Capacity Factors of a Point-to-point Network

0.22 MB·English

by  Yuan Li

Capacity Factors of a Point-to-point Network
Yuan Lia,1, Yue Zhaob, Haibin Kana,1,∗
aSchool of Computer Science, Fudan University, Shanghai 200433, China
bSchool of Mathematical Science, Fudan University, Shanghai 200433, China
2 cShanghai Key Lab of Intelligent Information Processing, Fudan University, Shanghai
1
200433, China
0
2

n
a
J
Abstract
3
1
In this paper, we investigate some properties on capacity factors, which were

] proposed to investigate the link failure problem from network coding. A
T
I capacity factor (CF) of a network is an edge set, deleting which will cause
.
s the maximum flow to decrease while deleting any proper subset will not.
c
[ Generally, a k-CF is a minimal (not minimum) edge set which will cause the

network maximum flow decrease by k.
2
v Under point to point acyclic scenario, we characterize all the edges which
5
are contained in some CF, and propose an efficient algorithm to classify. And
3
4 we show that all edges on some s-t path in an acyclic point-to-point acyclic
0
network are contained in some 2-CF. We also study some other properties of
.
1
CF of point to point network, and a simple relationship with CF in multicast
0
2 network.
1 On the other hand, some computational hardness results relating to ca-
:
v pacity factors are obtained. We prove that deciding whether there is a capac-
i
X ityfactorofacyclicnetworkwithsizenotlessagivennumberisNP-complete,
r and the time complexity of calculating the capacity rank is lowered bounded
a
by solving the maximal flow. Besides that, we propose the analogous defini-
tion of CF on vertices and show it captures edge capacity factors as a special
case.
Keywords: network coding, capacity factor, capacity rank, point-to-point
network, multicast network, NP-complete, vertex capacity factor.
∗
Corresponding author
Email addresses: [email protected](Yuan Li),
[email protected](Yue Zhao), [email protected](Haibin Kan)
Preprint submitted to Information Science January 16, 2012
1. Introduction
Reliability is a critical theme in the topology design of communication
networks. In traditional combinatorial network theory, we use the concept
edge-connectivity to evaluate the reliability of a network in case of edge
failures [2]. A lot of intense studies have been focused on the connectivity of
a network and extending concepts such as super connectivity and conditional
connectivity [2], [6], [8].
The more recent study of network coding shows that [1] by coding at
internal vertices, onecanachieve theoptimalcapacityofamulticast network,
whichisupperboundedbythemaximum flow(minimum cut)ofthenetwork.
However, this optimal capacity can not be achieved by traditional routing
scheme.
Here comes a question: for a network coding-based network, is it still ap-
propriatetoevaluateitsreliabilitybytraditionalconceptslikeedge-connectivity
as mentioned above? For traditional networks where only routing schemes
are adopted, the communication will not be ruptured in case of edge failures
as long as at least one path from source to the sink vertex still exists. How-
ever, in network coding-based network, the communication will be degraded
even if the failures of an edge set may reduce the number of disjoint paths
between source and sink vertices for the network capacity is decreased.
Koetter et al. [10] first mentioned the edge failure problems in network
coding-based networks. Cai and Fan [3] formally proposed the concept of
capacity factor and capacity rank. The capacity rank characterizes the crit-
icality of a link for the network communication. When there is no capacity
factor containing an edge, the capacity rank of this edge is defined as ∞.
Recently, we notice some related work in [5] about k-Route Cut, which is the
minimum number of edges to let the connectivity of every pair of source and
sink falls below k. They not only propose some approximation algorithms,
but also prove some computational hardness results. In fact, the generalized
definition of k-CF captures some definitions such as k-route cute when the
connectivity is defined as maximal number of edge-disjoint paths.
In [3], the authors proposed an open problem which is deciding the ca-
pacity rank of a given edge. In this paper, we obtain an equivalent condition
that deciding whether CR(e) = ∞. By this result, it is easy to develop an
algorithm in O(V3) to determine whether the capacity rank of a given edge is
2
finite or infinite, which partially answers the open problem in [3]. Although
neither can we find an efficient algorithm to compute the capacity rank of a
general network, nor can we prove the problem is NP-hard, we obtain some
computational hardness results relating to it. For example, we prove that
deciding whether there is a capacity factor with size not less than a given
number is NP-complete, and show the time complexity of computing the
capacity rank is lower bounded by that of solving maximal flow.
Even though there is no benefit of network coding on single-source single-
sink network, our results are mainly focused on single-source single-sink sce-
nario. There are mainly two reasons: the study on point-to-point network
maybringinsights intomulticast scenario; theCFisanaturalconcept, which
might be interesting on its own right.
This paper is organized as follows. In section II , we review some basic
definitions, notations and related results. In section III, we investigate some
properties of capacity factors, including both point-to-point scenario and
multicast scenario. In section IV, we propose an algorithm to calculate the
D-set and H-set with its correctness proved and time complexity analyzed.
In section V, we present some computational hardness results relating to
capacity factors.
2. Preliminaries
In this section, we review some basic definitions, notations and results,
which will be used in the sequel.
A communication network is a collection of directed links connecting
transmitters, switches, and receivers. It is often represented by a 4-tuple
N = (V,E,S,T), where V is the vertex (node) set, E the edge (link) set, S
the source vertex set and T the sink vertex set. A communication network N
is called a point-to-point communication network if |S| = |T| = 1, denoted
by N = (V,E,s,t), where s is the source vertex and t the sink vertex.
Without loss of generality, we may assume that all links in a network
have the same capacity, 1 bit per transmission slot. For u,v ∈ V, denote by
hu,vi the edge from u to v. If there are k edges from u to v, we denote by
hu,vi the set consisting of edges from u to v, or simply denote it by hu,vi
k
when there is no ambiguity. For an edge e = hu,vi, u is called the tail of e
and denoted by tail(e), and v is called the head of e and denoted by head(e).
If F ⊆ E, denote by N\F the network obtained by deleting edges in F
from N. If V′ ⊆ V, denote by N(V′) the network consisting of vertices in
3
V′ and the edges among V′ of N, calling the vertex-induced network of N
by V′. For V ,V ⊆ V, denote by [V ,V ] the set consisting of all links with
1 2 1 2
tails in V and heads in V . For a network N = (V,E,S,T), an S-T cut of
1 2
N is [V ,V ] such that V is a subset of V containing all vertices in S but not
1 1 1
containing any vertex in T. A minimal S-T cut of N is an S-T cut with the
minimal size, denoted by C (S,T).
N
It is well known that, for a point-to-point network N = (V,E,s,t), the
maximal flow from s to t is equal to the minimal s-t cut of N and a feasible
flow is a maximal flow if and only if there is no augmenting path in the
corresponding residual network (Max-flow Min-cut Theorem [4], [13]). If
each link in N has unit capacity, then the maximal flow f of N corresponds
to |f| edge-disjoint paths from s to t in N (Integrality Theorem [13]), where
f denotes a collection of edge-disjoint paths (a flow) and |f| denotes the
number of the paths. Throughout the paper, we always assume each link has
unit capacity.
Let N = (V,E,s,t) be a point-to-point network. For any vertex v ∈ V,
we can assume that there exists a path from s to t in N which passes the
vertex v. Otherwise, we can delete the vertex v because v is useless for the
communication between s and t in N. Similarly, for any edge e ∈ E, we can
assume that there exists a path form s to t which passes the edge e.
Definition 2.1. [3] Let N = (V,E,s,t) be a point-to-point network. A
nonempty subset F of E is a capacity factor of N if and only if the following
two conditions hold:
1. C (s,t) < C (s,t);
N\F N
2. C = C (s,t) for any proper subset F′ofF.
N\F′ N
N\F denotes the induced network formed by deleting F in N.
By this definition, for a capacity factor F, adding any one edge e ∈ F in
thepoint-to-pointnetworkN\F willincreasethemaximalflow. Sinceadding
one edge can increase the maximal flow by at most 1, we have C (s,t) =
N\F
C (s,t)−1.
N
Generally, we can define kth order capacity factor (k-CF) of as follows,
where the motivation will be clear in the multicast scenario.
Definition2.2. LetN = (V,E,s,t)be a point-to-pointnetwork. Anonempty
th
subset F of E is a k order capacity factor (k-CF) of N if and only if the
following two conditions hold:
4
1. C (s,t) ≤ C (s,t)−k;
N\F N
2. C > C (s,t)−k for any proper subset F′ofF.
N\F′ N
N\F denotes the induced network formed by deleting F in N.
Since the network coding capacity of a single-source multi-sink network
N = (V,E,s,T), where T = (t ,...,t ), is upper bounded by the minimal
1 m
of the maximal flow from s to t [1], i.e., the capacity region is
i
(C (s,t ),C (s,t ),...,C (s,t )).
N 1 N 2 N m
Therefore, we have the following definition of k-CF on a multicast network.
Definition2.3. N = (V,E,s,T)isa multicastnetworkand T = {t ,...,t },
1 m
−→
th
edge set F is a (k ,k ,...,k ) order capacity factor (k -CF) of N if and
1 2 m
only if:
(1) For all 1 ≤ i ≤ m,C (s,t ) ≤ C (s,t )−k ;
N\F i N i i
(2) For any F′ ( F, there exists i ∈ {1,...,m}, such that C (s,t ) >
N\F′ i
C (s,t )−k .
N i i
−→
When k 6= 0 and k ≤ 1, we say F is a CF of multicast network N.
i
One could easily generalize the above definition to multi-source multi-
sink network. This generalized definition captures the k-route cut problem
−→
of edge-connectivity version, i.e., k-route cut is the minimum k -CF where
−→
k = (f −k +1,...,f −k +1).
1 t
Following is the definition of D-set and H-set, which is simply the edge
union of all CFs and all the remaining ones.
Definition 2.4. [3] Let N = (V,E,s,T) be a point-to-point or multicast
communicationnetwork. Thecollectionof all its capacityfactors D = {F ,F ,...,F }
1 2 r
r
is called the capacity factor set of N. While D = F is called the D-set
Si=1 i
of N and H = E\D is called the H-set of N.
By the definition, it is not difficult to see that C (s,t) = C (s,t).
N\H N
Thus, the edge set of a point-to-point network can be decomposed into two
disjoint parts, namely D-set and H-set, which represent the relatively impor-
tant links and the unimportant links. However, this classification is a little
rough. The next definition gives a concept characterizing the criticality of a
link more precisely.
5
Definition 2.5. [3] Let N = (V,E,s,t) be a point-to-point network. The
capacity rank of a edge e ∈ E is the minimum size of the capacity factors
containing e, denoted by CR (e) or CR(e) when there is no ambiguity. If
N
there is no capacity factor containing e, we define CR(e) = ∞.
The links with smaller capacity ranks are of higher criticality. How to
calculate the capacity rank of a given edge? As far as we know, the problem
is still open. A direct idea is to find all the capacity factors and then decide
the capacity rank of each edge. The following example shows it’s impractical
since the number of capacity factors may grow exponentially with the size of
network.
e
2
e e
1 3
e
5
e e
4 6
e s′ e e e
s 0 7 8 9 t
e e
3n−2 3n
e
3n−1
Figure 1: Network with exponential number of 1-CFs
Consider the network N = (V,E,s,t) shown in Figure 1, where |V| =
2n + 3,|E| = 3n + 1. The network can be decomposed into n internally
disjoint paths from s′ to t and an individual edge e . The maximal flow of N
0
is1. Let F = {e ,e ,...,e }, 3(k−1)+1 ≤ i ≤ 3k fork = 1,2,...,n. Due
i1 i2 in k
to the simple structure of the network, it is easy to see that F is a capacity
factor of N and all capacity factors besides {e } can be written in the form
0
of F. Therefore, the total number of capacity factor of N is 3n +1, which
grows exponentially with |V|+|E|.
3. Some Properties of Capacity Factor
The following lemma is very useful, which guarantees the existence of
some k-CF containing a specific edge e when some conditions are satisfied.
6
Lemma 3.1. Let N = (V,E,s,t) be a point-to-point network. For an edge
e ∈ E and an integer 1 ≤ k ≤ C (s,t), if there exists a edge set F′ ⊆ E
N
containing e such that C (s,t) = f−k, C (s,t) > f−k, then there
N\F′ N\(F′\e)
exists a k-CF containing e.
Proof. Let Fk = {F′ ⊆ E | e ∈ F′, C (s,t) = f − k, C (s,t) >
e N\F′ N\(F′\e)
f − k}. By condition, Fk is not empty. Hence we can find a F ∈ Fk
e e
with minimal cardinality. In fact, F is what we want. Firstly, we have
C (s,t) = f − k. Secondly, ∀ e′ ∈ F, if e′ = e then we already have
N\F
C (s,t) > f − k otherwise if C (s,t) = f − k then F \ e′ still
N\(F\e) N\(F\e′)
belongs to Fk which contradicts with that F has the minimal cardinality. So
e
C (s,t) > f −k for all e′ ∈ F, which implies F is a k-CF.
N\(F\e′)
The following proposition only holds for acyclic network, and therefore
all properties depending on it only holds for acyclic network.
Proposition 3.2. Let N = (V,E,s,t) be an acyclic network. If N can be
decomposed into C (s,t) edge-disjoint paths, then for any e ∈ E, we have
N
C (s,t) = C (s,t)−1.
N\e N
Proof. Let N′ = N\e, m = C (s,t). Denote the m edge-disjoint paths by
N
p , p , ..., p . Since N can be decomposed into C (s,t) disjoint paths, e
1 2 m N
must be on one of the path. Without loss of generality, we assume e is on p
m
andp = (u ,u ,...,u ,u ...,u ), wherehu ,u i = eand1 ≤ k ≤ l−1.
m 1 2 k k+1 l k k+1
Clearly, there is a feasible flow f on N′, which is consisting of m − 1
edge-disjoint paths p ,p ,...,p and hence C (s,t) ≥ m−1. Recalling
1 2 m−1 N′
Max-flow min-cut theorem, we know that a flow f is a maximal flow if and
onlyifthereisnoaugmentingpathsontheresidualnetwork. Soitissufficient
to show the residual network N′ has no augmenting path.
f
Since N is acyclic, we assign each vertex an integer label by topological
order, such that hu,vi ∈ E implies L(u) < L(v).
According totheedgedirectioninnetworkN, alltheedgesintheresidual
network N′ can partitioned into two parts, forward edges and reversal edges.
Consider all the forward edges in N′ , which can be viewed as the union
f
of two paths, (s = u ,...,u ) ∪ (u ,...,u = t). By the label properties
1 k k+1 l
of vertices, for a forward edge (u,v) and a reversal edge (u′,v′), we have
L(u) < L(v) and L(u′) > L(v′) respectively. Since u < u < ··· < u ,
1 2 l
there is no reversal edge with head in {u ,...,u } and tail in {u ,...,u }.
k+1 l 1 k
Therefore, u (= s) and u (= t) are disconnected and there is no augmenting
1 l
path in the residual network N′ . This completes our proof.
f
7
The following result shows thatany k-CF is contained insome (k+1)-CF,
assuming the maximal flow of network is greater than k of course.
Proposition 3.3. Let N = (V,E,s,t) be an acyclic network, and assume F
is a k-CF of N, where k < C (s,t), there exists a (k +1)-CF F′ such that
N
F ⊆ F′.
Proof. Let f = C (s,t). Assume e is a cut-edge of the network N \F. Let
N
F′ = F ∪{e}, then F′ is a (k +1)-CF of N. Because C = f −(k +1)
N\F′
and ∀ F˜ ( F′, C (s,t) > f −k if e ∈/ F˜ otherwise C (s,t) ≥ f −k for
N\F˜ N\F˜
deleting one edge at most diminishes one flow.
The following theorem characterizes an edge which is contained in some
k-CF of a point-to-point acyclic network.
Theorem 3.4. Let N = (V,E,s,t) be an acyclic point-to-point network,
and integer 1 ≤ k ≤ C (s,t). For any edge e ∈ E, there is a k-CF F
N
containing e if and only if there exists an s-t path p containing e such that
C (s,t) ≥ C (s,t)−k.
N\p N
Proof. Let f = C (s,t). “ ⇒”: e ∈ F, F isa k-CF.Since C (s,t) = f−k,
N N\F
while adding e to N \ F the maximum flow increase by 1, we claim there
is a path p containing e such that C (s,t) = f − k which implies
(N\F)\p
C (s,t) ≥ f −k.
N\p
“ ⇐”: By Proposition 3.3, assume C (s,t) = f −k. Since C (s,t) =
N\p N\p
f −k, there is a feasible flow on N \p, which can be decomposed into f −k
f−k
paths p , p ..., p . Denote by F = E \ ( p ∪ p), by Proposition
1 2 f−k S i
i=1
f−k
3.2, E \ (F ∪ e) = (E \ F) \ e = p ∪ (p \ e) implies C (s,t) =
S i N\(F∪e)
i=1
C (s,t) = f−k. And C (s,t) = C (s,t) = f−k+1 > f−k.
f−k N\F f−k
S pi∪(p\e) S pi∪p
i=1 i=1
Apply Lemma 3.1, we know there is always a k-CF F containing e.
Recall that an edge of a network is either in the D-set is defined as the
union of all CFs and H-set consists of all the remainings. Taking k = 1
in the of preceding theorem, we obtain the following result, which gives an
equivalent condition to characterize edges in the D-set and H-set.
8
Corollary 3.5. Let N = (V,E,s,t) be a point-to-point acyclic network. For
any e ∈ E on some s-t path, e is in the H-set if and only if for any s-t path
containing e, C (s,t) ≤ C (s,t)−2; and e is in the D-set if and only if
N\P N
there exists some s-t path p containing e such that C (s,t) = C (s,t)−1.
N\P N
v v
1 e 3
3
e e
1 6
e
s 4 t
e e
2 7
e
v 5 v
2 4
Figure 2: Example for D-set and H-set
Consider the point-to-point acyclic network N = (V,E,s = v ,t = v )
1 6
shown in Figure 2. There are three different s-t path in total, which are
p = (s,v ,v ,t), p = (s,v ,v ,t) and p = (s,v ,v ,v ,v ,t). It’s clear
1 1 3 2 2 4 3 1 3 2 4
that C (s,t) = C (s,t) = 1, but C (s,t) = 0. Thus e = hv ,v i
N\p1 N\p2 N\p3 4 3 2
is the only edge satisfying the conditions in Corollary 3.5. Therefore H =
{e } and D = E\H = {e ,e ,e ,e ,e ,e }, which coincides with the direct
4 1 2 3 5 6 7
computation that D = F = {e }∪{e }∪{e }∪{e }∪{e }∪{e }∪
SCFF 1 2 2 3 5 6
{e } = {e ,e ,e ,e ,e ,e }.
7 1 2 3 5 6 7
Even though network N is assumed to be acyclic in traditional network
coding, we still want to know whether this characterization holds in a cyclic
network. In fact, if an edge is in D-set, then there exists a path p containing
it and satisfying C (s,t) = C (s,t) − 1. It’s easy to check that proof
N\p N
procedure of necessity in Theorem 3.4 also holds for cyclic network. But the
sufficiency proof does not work. Because Proposition 3.2 does not hold for
cyclic network. The following is a counterexample.
Consider a cyclic network N = (V,E,s,t) in Figure 3. Clearly, H =
{hv ,v i,hv ,v i}. However, path p = (s,v ,v ,t) covers hv ,v i and satisfy-
1 2 2 1 1 2 1 2
ing C = C (s,t)−1 = 1, and so dos path p = (s,v ,v ,t).
N\p N 2 1
The following result shows that any k-CF can be decomposed into an
m-CF and (k −m)-CF corresponding to different networks.
9
v
1
s t
v
2
Figure 3: A counterexample for cyclic network
Proposition 3.6. Let N = (V,E,s,t) be a point-to-point network. If F is
a k-CF of N, then ∀ 1 ≤ m ≤ k−1, ∃F′ ⊆ F such that F′ is a m-CF of N
and F \F′ is a (k −m)-CF of N \F′.
Proof. Let f = C (s,t). Denote F = {F′ ⊆ F | C (s,t) = f −m},m =
N m N\F′
0,1,...,k. Since deleting one edge will cause the maximum flow decrease by
at most 1, we can show F 6= ∅ for m = 1,2,...,k −1 by induction.
m
For ∀1 ≤ m ≤ k −1 we claim that F˜ ∈ F with |F˜| = min{|F′| | F′ ∈
m
F } is a m-CF of N. We only need to show ∀F′ ( F˜, C ≥ f −m+1,
m N\F′
which is true by the minimality of F′.
Finally, we should prove if F′ is an m-CF of N and F′ ⊆ F then F \F′ is
a (k−m)-CF of N \F′. Denote F′′ = F\F′. Notice (E\F′)\F′′ = E\(F′∪
F′′) = E\F, we have C (s,t) = C = f −k = (f −m)−(k−m).
(N\F′)\F′′ N\F
Since C (s,t) = f −m, we only need ∀F˜ ( F′, C (s,t) > f −k.
N\F′ (N\F′)\F˜
Otherwise, if ∃F˜ ( F′ such that C (s,t) = f − k, we may have
(N\F′)\F˜
F′ ∪F˜ ( F and C (s,t) = f −k which is impossible.
N\(F′∪F˜)
The following corollary is a direct generalization of the preceding result,
which says a k-CF can be decomposed arbitrary.
Corollary 3.7. Let N = (V,E,s,t) be a point-to-point network, and F is a
k-CFof N. Foranyn ∈ Z+ andk ∈ Z+ (i = 1,2,...,n) such that n k =
i Pi=1 i
n
k, then there exists pairwise disjoint sets Fi ⊆ F such that [Fi = F and Fi
i=1
i−1
is a k -CF of N \( F ).
i S j
j=1
It’s natural to ask whether the converse is true, i.e., if there exists k ∈
i
Z+ (i = 1,2,...,n) and pairwise disjoint sets F ⊆ E such that F is a k -CF
i i i
10