n n ≥ lim (1 − g(n))) = 1 − ) = e−α > 0 from Fact 2 n→∞ n
(20)
Pr[All nodes faulty;broadcast issue moot] = pn ¶ µ g(n) 1 n = ng(n) → 0 ≥ 1 − o( ) = (1 − g(n))n where n 1/n
(22)
n→∞
(21)
e) p ≥ 1 − o( n1 ) :
lim Pr[All nodes faulty; broadcast issue moot] ¶ µ ng(n) n n ) ≥ lim (1 − g(n))) = lim 1 − n→∞ n→∞ n
n→∞
= e− lim(ng(n)) = 1 from Fact 2
THEOREM 3. When p ≤ 21 − ε(0 < ε < 12 ), and node degree d ≤ c ln
reliable broadcast asymptotically fails with probability 1.
ln n
1 1 2p +ln 2(1−p)
(23)
(24) (25) (26)
(for suitable constant c),
Proof: Suppose failure probability p ≤ 21 − ε, where ε is an arbitrarily small constant. Choose an √ increasing function f (n) = o( n), and a constant 0 < c < 1 such that 2c ln n ≤ ln n − 3 ln ln n − 2 ln f (n), 1 1 n for sufficiently large n. Take n to be large enough so that ln 2p + ln 2(1−p) ≥ fln(n) . Thus we obtain d ≤ c f (n)) < f (n). To illustrate, we take f (n) = (ln n)2 , and n to be large enough so that 1 ln n 1 ≥ ln1n . Setting d ≤ c ln 1 +ln + ln 2(1−p) , for this choice of c, and large enough n, we obtain ln 2p 1 d
≤ c(ln n)2
< (ln n)2 .
2p
2(1−p)
Consider a particular node j in the network. Then, if j is non-faulty, but more than half of its neighbors are faulty, reliable broadcast fails with probability at least half (from Theorem 1). Given that there are d neighbors, and each may fail independently with probability p, let I jk (1 ≤ k ≤ d) denote the indicator variable corresponding to neighbor k of j (enumerated in some order), such that I jk = 1 if k is faulty, and 0 otherwise. Then Y j = ∑ I jk denotes the number of failed neighbors of j. Y takes values d ¡ ¢ from 0, 1, ..., d, and E[Y ] = pd. Pr[Y j ≥ d2 ] = ∑ di pi (1 − p)(d−i) . Let us simply consider the event d 2.
i= d2
Y j = Then we can apply the lower bound from Lemma 4. The variables I jk (1 ≤ k ≤ d) are drawn from χ = {0, 1} as per distribution Q = Be(p), and the distribution P corresponding to Y j = d2 is Be( 21 ) (we
shall refer to this as Q 1 ). |χ| = 2, and 2
1 (d+1)|χ|
=
1 (d+1)2
>
1
3 2 2d
= 23 e−2 ln d (since d ≥ 8). Thus, we obtain:
−d(D(Q 1 ||Q)) d 2 −d(D(Q 1 ||Q))−2 ln d d 1 1 −d(D(P||Q)) 2 2 Pr[Y j ≥ ] ≥ Pr[Y j = ] ≥ e > e e = 2 2 (d + 1)2 3 (d + 1)|χ| ln n
1
1
1
(27)
1
)( 2 ln 2p + 2 ln 2(1−p) )−4 ln ln n 1 2 −(c ln 2p1 +ln 2(1−p) > e 3 1 1 1 (since n is chosen large enough to ensure that ln + ln ≥ , and c < 1, 2p 2(1 − p) ln n leading to d ≤ c(ln n)2 < (ln n)2 )
(29)
from our choice of c
(31)
2 c = e− 2 ln n−4 ln ln n ≥ 3
Let us call this probability q.
2(ln n)3 3n
(28)
(30)
Pr[ j alive; at least half nbd( j) faulty ] ≥ (1 − p)q
(32)
(ln n)3
1 2(ln n)3
= (33) 2 3n 3n Let us mark out a subset of nodes j such that the neighborhoods of these nodes are all disjoint, as in Fig. 1. Then the number of such nodes that we may obtain is approximately k = b dn c ≥ dn − 1. Let I j be an indicator variable that takes value 1 if j is non-faulty but has at least half faulty neighbors. 3 Then Pr[I j = 1] ≥ (ln3nn) , and all I j ’s are independent. Consider the case where dn → ∞, as n → ∞. We 1 1 + ln 2(1−p) ≥ ln1n , i.e. d ≤ c(ln n)2 . Let X be a random have chosen n large enough to ensure that ln 2p variable indicating the number of non-faulty nodes with half or more faulty neighbors. Then ∑ I j , and ¢ 3 ¡ 3 E[X] = ∑ Pr[I j = 1] ≥ (ln3nn) dn − 1 ≈ (ln3dn) > ln3n → ∞ (as d < (ln n)2 ). Thus we can apply the Chernoff bound in Lemma 2 to obtain: >
Pr[X ≤ βE[X]] ≤ e−
lim Pr[X > βE[X]] > lim 1 − e
n→∞
n→∞
(1−β)2 E[X] − 2
(1−β)2 E[X] 2
(34)
= 1 ∵ E[X] → ∞
(35)
Thus, as n → ∞, the number of non-faulty nodes isolated by half or more faulty neighbors will also tend to infinity with probability 1. Since, each of them fails to receive broadcast with probability at least half, the probability that some nodes will indeed fail to receive the broadcast tends to 1: lim Pr[ reliable broadcast fails] → 1
n→∞
VIII. S UFFICIENT C ONDITION FOR R ELIABLE B ROADCAST We now present a sufficient condition for the asymptotic achievability of reliable broadcast. ln n THEOREM 4. When p < 21 , and node degree d ≥ max{dmin , 16 ln 1 +ln p
1 2(1−p)
} = max{dmin , 16 D(Qln1n||P) )} 2
(recall that dmin = 8 corresponding to r = 1), reliable broadcast is asymptotically achievable with probability 1.
1 1 + ln 2(1−p) Note that when ln 2p ≤ 16nln n , the degree exceeds total network size n, and thus the sufficient condition ceases to be relevant, merely indicating that having a single-hop network suffices for reliable broadcast (which is the trivial sufficient condition for the assumed radio network model). Thus the sufficient 1 1 + ln 2(1−p) condition is of interest only so long as ln 2p > 16nln n .
x=a−1 x=a+1
y=b+r
qnbdC (a, b)
y=b
y=b+r
qnbdD (a, b)
y=b
(a, b)
x=a−r
x=a
qnbdD0
qnbdC 0 y=b+1
(a, b)
y=b−1
qnbdB 0
qnbdB (a, b) qnbdA (a, b)
y=b−r
y=b−r
qnbdA0
x=a+r
x=a−r−1
Fig. 2.
y=b+1 y=b−1
Depiction of qnbdA , qnbdB , qnbdC , qnbdD Region qnbdA (a, b) qnbdB (a, b) qnbdC (a, b) qnbdD (a, b) qnbdA0 (a, b) qnbdB0 (a, b) qnbdC0 (a, b) qnbdD0 (a, b)
Fig. 3.
x-extent a ≤ x ≤ (a + r) (a − r) ≤ x ≤ (a − 1) (a − r) ≤ x ≤ a (a + 1) ≤ x ≤ (a + r) (a + 1) ≤ x ≤ (a + r) (a − r) ≤ x ≤ a (a − r) ≤ x ≤ (a − 1) a ≤ x ≤ (a + r)
x=a
x=a+r+1
Depiction of qnbdA0 , qnbdB0 , qnbdC0 , qnbdD0 y-extent (b − r) ≤ y ≤ (b − 1) (b − r) ≤ y ≤ b (b + 1) ≤ y ≤ (b + r) b ≤ y ≤ (b + r) (b − r) ≤ y ≤ b (b − r) ≤ y ≤ (b − 1) b ≤ y ≤ (b + r) (b + 1) ≤ y ≤ (b + r)
TABLE I
S PATIAL E XTENTS OF Q UARTER N EIGHBORHOODS
a) p ≤ o( 1n ): When the failure probability is so small as to fall in this range, the probability of even a single node failing approaches 0 asymptotically, and thus reliable broadcast is trivially ensured even with the minimum transmission range of 1. This may be seen thus: Pr[No failures;trivial broadcast] = (1 − p)n µ ¶ 1 n ≥ 1 − o( ) n lim Pr[No failures;trivial broadcast]
≥ lim
n→∞
µ
n→∞ ¶n
1 1 − o( ) n
1
= e− lim(no( n )) = 1 from Fact 2
(36) (37) (38) (39)
b) p = Ω( n1 ): We define a term called quarter-neighborhood of a node (x, y), and denote it by qnbd(x, y). We associate eight quarter-neighborhoods with each node: qnbd A , qnbdB , qnbdC , qnbdD , qnbdA0 , qnbdB0 , qnbdC0 , qnbdD0 . The quarter-neighborhoods for a node (a, b) are depicted in Fig. 2 and 3, and their spatial extents are tabulated in Table I. Observe that qnbd B (a, b) = qnbdA0 (a − r − 1, b), qnbdC (a, b) = qnbdA (a − r, b + r + 1), and qnbdD (a, b) = qnbdA0 (a, b + r + 1). Similarly, qnbdB0 (a, b) = qnbdA (a − r − 1, b), qnbdC0 (a, b) = qnbdA0 (a − r − 1, b + r), and qnbdD0 (a, b) = qnbdA (a, b + r + 1) Thus if we simply consider qnbdA (u) and qnbdA0 (u)∀ nodes u, we will have considered all quarter-neighborhoods,
i.e. the number of distinct (but not disjoint) quarter-neighborhoods is 2n. Henceforth, we shall sometimes use Q(x, y) to refer to qnbdA (x, y), and Q0 (x, y) to refer to qnbdA0 (x, y). The population of any qnbd is r(r + 1), and since d = 4r 2 + 4r = 4r(r + 1), the qnbd population = d4 . We now state and prove the following result which is crucial to proving our sufficient condition for reliable broadcast: THEOREM 5. If p < 12 , d ≥ max{dmin , 16 ln
ln n
1 1 2p +ln 2(1−p)
} = max{dmin , 16 D(Qln1n||P) )}, then: 2
d faults in 8 Q(x, y) and Q0 (x, y)] → 1
lim Pr[ ∀(x, y) less than
n→∞
Proof: As shown above, the population of any qnbd is d4 . Each node may fail independently with probability p.Let Y(x,y) be a random variable denoting the number of faulty nodes in Q(x, y). Then 1 − 1, we may then apply the relative entropy form of the Chernoff bound E[Y(x,y) ] = p d4 . Using δ = 2p ln n ln n } ≥ 16 ln 1 +ln . Thus, we (Lemma 3) to Y(x,y) = ∑ I j . Note that d ≥ max{dmin , 16 ln 1 +ln 1 1 obtain:
2p
j∈nbd(x,y)
2(1−p)
d − d ( 1 ln 1 + 1 ln 1 ) Pr[Y(x,y) ≥ ] ≤ e 4 2 2p 2 2(1−p) 8 16 ln n 1 +ln 1 4(ln 2p 2(1−p)
2(1−p)
(40)
1 1 ) ))( 21 ln 2p + 21 ln 2(1−p)
(41) 1 = e−2 ln n = 2 (42) n be a random variable denoting the number of faulty nodes in Q 0 (x, y), we obtain ≤e
0 Similarly, setting Y(x,y) that:
−(
2p
d 1 0 ≥ ]≤ 2 Pr[Y(x,y) 8 n
(43)
0 ’s are not independent, as they are not all disjoint. However, it may be seen that where The Y(x,y) ’s and Y(x,y) dependence exists, it is that of positive correlation (Lemma 5). Thus Pr[Y(x0 ,y0 ) < d8 |Y(x,y) < d8 ] ≥ Pr[Y(x0 ,y0 ) < d 0 d d d d d 0 8 ], and Pr[Y(x0 ,y0 ) < 8 |Y(x,y) < 8 ] ≥ Pr[Y(x0 ,y0 ) < 8 ]. Similarly, we obtain that: Pr[Y(x0 ,y0 ) < 8 |Y(x,y) < 8 ] ≥ 0 Pr[Y(x0 0 ,y0 ) < d8 ], and Pr[Y(x0 0 ,y0 ) < d8 |Y(x,y) < d8 ] ≥ Pr[Y(x0 0 ,y0 ) < d8 ] Hence:
d d and Y 0 (x, y) < ] 8 8 d d ≥ ∏ Pr[Y(x0 ,y0 ) < ] ∏ Pr[Y(x0 0 ,y0 ) < ] µ8 ¶ µ ¶8 1 n 1 n = 1− 2 1− 2 n n ¶ µ 1 2 = 1− 2 n n d d ∴ lim Pr[∀(x, y),Y (x, y) < and Y 0 (x, y) < ] n→∞ 8 8 µ ¶2 2 1 ≥ lim 1 − 2 n = e− lim( n ) = 1 from Fact 2 n→∞ n Pr[∀(x, y),Y (x, y)
0 (independent of n), then for sufficiently large n, the necessary condition would hold for all p. Also note that 1 − ln1n < 1nc for large n (from Fact 3). Thus, the values of p for which our necessary n 1 condition holds are those in which the transmission range remains less than D. When p ≥ 1 − n1+ε , all 1 nodes r are faulty with probability approaching 1, and the issue of connectivity is moot. When p ≤ nc , r=
1.
c ln n ln 1p
≤ 1, and for this range of p, the necessary condition lapses to having the minimum range of
a) p ≤ 1 − ln1n : Consider a particular node j in the network. Then, if j is non-faulty, but all its neighbors are faulty, we have a potential disconnection event. Given that there are d neighbors, and each may fail independently with probability p, the probability that j does not fail, but all nodes in nbd( j) fail, is (1 − p)pd . We choose a constant 0 < c < 1 such that c ln n ≤ ln n − 4 ln ln n, for sufficiently large
Fig. 6.
Nodes having disjoint neighborhoods
n. In general, c can be chosen very close to 1, e.g., 1 − ε(0 < ε < 1), and the rcondition will hold for ln n 1 n > no , for some no . Since p ≤ 1 − ln1n , we obtain that 1−p ≤ ln n. Let r ≤ 8cln 1 . The node degree d=
4r2 + 4r
≤
4r2 + 4r2
=
8r2 ,
p
ln n for n ≥ 1. Thus, for our choice of r, it turns out that d ≤ c ln 1 . Then, it p
may be seen that:
Pr[ A given node j is alive, but isolated] ≥ Pr[ j is alive and all neighbors of j are faulty ] ln n
1 c ln 1p p ln n 1 1 1 = = c c ln n n n ln n
= (1 − p)pd >
≥
(ln n)3 (from our choice of c) n
(50) (51) (52) (53) (54)
Let us mark out a subset of nodes j such that the neighborhoods of these are all disjoint, ³ √ nodes ´ n 2 as in Fig. 6. Then the number of such nodes that we may obtain = b 2r+1 c ≥ 9rn2 − 1 (since √ n may not be multiple of 2r + 1). Let I j be an indicator variable that takes value 1 if j is alive 3 but isolated. Then Pr[I j = 1] ≥ (lnnn) , and all I j ’s are independent. Let X be a random variable denoting the number of nodes from the chosen set that are alive and isolated. Then X = ∑ I j , and µ 1 ¶ n ln 1 1 n ln p 1− ln n (ln n)3 (ln n)3 E[X] ≥ n ≥ 19 (ln n)2 ln 1−1 1 ≥ 19 ln n → ∞. We can thus apply the 9c ln n − 1 ≥ n 9 ln n ln n
Chernoff bound from Lemma 2: Thus, with suitable 0 < c < 1 and β = E[X]−1 E[X] , we obtain that for r ln n p < 1 − ln1n , if r ≤ 8cln 1 , then E[X] → ∞, and hence lim Pr[ At least two alive nodes are isolated] = 1. p
n→∞
Observe that actually the necessary condition would hold for all p such that E[X] → ∞. For instance, the above analysis holds for all p ≤ 1 − (ln1n)b (where b is a constant), with a corresponding suitably
(b+2)
varying choice of c to ensure that Pr[I j = 1] ≥ (ln n)n . Besides, if E[X] → γ > 0, the asymptotic disconnection probability is still a positive finite quantity, and the condition is still necessary for asymptotic connectedness probability to approach 1. 1 b) p ≥ 1 − n1+ε : When the failure probability becomes so high as to fall in this range, we obtain:
µ
= lim 1 − 1 − n→∞
1
n1+ε
lim Pr[ Any node is alive] = 1 − pn
n→∞ ¶ n
(55)
1
= 1 − e− lim( nε ) = 0 from Fact 2
(56)
Thus the network is trivially connected by definition, regardless of degree. XV. N ECESSARY C ONDITION FOR C OVERAGE We now show that for the network to be asymptotically covered r with probability approaching 1, it ln n is necessary that the transmission range r satisfy: r ≥ max{1, Ω( ln 1 )}, i.e., the node degree be d ≥ p
ln n max{1, Ω( ln 1 )}. p
ln n THEOREM 9. For p < 1 − ln1n , for a suitable constant 0 < c < 1, if d < c ln 1 : p
lim Pr[Some point is not covered] → 1
n→∞
Proof: As in the case of connectivity it is obvious that r must be at least 1, else some points will not be covered. We handle two subranges of p separately. a) p < 1 − ln1n : The proof relies on subdivision of the network into disjoint neighborhoods, as in Fig. 6. If there exists at least one neighborhood with absolutely no nodes alive (neither the neighborhood center nor its neighbors), then the center of that neighborhood is not covered. Thus we seek to determine the probability of such an event. We choose a constant 0 < c < 1 such that ensures that
1 nc
≥
(ln n)3 n
for large n. Let r
9 ln n ≤ ln n − 8 cr ln n ≤ 8cln 1 . The p
3 ln ln n, for sufficiently large n. This neighborhood population is given by
ln n d + 1 = 4r2 + 4r + 1 ≤ 4r2 + 4r2 + r2 = 9r2 , for n ≥ 1. Thus, d + 1 ≤ 89 c ln 1 . Let I j be an indicator p
variable that takes value 1 if there is no alive node in the neighborhood centered at node j, and value 0 otherwise. Then Pr[X j = 1] = pd+1 = p
9 ln n 8 c ln 1 p
=
(ln n)3 n
(from our choice of c). Let X = ∑ I j be a random 3
8(ln n)2 ln 1
p variable indicating the number of neighborhoods with no alive node. Then E[X] = (ln9rn)2 = 9c (after plugging in the chosen value of r). If p < 1 − ln1n , then E[X] ≥ ln n(ln n ln 1−1 1 ) > ln n → ∞ (from ln n
Fact 1), and application of the Chernoff bound from Lemma 2 yields that Pr[X = 0] ≤ exp(− E[X] 2 ) → 0. Thus there is some uncovered region with probability 1. Similar to the necessary condition for connectivity, observe that this necessary condition would hold for all p such that E[X] → ∞. In particular, the above analysis holds for all p ≤ 1 − (ln1n)b (where b is a constant), with a corresponding suitably varying choice of c to ensure that Pr[I j = 1] ≥
(ln n)(b+2) . n
Also, if E[X] → γ > 0, the asymptotic probability of some point being uncovered is a positive finite quantity, and the condition is still necessary for asymptotic coverage probability to approach 1. ³ ´n 1 1 n b) p ≥ 1 − n1+ε (0 < ε < 1): We obtain that Pr[ no nodes alive ] = p ≥ 1 − n1+ε . As n → ∞, the following holds: lim Pr[some point not covered] ≥ Pr[no node alive] µ ¶n 1 1 = e− lim( nε ) = 1 from Fact 2 = lim 1 − 1+ε n→∞ n
(57)
n→∞
(58)
Thus the network is trivially not covered, regardless of transmission range. XVI. S UFFICIENT C ONDITION FOR C ONNECTIVITY AND C OVERAGE We now present a sufficient condition for the asymptotic existence of both connectivity and coverage. ln n THEOREM 10. When d ≥ 32 ln 1 , the network is asymptotically connected and covered with probability
1.
p
1 a) p ≤ n1+ε : When the failure probability is so small as to fall in this range, the probability of even a single node failing approaches 0 asymptotically, and thus connectivity and coverage is trivially ensured even with the minimum transmission range of 1. This may be seen thus:
Pr[No failures;full connectivity/coverage] = (1 − p)n ¶n µ 1 ≥ 1 − 1+ε n
(59) (60)
lim Pr[No failures;full connectivity/coverage] ¶n µ 1 1 = e− lim( nε ) = 1 from Fact 2 ≥ lim 1 − 1+ε n→∞ n
(61)
n→∞
(62)
Proof: b) p = Ω( n1 ): Consider the subdivision of the grid as depicted in Fig. 7, so that the resulting cells have x-extents (y-extents) 0 to a, a + 1 to a + b, a + b + 1 to 2a + b + 1, and so on. Here a = b 2r c and b = r − a = r − b 2r c. Then, each node is within range of all other nodes in the cells adjoining its own. Thus it is obvious that if each square has at least one non-faulty node, there exists a connected backbone that covers all points, and hence all nodes. Thus all non-faulty nodes are connected to each other via this backbone. The dimensions 2 of the cells thus obtained can be (a + 1)2 , (a + 1)b or b2 . Thus the population k of any cell satisfies k ≥ r4 , and the maximum possible number of cells m ≤ 4n . Then: r2 r2
Pr[ at least one node alive in a given cell ] = 1 − pk ≥ 1 − p 4 µ ¶ 4n2 r r2 ∴ Pr[ at least 1 node alive in each cell] ≥ 1 − p 4
(63) (64)
2a+b+1 a+b
Pop. (a+1)b
a
Pop. 2
b
Pop. Pop. (a+1) 2 (a+1)b
0 0
a
Fig. 7.
Let us choose r ≥
r
8 ln n . ln 1p
a+b
2a+b+1
Subdivision of network into cells
Then: 1
µ ¶ n ln p 2 ln n r2 Pr[at least 1 node alive in each cell] ≥ 1 − p 4
Since p ≥ α 1n for some constant α , ln 1p ≤ ln n − ln α. Hence:
1
1
µ
Pr[at least 1 node alive in each cell] ≥ 1 − p
¶ n ln p 2
r 4
2 ln n
=
(65)
Ã
1− p
! n ln p
¶ n (1− ln α )
(66)
= e− lim( 2n ) = 1 from Fact 2
(68)
2 ln n ln 1p
2 ln n
1 ≥ 1− 2 n µ
2
ln n
(67)
Thus, by application of Fact 2, we obtain: lim Pr[at least 1 node alive in each cell] ≥ lim
n→∞
n→∞
µ
1 1− 2 n
¶ n (1− ln α ) 2
ln n
1
Since this condition ensures connectivity and coverage, we obtain that: lim Pr[network is connected and covered] → 1
n→∞
(69)
XVII. C ONDITIONS IN E UCLIDEAN M ETRIC We show that our results derived for L∞ metric continue to hold for L2 metric, with only the constants in the theta notation changing. LEMMA 8. If the network is asymptotically connected (covered) in L ∞ for all r ≥ rmin , then the network √ is connected (covered) asymptotically in L2 for all r ≥ rmin 2. Proof: The proof is by contradiction. Suppose that, for a given failure configuration, the network √ is asymptotically connected in L∞ for all r ≥ rmin but is not asymptotically connected for all r ≥ rmin 2
r
r √ r 2 r
Fig. 8.
r
Relationship between L∞ and L2 neighborhoods
in L2 . Observe √ that it is possible to circumscribe a L∞ neighborhood of range r by√a L2 neighborhood of range r 2 (Fig. 8). Hence the nodes in an L2 network of transmission range r 2 can be made to simulate the operation of nodes in a L∞ network with range r (as the L∞ neigborhood is fully contained within the L2 neighborhood). This √ implies that if the L∞ network of range r is connected (covered), so must be the L2 network of range r 2. If there is some r ≥ rmin for√which the L∞ network of range r is connected (covered) asymptotically, but the L2 network of range r 2 is not, we obtain a contradiction, as connectedness (coverage) of the L∞ network would imply connectedness (coverage) of the L2 network. LEMMA 9. If the network is asymptotically disconnected (not covered) in L ∞ for all r ≤ rmin , then the network is disconnected (not covered) asymptotically in L2 for all r ≤ rmin . Proof: The proof is by contradiction. Suppose that the network is asymptotically disconnected (not covered) in L∞ for range r, but is not disconnected (not covered) in L2 for range r. Observe that an L∞ neighborhood of transmission range r circumscribes an L2 neighborhood of range r (Fig. 8). Thus, for any given random failure configuration, if the L2 network of range r were connected (covered), so would be the L∞ network of radius r, as we could simply make the nodes in the L ∞ network simulate the behavior of nodes in the L2 network, and obtain connectedness (coverage). Hence, if the L 2 network of range r ≤ rmin is not asymptotically disconnected (not covered), the L∞ network of range r ≤ rmin must also not be disconnected (not covered). This yields a contradiction. XVIII. D ISCUSSION It is interesting to note that in case of a grid network, the necessary and sufficient node degree turns µ ¶ log n out to be Θ log ) (when expressed in our notation) for the case of a , as compared to Θ( log (n(1−p)) 1 1−p p
randomly deployed network, where sensors are active with probability 1 − p [14]. However, it is not difficult to see that such a difference is to be expected. In a grid network, as failure(or sleep) probability p → 0, the network tends towards a deterministic topology, whereas in a random network, if failure or sleep probability p → 0, the network can only tend towards a denser but still random network. Thus, at small values of p, a very small degree will suffice for a grid network, but may not for a random network. At larger p values, the grid network exhibits increasing randomness and begins to resemble a network with random deployment. Thus, one may see that the two expressions are within a small range of each other when p is large (given sufficiently large n), but diverge as p → 0. Another observation is that the form of the results is very similar to results obtained by us for reliable broadcast in a grid network with Byzantine failures. For Byzantine failures, we have ¶ µ obtained that the necessary and sufficient conditions for reliable broadcast entail a node degree of Θ
ln n 1 1 ln 2p +ln 2(1−p)
,
which may be re-stated as Θ
µ
ln n D(Q 1 ||P) 2
¶
where Q 1 denotes a distribution with failure probability 2
1 2,
P
denotes the actual distribution with failure probability p, and D(Q||P) denotes the relative entropy (or Kullback-Leibler distance) ´between distributions Q and P. Similarly, one may view the node degree for ³ ln n connectivity as Θ D(Q1 ||P) , where Q is the distribution with failure probability 1, and P is the actual failure distribution. XIX. N ON -T OROIDAL N ETWORKS We have made the assumption that the network is toroidal, so as to avoid edge effects. However, we can see that the degree of any node at the outermost edge is no more than d, and at least d4 (where d is the uniform degree that each node would have in the toroidal case). Thus, the necessary condition would continue to hold as is (since some nodes having a lesser degree can only increase the probability of disconnection). The construction used to prove the sufficient condition also continues to hold as is, since all full-cells in the tiling will have at least one active node each, and even if there are regions at the fringes left-over, they will still fall within range of some active node in the nearest full tile (due to the chosen dimensions of the cells). Thus, the results are not affected. A similar argument leads to the conclusion that the coverage results are not affected. R EFERENCES [1] G. B. Thomas, Jr. and R. L. Finney, Calculus and Analytic Geometry. Addison-Wesley Publishing Company, 1992. [2] K. Jogdeo and S. M. Samuels, “Monotone convergence of binomial probabilities and a generalization of ramanujan’s equation,” The Annals of Mathematical Statistics, vol. 39, no. 4, pp. 1191–1195, August 1968. [3] W. Hoeffding, “Probability inequalities for sums of bounded random variables,” Journal of the American Statistical Association, vol. 58, no. 301, pp. 13–30, Mar. 1963. [4] T. M. Cover and J. A. Thomas, Elements of Information Theory. John Wiley & Sons, Inc., 1991. [5] E. Kreyszig, Advanced Engineering Mathematics, 7th ed. John Wiley & Sons, 1993. [6] C.-Y. Koo, “Broadcast in radio networks tolerating byzantine adversarial behavior,” in Proceedings of the twenty-third annual ACM symposium on Principles of distributed computing. ACM Press, 2004, pp. 275–282. [7] V. Bhandari and N. H. Vaidya, “On reliable broadcast in a radio network,” in PODC ’05: Proceedings of the twenty-fourth annual ACM SIGACT-SIGOPS symposium on Principles of distributed computing. ACM Press, 2005, pp. 138–147. [8] E. Kranakis, D. Krizanc, and A. Pelc, “Fault-tolerant broadcasting in radio networks,” J. Algorithms, vol. 39, no. 1, pp. 47–67, 2001. [9] A. Pelc and D. Peleg, “Feasibility and complexity of broadcasting with random transmission failures,” in PODC ’05: Proceedings of the twenty-fourth annual ACM SIGACT-SIGOPS symposium on Principles of distributed computing. ACM Press, 2005, pp. 334–341. [10] V. Bhandari and N. H. Vaidya, “On reliable broadcast in a radio network: A simplified characterization,” Technical Report, CSL, UIUC, May 2005. [11] A. Pelc and D. Peleg, “Broadcasting with locally bounded byzantine faults,” Information Processing Letters, vol. 93, no. 3, pp. 109–115, Feb 2005. [12] P. Gupta and P. R. Kumar, “Critical power for asymptotic connectivity in wireless networks,” in Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W.H. Fleming, W. M. McEneany, G. Yin, and Q. Zhang, Eds. Boston: Birkhauser, 1998, pp. 547–566. [13] F. Xue and P. R. Kumar, “The number of neighbors needed for connectivity of wireless networks,” Wirel. Netw., vol. 10, no. 2, pp. 169–181, 2004. [14] D. Kim, C. Hsin, and M. Liu, “Asymptotic connectivity of low duty-cycled wireless sensor networks,” in Proc. MILCOM, 2005. [15] S. Shakkottai, R. Srikant, and N. Shroff, “Unreliable sensor grids: Coverage, connectivity, and diameter,” in Proc. of Infocom 2003, 2003. [16] S. Kumar, T. H. Lai, and J. Balogh, “On k-coverage in a mostly sleeping sensor network,” in MobiCom ’04: Proceedings of the 10th annual international conference on Mobile computing and networking. New York, NY, USA: ACM Press, 2004, pp. 144–158. [17] S. Shakkottai, R. Srikant, and N. Shroff, “Correction to unreliable sensor grids: Coverage, connectivity, and diameter,” Personal Communication, 2005. [18] X. Liu and R. Srikant, “An information-theoretic view of connectivity in wireless sensor networks,” in the first IEEE Communications Society Conference on Sensor and Ad Hoc Communications and Networks (SECON). Santa Clara, CA: IEEE, Oct. 4-7 2004.
Vartika Bhandari
Nitin H. Vaidya
Dept. of Electrical and Computer Eng., and Coordinated Science Laboratory University of Illinois at Urbana-Champaign [email protected]
Dept. of Computer Science, and Coordinated Science Laboratory University of Illinois at Urbana-Champaign [email protected]
Abstract We consider a wireless grid network in which nodes are prone to failure. In the considered failure mode, each node has an independent probability of failure p, and failures may be either Byzantine or crash-stop in nature. All nodes are assumed to have a common transmission range r, and a resultant common degree d. We establish necessary and sufficient conditions for the degree of each node as a function of the total network size n and the failure probability p, so as to ensure that reliable broadcast succeeds with probability 1, as n → ∞. Our results 1 indicate that reliable broadcast ¶ is asymptotically achievable with Byzantine failures if p < 2 , and the degree of µ ln n 1 1 +ln ln 2p 2(1−p)
. These results exhibit similarity of form to results obtained for crash-stop failures µ ¶ that indicate a required degree of Θ ln n1 for p < 1.
each node is Θ
ln p
I. I NTRODUCTION We consider the problem of reliable broadcast in a wireless grid network prone to probabilistic failures. The node failures are assumed i.i.d. with probability p. Two separate failure types are considered, viz., Byzantine and crash-stop. We show that when nodes failures, reliable broadcast requires µ exhibit Byzantine ¶ ln n that p < 21 , and the node degree must be Θ ln 1 +ln for asymptotic achievability of reliable 1 2p 2(1−p) µ ¶ broadcast. This may alternatively be stated as Θ D(Qln1n||P) where Q 1 denotes a distribution with failure 1 2,
2
2
probability P denotes the actual distribution with failure probability p, and D(Q||P) denotes the relative entropy (or Kullback-Leibler distance) between distributions Q and P. For crash-stop failures, the problem µ ¶
ln n of reliable broadcast is equivalent to connectivity. For this case, we show that node degree must be Θ ln 1 p ³ ´ ln n for p < 1, or alternatively stated, Θ D(Q1 ||P) , where Q1 is the distribution with failure probability 1. This report comprises two independent parts. We consider the case of Byzantine failures in the first part. In the second part, we address the issue of crash-stop failures as a connectivity question. Along with connectivity, we also obtain conditions for coverage that point toward the same expression, except for the constants involved.
Minor Corrections made in Feb 2006. This report supercedes an earlier report ”Connectivity and Coverage in Failure-Prone Wireless Grid Networks”, dated September 2005. This research is supported in part by the National Science Foundation, and a Vodafone Graduate Fellowship.
II. S OME U SEFUL M ATHEMATICAL R ESULTS We state some mathematical results that have been used in our proofs: 1 ≥x FACT 1. ∀x ∈ [0, 1] : ln 1−x
1
FACT 2. If f (n) ≤ n 2 −ε (0 < ε < 12 ): µ ¶ f (n) n ( lim f (n)) lim 1 + = e n→∞ n→∞ n 1
Proof: Let f (n) ≤ n 2 −ε , where 0 < ε < 12 . Let g(n) = (1 +
f (n) n n ) .
Then:
µ ¶ f (n) 1 f (n) 2 1 f (n) 3 f (n) )=n − ( ) + ( ) − .... [1] ln g = n ln (1 + n n 2 n 3 n ∞ ∞ 1 f (n) k 1 f (n)k = n ∑ (−1)k−1 ( ) = f + ∑ (−1)k−1 ( k−1 ) k n k n k=1 k=2 ∞ ∞ 1 f (n) k−1 1 ≤ f (n) + f (n) ∑ ( ) < f (n) + f (n) ∑ ( √ )k−1 n n k=2 k ! ! Ã k=2 Ã ∞ 1 1 k = f (n) 1 + ∑ ( √ ) = f (n) 1 + n 1 − √1 k=1 n
µ
∴ 1+
f (n) n
¶n
≤ 2 f for n ≥ 4
≤ e2 f (n) for n ≥ 4
µ ¶ ∞ 1 f (n) k f (n) f (n) 1 f (n) 2 1 f (n) 3 )=n − ( ) + ( ) − .... [1] = n ∑ (−1)k−1 ( ) ln g = n ln (1 + n n 2 n 3 n k n k=1
∞ 1 f (n)k = f (n) + ∑ (−1)k−1 ( k−1 ) k n k=2
∞ 1 f (n)k lim ln g = lim f (n) + ∑ (−1)k−1 ( k−1 ) = lim f (n) n→∞ n→∞ n→∞ k n k=2 ( lim f (n))
∴ lim g(n) = e n→∞ n→∞
FACT 3. If c > 0 is a positive constant independent of n, and b ≥ 1 is another positive constant independent of n, then ∃no ∈ N such that: 1 − (ln1n)b ≤ 1nc for n > no n
Proof: ∵ ∴ 1−
1
1
1 − (ln1n)b
≥ e (ln n)b (from Fact 1 )
− 1 b 1 (ln n) = ≤ e (ln n)b
1 e
1 (ln n)b
1
=
ln n
e (ln n)(b+1)
1 c for large n nn n 1 c ∵ ∃no ∈ N s.t. , ∀n > no ≥ (ln n)(b+1) n =
1
1 (ln n)(b+1)
≤
LEMMA 1. (Jogdeo & Samuels [2]) Given X = Y1 + Y2 + ..., +Yn where ∀i,Yi = Bernoulli(pi ), and ∑ pi = np, the median m of the distribution is either bnpcordnpe, i.e., Pr[X ≤ m] ≥ 21 and Pr[X ≥ m] ≥ 21 . Corollary 1. Given X = Y1 + Y2 + ..., +Yn where ∀i,Yi = Bernoulli(p), the median m of the distribution is either bnpcordnpe, i.e., Pr[X ≤ m] ≥ 12 and Pr[X ≥ m] ≥ 12 . Proof: The proof proceeds by setting p1 = p2 = ... = pn = p and applying the above-stated Lemma. Corollary 2. Given X = Y1 + Y2 + ..., +Yn where n is even, and ∀i,Yi = Bernoulli(p) where p ≥ 12 , the median m of the distribution satisfies m ≥ n2 . Proof: We know that m is either bnpcordnpe. When p = 12 , m = m ≥ bnpc ≥ b n2 c = n2 .
n 2
(as n is even). For p > 12 ,
n
LEMMA 2. (Chernoff Bound) If X = ∑ Xi , where each Xi is Bernoulli(p), then for 0 ≤ β ≤ 1: i=1
Pr[X ≤ (1 − β)E[X]] ≤ exp(−
β2 E[X]) 2
(1) n
LEMMA 3. (Relative Entropy Form of Chernoff-Hoeffding Bound[3]) If X = ∑ Xi , where each Xi is Bernoulli(p), then for 0 ≤ β ≤ 1:
i=1
β
1−β
Pr[X ≥ βn] ≤ e−n(β ln p +(1−β) ln 1−p )
(2)
LEMMA 4. [4] If X1 , X2 ,..., Xn are drawn i.i.d. from alphabet χ according to Q(x), then probability of x is given by: Q(n) (x) = e−n(H(Px )+D(Px ||Q)) (3) where H and P denote the entropy and relative entropy functions (here considered w.r.t base e). Also, for any distributions P and Q, the size of type class T (P) satisfies: 1 enH(P) ≤ |T (p)| ≤ enH(P) |χ| (n + 1)
(4)
and, the probability of the type class T (P) under Q is governed by: 1 e−n(D(P||Q)) ≤ Q(n) (T (p)) ≤ e−n(D(P||Q)) (n + 1)|χ|
(5)
LEMMA 5. Suppose S1 and S2 are sets of Bernoulli random variables, such that S1 = {I1 , I1 , ..., Im } and S2 = {Ik+1 , ..., Ik+m }, where ∀i, Ii = Bernoulli(p). If N1 = ∑ I j and N2 = ∑ I j then: I j ∈S1
I j ∈S2
(6)
Pr[N2 < a|N1 < a] ≥ Pr[N2 < a] Proof: We know that S1 ∩ S2 = {Ik+1 , ..., Im }. Let M1 =
M1 = N1 − b where b = Pr[M1 (9) 2 Consider any set S ∈ P (nbd(u)). Then Pr[ at least half nodes in S faulty] ≥ 21 (from Lemma 1). Hence for any subset S, the probability of obtaining an erroneous value from S is at least 12 . Iteratively sampling over many such subsets is also not useful, as on sampling a sequence of m sets S 1 , S2 , ..., Sm , the probability that at least half the Si ’s had half or more faults, is at least 12 . An alternative way to view this is that corresponding to each fault configuration C1 with t ≥ d2 in nbd(u), there is another configuration C2 with t faults, such that all non-faulty nodes in C1 are faulty in C2 , while the non-faulty nodes in C2 were all faulty in C1 . Then, the faulty nodes can modulate their behavior so that u is unable to distinguish between the case where the correct broadcast value was v 1 and configuration was C1 and the case when the correct value was v2 and the configuration was C2 . THEOREM 2. When failure probability p ≥ 21 , and dn → ∞ (this happens when d = o(n)), lim Pr[ reliable broadcast fails] > η > 0 (for some positive constant η ≤ 1). In particular, if n(1−p) → ∞, d
n→∞
or if p ≥ 1 − o( n1 ), then: lim Pr[ reliable broadcast fails] = 1. n→∞
Proof: Suppose failure probability p ≥ 21 .
Fig. 1.
Division of network into disjoint neighborhoods
a) 12 ≤ p ≤ 1 − γ(0 < γ < 12 ): Note that in this case, γ can be an arbitrarily small constant, but must be independent of n. Consider a particular node j in the network. Then, if j is non-faulty, but more than half of its neighbors are faulty, reliable broadcast fails with probability at least half, as this node cannot get a correct view. Given that there are d neighbors, and each may fail independently with probability p, let Y j denote the number of failed neighbors of j. Then, Y takes values from 0, 1, ..., d, and E[Y ] ≥ d2 . Thus bE[Y ]c ≥ b d2 c = d2 (since d = 4r2 + 4r is always even). Thus, Pr[Y ≥ d2 ] ≥ Pr[Y ≥ bE[Y ]c] ≥ 21 (from lemma 1). Let us call this probability q. When p ≤ 1 − γ (for arbitrarily small constant γ), we have 1 − p ≥ γ > 0. Thus: γ Pr[ j alive; at least half nbd( j) faulty ] ≥ (1 − p)q ≥ > 0 (10) 2 Let us mark out a subset of nodes j such that the neighborhoods of these nodes are all disjoint, as in Fig. 1. Then the number of such nodes that we may obtain is approximately k = b dn c ≥ dn − 1 (a √ more precise number would be (b rn c)2 (where d = 4r2 + 4r), but the loss of precision is negligible for large n). Let I j be an indicator variable that takes value 1 if j is non-faulty but has at least half faulty neighbors. Then Pr[I j = 1] ≥ 2γ > 0, and all I j ’s are independent. Consider the case where dn → ∞. Let X be a random variable indicating ¢ the number of non-faulty nodes with at least half faulty neighbors. Then ¡ E[X] = ∑ Pr[I j = 1] ≥ 2γ dn − 1 → ∞. Thus from the Chernoff Bound in Lemma 2: Pr[X ≤ βE[X]] ≤ e−
(1−β)2 E[X] 2
lim Pr[X > βE[X]] > lim 1 − e−
n→∞
n→∞
(0 < β < 1)
(1−β)2 E[X] 2
= 1(∵ E[X] → ∞)
(11)
Thus, as n → ∞, the number of non-faulty nodes isolated by half or more faulty neighbors will also tend to infinity with probability 1. Given that each of these fails to receive broadcast with probability at least half, application of Chernoff bound leads to the conclusion that the probability of broadcast failure tends to 1.
b) 1 − γ < p ≤ 1 − ω( dn ): This is relevant if d is an increasing function of n and/or p. Once again, consider a particular node j in the network. Then, if j is non-faulty, but more than half of its neighbors are faulty, reliable broadcast fails. Given that there are d neighbors, and each may fail independently with probability p, let Y j denote the number of failed neighbors of j. Then, Y takes values from 0, 1, ..., d, and 1 E[Y ] = pd > (1 − γ)d > 12 d. We set β = 1 − 2(1−γ) and apply the Chernoff bound in Lemma 2. This yields: ³ ´2 1 d 1 − 2(1−γ) Pr[Y j ≤ ) ≤ exp − (1 − γ)d 2 2 ´ ³ Ã ! 1 1 − γ 1 4(1−γ) d < exp − 4 d if 0 < γ < ≤ exp − 2 2 16 1 (as d ≥ 8) e d 1 ∴ Pr[Y j ≥ ] > 1 − 2 e ≤
(12)
(13) (14) (15)
Since 1 − γ < p ≤ 1 − ω( dn ), we have dn (1 − p) → ∞. Also: 1 Pr[ j fault-free; at least half nbd( j) faulty ] ≥ (1 − p)(1 − ) e
(16)
Let us again mark out a subset of nodes j such that the neighborhoods of these nodes are all disjoint, as in Fig. 1. Then the number of such nodes obtained is approximately k = b dn c ≥ dn − 1. Let I j be an indicator variable that takes value 1 if j is non-faulty but has at least half faulty neighbors. Then Pr[I j = 1] ≥ (1 − p)(1 − 1e ), and all I j ’s are independent. Let X = ∑ I j be a random variable denoting ¡ ¢ n(1−p)(1− 1e ) → ∞ if dn → ∞. number of alive but isolated nodes. Then E[X] ≥ (1 − p)(1 − 1e ) dn − 1 ≈ d Thus from the Chernoff Bound in Lemma 2, for 0 < β < 1 (e.g. β = 31 ) : Pr[X ≤ βE[X]] ≤ e−
(1−β)2 E[X] 2
lim Pr[X > βE[X]] > lim 1 − e−
n→∞
n→∞
(1−β)2 E[X] 2
= 1(∵ E[X] → ∞)
(17)
Thus, as n → ∞, the number of non-faulty nodes isolated by half or more faulty neighbors will also tend to infinity with probability 1. c) 1 − ω( dn ) < p ≤ 1 − ω( 1n ): Note that n(1 − p) → ∞. Thus, it is easily seen there will still be a large number of fault-free nodes in the network (and this number will also tend to infinity as n increases). The cases of interest are those in which at least two non-neighboring nodes in the entire network are alive (else the broadcast issue is either trivialized or moot), and as n(1 − p) → ∞ in this range, there will indeed be at least two such alive nodes with probability q → 1 (as may be verified by application of the Chernoff bound from Lemma 2). Then, consider each of these alive nodes, say A and B. The probability that half or more of A’s neighbors are faulty can be no less that that in the previous case, i.e., Pr[ A has half or more faulty neighbors] ≥ 1 − 1e . Similarly, Pr[ B has half or more faulty neighbors] ≥ 1− 1e . Then Pr[at least A or B has half or more faulty neighbors] ≥ 1− 1e > 0. From Theorem 1, we know that having half faulty neighbors leads to failure with probability at least half. Hence reliable broadcast fails with a significant positive probability.
d) p = 1 − Θ( 1n ): Pr[All nodes faulty;broadcast issue moot] = pn ¶ µ α 1 n ≥ 1 − Θ( ) = (1 − g(n))n where g(n) = n n
(18) (19)
lim Pr[All nodes faulty; broadcast issue moot] ³ α ´n n ≥ lim (1 − g(n))) = 1 − ) = e−α > 0 from Fact 2 n→∞ n
(20)
Pr[All nodes faulty;broadcast issue moot] = pn ¶ µ g(n) 1 n = ng(n) → 0 ≥ 1 − o( ) = (1 − g(n))n where n 1/n
(22)
n→∞
(21)
e) p ≥ 1 − o( n1 ) :
lim Pr[All nodes faulty; broadcast issue moot] ¶ µ ng(n) n n ) ≥ lim (1 − g(n))) = lim 1 − n→∞ n→∞ n
n→∞
= e− lim(ng(n)) = 1 from Fact 2
THEOREM 3. When p ≤ 21 − ε(0 < ε < 12 ), and node degree d ≤ c ln
reliable broadcast asymptotically fails with probability 1.
ln n
1 1 2p +ln 2(1−p)
(23)
(24) (25) (26)
(for suitable constant c),
Proof: Suppose failure probability p ≤ 21 − ε, where ε is an arbitrarily small constant. Choose an √ increasing function f (n) = o( n), and a constant 0 < c < 1 such that 2c ln n ≤ ln n − 3 ln ln n − 2 ln f (n), 1 1 n for sufficiently large n. Take n to be large enough so that ln 2p + ln 2(1−p) ≥ fln(n) . Thus we obtain d ≤ c f (n)) < f (n). To illustrate, we take f (n) = (ln n)2 , and n to be large enough so that 1 ln n 1 ≥ ln1n . Setting d ≤ c ln 1 +ln + ln 2(1−p) , for this choice of c, and large enough n, we obtain ln 2p 1 d
≤ c(ln n)2
< (ln n)2 .
2p
2(1−p)
Consider a particular node j in the network. Then, if j is non-faulty, but more than half of its neighbors are faulty, reliable broadcast fails with probability at least half (from Theorem 1). Given that there are d neighbors, and each may fail independently with probability p, let I jk (1 ≤ k ≤ d) denote the indicator variable corresponding to neighbor k of j (enumerated in some order), such that I jk = 1 if k is faulty, and 0 otherwise. Then Y j = ∑ I jk denotes the number of failed neighbors of j. Y takes values d ¡ ¢ from 0, 1, ..., d, and E[Y ] = pd. Pr[Y j ≥ d2 ] = ∑ di pi (1 − p)(d−i) . Let us simply consider the event d 2.
i= d2
Y j = Then we can apply the lower bound from Lemma 4. The variables I jk (1 ≤ k ≤ d) are drawn from χ = {0, 1} as per distribution Q = Be(p), and the distribution P corresponding to Y j = d2 is Be( 21 ) (we
shall refer to this as Q 1 ). |χ| = 2, and 2
1 (d+1)|χ|
=
1 (d+1)2
>
1
3 2 2d
= 23 e−2 ln d (since d ≥ 8). Thus, we obtain:
−d(D(Q 1 ||Q)) d 2 −d(D(Q 1 ||Q))−2 ln d d 1 1 −d(D(P||Q)) 2 2 Pr[Y j ≥ ] ≥ Pr[Y j = ] ≥ e > e e = 2 2 (d + 1)2 3 (d + 1)|χ| ln n
1
1
1
(27)
1
)( 2 ln 2p + 2 ln 2(1−p) )−4 ln ln n 1 2 −(c ln 2p1 +ln 2(1−p) > e 3 1 1 1 (since n is chosen large enough to ensure that ln + ln ≥ , and c < 1, 2p 2(1 − p) ln n leading to d ≤ c(ln n)2 < (ln n)2 )
(29)
from our choice of c
(31)
2 c = e− 2 ln n−4 ln ln n ≥ 3
Let us call this probability q.
2(ln n)3 3n
(28)
(30)
Pr[ j alive; at least half nbd( j) faulty ] ≥ (1 − p)q
(32)
(ln n)3
1 2(ln n)3
= (33) 2 3n 3n Let us mark out a subset of nodes j such that the neighborhoods of these nodes are all disjoint, as in Fig. 1. Then the number of such nodes that we may obtain is approximately k = b dn c ≥ dn − 1. Let I j be an indicator variable that takes value 1 if j is non-faulty but has at least half faulty neighbors. 3 Then Pr[I j = 1] ≥ (ln3nn) , and all I j ’s are independent. Consider the case where dn → ∞, as n → ∞. We 1 1 + ln 2(1−p) ≥ ln1n , i.e. d ≤ c(ln n)2 . Let X be a random have chosen n large enough to ensure that ln 2p variable indicating the number of non-faulty nodes with half or more faulty neighbors. Then ∑ I j , and ¢ 3 ¡ 3 E[X] = ∑ Pr[I j = 1] ≥ (ln3nn) dn − 1 ≈ (ln3dn) > ln3n → ∞ (as d < (ln n)2 ). Thus we can apply the Chernoff bound in Lemma 2 to obtain: >
Pr[X ≤ βE[X]] ≤ e−
lim Pr[X > βE[X]] > lim 1 − e
n→∞
n→∞
(1−β)2 E[X] − 2
(1−β)2 E[X] 2
(34)
= 1 ∵ E[X] → ∞
(35)
Thus, as n → ∞, the number of non-faulty nodes isolated by half or more faulty neighbors will also tend to infinity with probability 1. Since, each of them fails to receive broadcast with probability at least half, the probability that some nodes will indeed fail to receive the broadcast tends to 1: lim Pr[ reliable broadcast fails] → 1
n→∞
VIII. S UFFICIENT C ONDITION FOR R ELIABLE B ROADCAST We now present a sufficient condition for the asymptotic achievability of reliable broadcast. ln n THEOREM 4. When p < 21 , and node degree d ≥ max{dmin , 16 ln 1 +ln p
1 2(1−p)
} = max{dmin , 16 D(Qln1n||P) )} 2
(recall that dmin = 8 corresponding to r = 1), reliable broadcast is asymptotically achievable with probability 1.
1 1 + ln 2(1−p) Note that when ln 2p ≤ 16nln n , the degree exceeds total network size n, and thus the sufficient condition ceases to be relevant, merely indicating that having a single-hop network suffices for reliable broadcast (which is the trivial sufficient condition for the assumed radio network model). Thus the sufficient 1 1 + ln 2(1−p) condition is of interest only so long as ln 2p > 16nln n .
x=a−1 x=a+1
y=b+r
qnbdC (a, b)
y=b
y=b+r
qnbdD (a, b)
y=b
(a, b)
x=a−r
x=a
qnbdD0
qnbdC 0 y=b+1
(a, b)
y=b−1
qnbdB 0
qnbdB (a, b) qnbdA (a, b)
y=b−r
y=b−r
qnbdA0
x=a+r
x=a−r−1
Fig. 2.
y=b+1 y=b−1
Depiction of qnbdA , qnbdB , qnbdC , qnbdD Region qnbdA (a, b) qnbdB (a, b) qnbdC (a, b) qnbdD (a, b) qnbdA0 (a, b) qnbdB0 (a, b) qnbdC0 (a, b) qnbdD0 (a, b)
Fig. 3.
x-extent a ≤ x ≤ (a + r) (a − r) ≤ x ≤ (a − 1) (a − r) ≤ x ≤ a (a + 1) ≤ x ≤ (a + r) (a + 1) ≤ x ≤ (a + r) (a − r) ≤ x ≤ a (a − r) ≤ x ≤ (a − 1) a ≤ x ≤ (a + r)
x=a
x=a+r+1
Depiction of qnbdA0 , qnbdB0 , qnbdC0 , qnbdD0 y-extent (b − r) ≤ y ≤ (b − 1) (b − r) ≤ y ≤ b (b + 1) ≤ y ≤ (b + r) b ≤ y ≤ (b + r) (b − r) ≤ y ≤ b (b − r) ≤ y ≤ (b − 1) b ≤ y ≤ (b + r) (b + 1) ≤ y ≤ (b + r)
TABLE I
S PATIAL E XTENTS OF Q UARTER N EIGHBORHOODS
a) p ≤ o( 1n ): When the failure probability is so small as to fall in this range, the probability of even a single node failing approaches 0 asymptotically, and thus reliable broadcast is trivially ensured even with the minimum transmission range of 1. This may be seen thus: Pr[No failures;trivial broadcast] = (1 − p)n µ ¶ 1 n ≥ 1 − o( ) n lim Pr[No failures;trivial broadcast]
≥ lim
n→∞
µ
n→∞ ¶n
1 1 − o( ) n
1
= e− lim(no( n )) = 1 from Fact 2
(36) (37) (38) (39)
b) p = Ω( n1 ): We define a term called quarter-neighborhood of a node (x, y), and denote it by qnbd(x, y). We associate eight quarter-neighborhoods with each node: qnbd A , qnbdB , qnbdC , qnbdD , qnbdA0 , qnbdB0 , qnbdC0 , qnbdD0 . The quarter-neighborhoods for a node (a, b) are depicted in Fig. 2 and 3, and their spatial extents are tabulated in Table I. Observe that qnbd B (a, b) = qnbdA0 (a − r − 1, b), qnbdC (a, b) = qnbdA (a − r, b + r + 1), and qnbdD (a, b) = qnbdA0 (a, b + r + 1). Similarly, qnbdB0 (a, b) = qnbdA (a − r − 1, b), qnbdC0 (a, b) = qnbdA0 (a − r − 1, b + r), and qnbdD0 (a, b) = qnbdA (a, b + r + 1) Thus if we simply consider qnbdA (u) and qnbdA0 (u)∀ nodes u, we will have considered all quarter-neighborhoods,
i.e. the number of distinct (but not disjoint) quarter-neighborhoods is 2n. Henceforth, we shall sometimes use Q(x, y) to refer to qnbdA (x, y), and Q0 (x, y) to refer to qnbdA0 (x, y). The population of any qnbd is r(r + 1), and since d = 4r 2 + 4r = 4r(r + 1), the qnbd population = d4 . We now state and prove the following result which is crucial to proving our sufficient condition for reliable broadcast: THEOREM 5. If p < 12 , d ≥ max{dmin , 16 ln
ln n
1 1 2p +ln 2(1−p)
} = max{dmin , 16 D(Qln1n||P) )}, then: 2
d faults in 8 Q(x, y) and Q0 (x, y)] → 1
lim Pr[ ∀(x, y) less than
n→∞
Proof: As shown above, the population of any qnbd is d4 . Each node may fail independently with probability p.Let Y(x,y) be a random variable denoting the number of faulty nodes in Q(x, y). Then 1 − 1, we may then apply the relative entropy form of the Chernoff bound E[Y(x,y) ] = p d4 . Using δ = 2p ln n ln n } ≥ 16 ln 1 +ln . Thus, we (Lemma 3) to Y(x,y) = ∑ I j . Note that d ≥ max{dmin , 16 ln 1 +ln 1 1 obtain:
2p
j∈nbd(x,y)
2(1−p)
d − d ( 1 ln 1 + 1 ln 1 ) Pr[Y(x,y) ≥ ] ≤ e 4 2 2p 2 2(1−p) 8 16 ln n 1 +ln 1 4(ln 2p 2(1−p)
2(1−p)
(40)
1 1 ) ))( 21 ln 2p + 21 ln 2(1−p)
(41) 1 = e−2 ln n = 2 (42) n be a random variable denoting the number of faulty nodes in Q 0 (x, y), we obtain ≤e
0 Similarly, setting Y(x,y) that:
−(
2p
d 1 0 ≥ ]≤ 2 Pr[Y(x,y) 8 n
(43)
0 ’s are not independent, as they are not all disjoint. However, it may be seen that where The Y(x,y) ’s and Y(x,y) dependence exists, it is that of positive correlation (Lemma 5). Thus Pr[Y(x0 ,y0 ) < d8 |Y(x,y) < d8 ] ≥ Pr[Y(x0 ,y0 ) < d 0 d d d d d 0 8 ], and Pr[Y(x0 ,y0 ) < 8 |Y(x,y) < 8 ] ≥ Pr[Y(x0 ,y0 ) < 8 ]. Similarly, we obtain that: Pr[Y(x0 ,y0 ) < 8 |Y(x,y) < 8 ] ≥ 0 Pr[Y(x0 0 ,y0 ) < d8 ], and Pr[Y(x0 0 ,y0 ) < d8 |Y(x,y) < d8 ] ≥ Pr[Y(x0 0 ,y0 ) < d8 ] Hence:
d d and Y 0 (x, y) < ] 8 8 d d ≥ ∏ Pr[Y(x0 ,y0 ) < ] ∏ Pr[Y(x0 0 ,y0 ) < ] µ8 ¶ µ ¶8 1 n 1 n = 1− 2 1− 2 n n ¶ µ 1 2 = 1− 2 n n d d ∴ lim Pr[∀(x, y),Y (x, y) < and Y 0 (x, y) < ] n→∞ 8 8 µ ¶2 2 1 ≥ lim 1 − 2 n = e− lim( n ) = 1 from Fact 2 n→∞ n Pr[∀(x, y),Y (x, y)
0 (independent of n), then for sufficiently large n, the necessary condition would hold for all p. Also note that 1 − ln1n < 1nc for large n (from Fact 3). Thus, the values of p for which our necessary n 1 condition holds are those in which the transmission range remains less than D. When p ≥ 1 − n1+ε , all 1 nodes r are faulty with probability approaching 1, and the issue of connectivity is moot. When p ≤ nc , r=
1.
c ln n ln 1p
≤ 1, and for this range of p, the necessary condition lapses to having the minimum range of
a) p ≤ 1 − ln1n : Consider a particular node j in the network. Then, if j is non-faulty, but all its neighbors are faulty, we have a potential disconnection event. Given that there are d neighbors, and each may fail independently with probability p, the probability that j does not fail, but all nodes in nbd( j) fail, is (1 − p)pd . We choose a constant 0 < c < 1 such that c ln n ≤ ln n − 4 ln ln n, for sufficiently large
Fig. 6.
Nodes having disjoint neighborhoods
n. In general, c can be chosen very close to 1, e.g., 1 − ε(0 < ε < 1), and the rcondition will hold for ln n 1 n > no , for some no . Since p ≤ 1 − ln1n , we obtain that 1−p ≤ ln n. Let r ≤ 8cln 1 . The node degree d=
4r2 + 4r
≤
4r2 + 4r2
=
8r2 ,
p
ln n for n ≥ 1. Thus, for our choice of r, it turns out that d ≤ c ln 1 . Then, it p
may be seen that:
Pr[ A given node j is alive, but isolated] ≥ Pr[ j is alive and all neighbors of j are faulty ] ln n
1 c ln 1p p ln n 1 1 1 = = c c ln n n n ln n
= (1 − p)pd >
≥
(ln n)3 (from our choice of c) n
(50) (51) (52) (53) (54)
Let us mark out a subset of nodes j such that the neighborhoods of these are all disjoint, ³ √ nodes ´ n 2 as in Fig. 6. Then the number of such nodes that we may obtain = b 2r+1 c ≥ 9rn2 − 1 (since √ n may not be multiple of 2r + 1). Let I j be an indicator variable that takes value 1 if j is alive 3 but isolated. Then Pr[I j = 1] ≥ (lnnn) , and all I j ’s are independent. Let X be a random variable denoting the number of nodes from the chosen set that are alive and isolated. Then X = ∑ I j , and µ 1 ¶ n ln 1 1 n ln p 1− ln n (ln n)3 (ln n)3 E[X] ≥ n ≥ 19 (ln n)2 ln 1−1 1 ≥ 19 ln n → ∞. We can thus apply the 9c ln n − 1 ≥ n 9 ln n ln n
Chernoff bound from Lemma 2: Thus, with suitable 0 < c < 1 and β = E[X]−1 E[X] , we obtain that for r ln n p < 1 − ln1n , if r ≤ 8cln 1 , then E[X] → ∞, and hence lim Pr[ At least two alive nodes are isolated] = 1. p
n→∞
Observe that actually the necessary condition would hold for all p such that E[X] → ∞. For instance, the above analysis holds for all p ≤ 1 − (ln1n)b (where b is a constant), with a corresponding suitably
(b+2)
varying choice of c to ensure that Pr[I j = 1] ≥ (ln n)n . Besides, if E[X] → γ > 0, the asymptotic disconnection probability is still a positive finite quantity, and the condition is still necessary for asymptotic connectedness probability to approach 1. 1 b) p ≥ 1 − n1+ε : When the failure probability becomes so high as to fall in this range, we obtain:
µ
= lim 1 − 1 − n→∞
1
n1+ε
lim Pr[ Any node is alive] = 1 − pn
n→∞ ¶ n
(55)
1
= 1 − e− lim( nε ) = 0 from Fact 2
(56)
Thus the network is trivially connected by definition, regardless of degree. XV. N ECESSARY C ONDITION FOR C OVERAGE We now show that for the network to be asymptotically covered r with probability approaching 1, it ln n is necessary that the transmission range r satisfy: r ≥ max{1, Ω( ln 1 )}, i.e., the node degree be d ≥ p
ln n max{1, Ω( ln 1 )}. p
ln n THEOREM 9. For p < 1 − ln1n , for a suitable constant 0 < c < 1, if d < c ln 1 : p
lim Pr[Some point is not covered] → 1
n→∞
Proof: As in the case of connectivity it is obvious that r must be at least 1, else some points will not be covered. We handle two subranges of p separately. a) p < 1 − ln1n : The proof relies on subdivision of the network into disjoint neighborhoods, as in Fig. 6. If there exists at least one neighborhood with absolutely no nodes alive (neither the neighborhood center nor its neighbors), then the center of that neighborhood is not covered. Thus we seek to determine the probability of such an event. We choose a constant 0 < c < 1 such that ensures that
1 nc
≥
(ln n)3 n
for large n. Let r
9 ln n ≤ ln n − 8 cr ln n ≤ 8cln 1 . The p
3 ln ln n, for sufficiently large n. This neighborhood population is given by
ln n d + 1 = 4r2 + 4r + 1 ≤ 4r2 + 4r2 + r2 = 9r2 , for n ≥ 1. Thus, d + 1 ≤ 89 c ln 1 . Let I j be an indicator p
variable that takes value 1 if there is no alive node in the neighborhood centered at node j, and value 0 otherwise. Then Pr[X j = 1] = pd+1 = p
9 ln n 8 c ln 1 p
=
(ln n)3 n
(from our choice of c). Let X = ∑ I j be a random 3
8(ln n)2 ln 1
p variable indicating the number of neighborhoods with no alive node. Then E[X] = (ln9rn)2 = 9c (after plugging in the chosen value of r). If p < 1 − ln1n , then E[X] ≥ ln n(ln n ln 1−1 1 ) > ln n → ∞ (from ln n
Fact 1), and application of the Chernoff bound from Lemma 2 yields that Pr[X = 0] ≤ exp(− E[X] 2 ) → 0. Thus there is some uncovered region with probability 1. Similar to the necessary condition for connectivity, observe that this necessary condition would hold for all p such that E[X] → ∞. In particular, the above analysis holds for all p ≤ 1 − (ln1n)b (where b is a constant), with a corresponding suitably varying choice of c to ensure that Pr[I j = 1] ≥
(ln n)(b+2) . n
Also, if E[X] → γ > 0, the asymptotic probability of some point being uncovered is a positive finite quantity, and the condition is still necessary for asymptotic coverage probability to approach 1. ³ ´n 1 1 n b) p ≥ 1 − n1+ε (0 < ε < 1): We obtain that Pr[ no nodes alive ] = p ≥ 1 − n1+ε . As n → ∞, the following holds: lim Pr[some point not covered] ≥ Pr[no node alive] µ ¶n 1 1 = e− lim( nε ) = 1 from Fact 2 = lim 1 − 1+ε n→∞ n
(57)
n→∞
(58)
Thus the network is trivially not covered, regardless of transmission range. XVI. S UFFICIENT C ONDITION FOR C ONNECTIVITY AND C OVERAGE We now present a sufficient condition for the asymptotic existence of both connectivity and coverage. ln n THEOREM 10. When d ≥ 32 ln 1 , the network is asymptotically connected and covered with probability
1.
p
1 a) p ≤ n1+ε : When the failure probability is so small as to fall in this range, the probability of even a single node failing approaches 0 asymptotically, and thus connectivity and coverage is trivially ensured even with the minimum transmission range of 1. This may be seen thus:
Pr[No failures;full connectivity/coverage] = (1 − p)n ¶n µ 1 ≥ 1 − 1+ε n
(59) (60)
lim Pr[No failures;full connectivity/coverage] ¶n µ 1 1 = e− lim( nε ) = 1 from Fact 2 ≥ lim 1 − 1+ε n→∞ n
(61)
n→∞
(62)
Proof: b) p = Ω( n1 ): Consider the subdivision of the grid as depicted in Fig. 7, so that the resulting cells have x-extents (y-extents) 0 to a, a + 1 to a + b, a + b + 1 to 2a + b + 1, and so on. Here a = b 2r c and b = r − a = r − b 2r c. Then, each node is within range of all other nodes in the cells adjoining its own. Thus it is obvious that if each square has at least one non-faulty node, there exists a connected backbone that covers all points, and hence all nodes. Thus all non-faulty nodes are connected to each other via this backbone. The dimensions 2 of the cells thus obtained can be (a + 1)2 , (a + 1)b or b2 . Thus the population k of any cell satisfies k ≥ r4 , and the maximum possible number of cells m ≤ 4n . Then: r2 r2
Pr[ at least one node alive in a given cell ] = 1 − pk ≥ 1 − p 4 µ ¶ 4n2 r r2 ∴ Pr[ at least 1 node alive in each cell] ≥ 1 − p 4
(63) (64)
2a+b+1 a+b
Pop. (a+1)b
a
Pop. 2
b
Pop. Pop. (a+1) 2 (a+1)b
0 0
a
Fig. 7.
Let us choose r ≥
r
8 ln n . ln 1p
a+b
2a+b+1
Subdivision of network into cells
Then: 1
µ ¶ n ln p 2 ln n r2 Pr[at least 1 node alive in each cell] ≥ 1 − p 4
Since p ≥ α 1n for some constant α , ln 1p ≤ ln n − ln α. Hence:
1
1
µ
Pr[at least 1 node alive in each cell] ≥ 1 − p
¶ n ln p 2
r 4
2 ln n
=
(65)
Ã
1− p
! n ln p
¶ n (1− ln α )
(66)
= e− lim( 2n ) = 1 from Fact 2
(68)
2 ln n ln 1p
2 ln n
1 ≥ 1− 2 n µ
2
ln n
(67)
Thus, by application of Fact 2, we obtain: lim Pr[at least 1 node alive in each cell] ≥ lim
n→∞
n→∞
µ
1 1− 2 n
¶ n (1− ln α ) 2
ln n
1
Since this condition ensures connectivity and coverage, we obtain that: lim Pr[network is connected and covered] → 1
n→∞
(69)
XVII. C ONDITIONS IN E UCLIDEAN M ETRIC We show that our results derived for L∞ metric continue to hold for L2 metric, with only the constants in the theta notation changing. LEMMA 8. If the network is asymptotically connected (covered) in L ∞ for all r ≥ rmin , then the network √ is connected (covered) asymptotically in L2 for all r ≥ rmin 2. Proof: The proof is by contradiction. Suppose that, for a given failure configuration, the network √ is asymptotically connected in L∞ for all r ≥ rmin but is not asymptotically connected for all r ≥ rmin 2
r
r √ r 2 r
Fig. 8.
r
Relationship between L∞ and L2 neighborhoods
in L2 . Observe √ that it is possible to circumscribe a L∞ neighborhood of range r by√a L2 neighborhood of range r 2 (Fig. 8). Hence the nodes in an L2 network of transmission range r 2 can be made to simulate the operation of nodes in a L∞ network with range r (as the L∞ neigborhood is fully contained within the L2 neighborhood). This √ implies that if the L∞ network of range r is connected (covered), so must be the L2 network of range r 2. If there is some r ≥ rmin for√which the L∞ network of range r is connected (covered) asymptotically, but the L2 network of range r 2 is not, we obtain a contradiction, as connectedness (coverage) of the L∞ network would imply connectedness (coverage) of the L2 network. LEMMA 9. If the network is asymptotically disconnected (not covered) in L ∞ for all r ≤ rmin , then the network is disconnected (not covered) asymptotically in L2 for all r ≤ rmin . Proof: The proof is by contradiction. Suppose that the network is asymptotically disconnected (not covered) in L∞ for range r, but is not disconnected (not covered) in L2 for range r. Observe that an L∞ neighborhood of transmission range r circumscribes an L2 neighborhood of range r (Fig. 8). Thus, for any given random failure configuration, if the L2 network of range r were connected (covered), so would be the L∞ network of radius r, as we could simply make the nodes in the L ∞ network simulate the behavior of nodes in the L2 network, and obtain connectedness (coverage). Hence, if the L 2 network of range r ≤ rmin is not asymptotically disconnected (not covered), the L∞ network of range r ≤ rmin must also not be disconnected (not covered). This yields a contradiction. XVIII. D ISCUSSION It is interesting to note that in case of a grid network, the necessary and sufficient node degree turns µ ¶ log n out to be Θ log ) (when expressed in our notation) for the case of a , as compared to Θ( log (n(1−p)) 1 1−p p
randomly deployed network, where sensors are active with probability 1 − p [14]. However, it is not difficult to see that such a difference is to be expected. In a grid network, as failure(or sleep) probability p → 0, the network tends towards a deterministic topology, whereas in a random network, if failure or sleep probability p → 0, the network can only tend towards a denser but still random network. Thus, at small values of p, a very small degree will suffice for a grid network, but may not for a random network. At larger p values, the grid network exhibits increasing randomness and begins to resemble a network with random deployment. Thus, one may see that the two expressions are within a small range of each other when p is large (given sufficiently large n), but diverge as p → 0. Another observation is that the form of the results is very similar to results obtained by us for reliable broadcast in a grid network with Byzantine failures. For Byzantine failures, we have ¶ µ obtained that the necessary and sufficient conditions for reliable broadcast entail a node degree of Θ
ln n 1 1 ln 2p +ln 2(1−p)
,
which may be re-stated as Θ
µ
ln n D(Q 1 ||P) 2
¶
where Q 1 denotes a distribution with failure probability 2
1 2,
P
denotes the actual distribution with failure probability p, and D(Q||P) denotes the relative entropy (or Kullback-Leibler distance) ´between distributions Q and P. Similarly, one may view the node degree for ³ ln n connectivity as Θ D(Q1 ||P) , where Q is the distribution with failure probability 1, and P is the actual failure distribution. XIX. N ON -T OROIDAL N ETWORKS We have made the assumption that the network is toroidal, so as to avoid edge effects. However, we can see that the degree of any node at the outermost edge is no more than d, and at least d4 (where d is the uniform degree that each node would have in the toroidal case). Thus, the necessary condition would continue to hold as is (since some nodes having a lesser degree can only increase the probability of disconnection). The construction used to prove the sufficient condition also continues to hold as is, since all full-cells in the tiling will have at least one active node each, and even if there are regions at the fringes left-over, they will still fall within range of some active node in the nearest full tile (due to the chosen dimensions of the cells). Thus, the results are not affected. A similar argument leads to the conclusion that the coverage results are not affected. R EFERENCES [1] G. B. Thomas, Jr. and R. L. Finney, Calculus and Analytic Geometry. Addison-Wesley Publishing Company, 1992. [2] K. Jogdeo and S. M. Samuels, “Monotone convergence of binomial probabilities and a generalization of ramanujan’s equation,” The Annals of Mathematical Statistics, vol. 39, no. 4, pp. 1191–1195, August 1968. [3] W. Hoeffding, “Probability inequalities for sums of bounded random variables,” Journal of the American Statistical Association, vol. 58, no. 301, pp. 13–30, Mar. 1963. [4] T. M. Cover and J. A. Thomas, Elements of Information Theory. John Wiley & Sons, Inc., 1991. [5] E. Kreyszig, Advanced Engineering Mathematics, 7th ed. John Wiley & Sons, 1993. [6] C.-Y. Koo, “Broadcast in radio networks tolerating byzantine adversarial behavior,” in Proceedings of the twenty-third annual ACM symposium on Principles of distributed computing. ACM Press, 2004, pp. 275–282. [7] V. Bhandari and N. H. Vaidya, “On reliable broadcast in a radio network,” in PODC ’05: Proceedings of the twenty-fourth annual ACM SIGACT-SIGOPS symposium on Principles of distributed computing. ACM Press, 2005, pp. 138–147. [8] E. Kranakis, D. Krizanc, and A. Pelc, “Fault-tolerant broadcasting in radio networks,” J. Algorithms, vol. 39, no. 1, pp. 47–67, 2001. [9] A. Pelc and D. Peleg, “Feasibility and complexity of broadcasting with random transmission failures,” in PODC ’05: Proceedings of the twenty-fourth annual ACM SIGACT-SIGOPS symposium on Principles of distributed computing. ACM Press, 2005, pp. 334–341. [10] V. Bhandari and N. H. Vaidya, “On reliable broadcast in a radio network: A simplified characterization,” Technical Report, CSL, UIUC, May 2005. [11] A. Pelc and D. Peleg, “Broadcasting with locally bounded byzantine faults,” Information Processing Letters, vol. 93, no. 3, pp. 109–115, Feb 2005. [12] P. Gupta and P. R. Kumar, “Critical power for asymptotic connectivity in wireless networks,” in Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W.H. Fleming, W. M. McEneany, G. Yin, and Q. Zhang, Eds. Boston: Birkhauser, 1998, pp. 547–566. [13] F. Xue and P. R. Kumar, “The number of neighbors needed for connectivity of wireless networks,” Wirel. Netw., vol. 10, no. 2, pp. 169–181, 2004. [14] D. Kim, C. Hsin, and M. Liu, “Asymptotic connectivity of low duty-cycled wireless sensor networks,” in Proc. MILCOM, 2005. [15] S. Shakkottai, R. Srikant, and N. Shroff, “Unreliable sensor grids: Coverage, connectivity, and diameter,” in Proc. of Infocom 2003, 2003. [16] S. Kumar, T. H. Lai, and J. Balogh, “On k-coverage in a mostly sleeping sensor network,” in MobiCom ’04: Proceedings of the 10th annual international conference on Mobile computing and networking. New York, NY, USA: ACM Press, 2004, pp. 144–158. [17] S. Shakkottai, R. Srikant, and N. Shroff, “Correction to unreliable sensor grids: Coverage, connectivity, and diameter,” Personal Communication, 2005. [18] X. Liu and R. Srikant, “An information-theoretic view of connectivity in wireless sensor networks,” in the first IEEE Communications Society Conference on Sensor and Ad Hoc Communications and Networks (SECON). Santa Clara, CA: IEEE, Oct. 4-7 2004.
