<<

. 82
( 87 .)



>>

An FPTAS is stronger than a polynomial time-approximation scheme (PTAS) that also runs in polynomial time
in terms of the original input, with the same error bound, but may be exponential in terms of 1/ . Thus, error re-
duction can be exponentially costly for a PTAS, but not for an FPTAS.
432 ALGORITHMIC CHALLENGES IN AD HOC NETWORKS


Example 2: Minimum Energy Path. Let F contain each directed edge in a directed
graph with multiplicity 2, as well as each pair of directed edges that have a common
tail, with multiplicity 1. That is, the two-element edge set {e, f } is in F if and only if
e = (u, v) and f = (u, w) for some vertices u, v, w; v w. It is not difficult to see that
in this case F(P) is precisely the metric that we used in the minimum energy path
problem, in the special case when transmission and reception both need unit energy.
By changing the multiplicities in F, one can incorporate different transmit and re-
ceive energy levels.
Example 3: Minimum Weight k-hop Path. For arbitrary nonnegative integers wi, let F
contain wi copies of edge ei for i = 1, . . . , m, where m is the number of edges. Fur-
ther, for a given integer 1 k m, let F also contain all edge sets H that have the
following properties: |H| = m “ k and the complement of H is a u“v path. Each such
edge set H is included in F with multiplicity M = 1 + m wi. i=1
Now we can observe that the singleton sets in F will contribute w(P) to F(P),
where w(P) is the weight of the path P with edge weights wi. The edge sets of cardi-
nality m “ k in F behave such that if the path has more than k edges, then it intersects
every such set (since each misses only k edges). On the other hand, if the path has
exactly k edges, then precisely one of the different m “ k-element sets is disjoint
from P, the one that has P as its complement. Finally, if P has less than k edges, then
it again intersects all these sets. To see this, observe that a simple u“v path cannot be
a proper subset of another simple u“v path (this claim will be formally proven in the
proof of Theorem 1 below). Therefore, for any set H F, the path P cannot be fully
contained in the complement of H, so P H 0 must hold. Thus, if the number of
/
k-hop u“v paths is denoted by R, then we have

w(P) + RM if |P| k
F(P) =
w(P) + (R “ 1)M if |P| = k

Since M is chosen such that M > w(P) holds for every path, therefore, the path for
which F(P) is minimum will be precisely the one that has minimum weight w(P)
among the u“v paths that have exactly k edges, if such a path exists. If no such path
exists, then F(P) > RM.

(Excercise: Modify the above construction such that the path that minimizes F(P)
will be a minimum weight path among those paths that have at most k edges, if such
a path exists.)

Example 4: Disjoint Connecting Paths with Minimum Total Hops. Let us consider the
following problem of finding multiple paths. Assume that two source nodes s1 and
s2, and two terminal nodes t1 and t2 are given in a directed graph. We would like to
find two edge-disjoint directed paths P1 and P2 such that P1 connects s1 to t1 and P2
connects s2 to t2 and |P1| + |P2| is minimum. The problem is known to be NP-hard. In
fact, just to decide if it is possible to connect the given terminals by edge-disjoint di-
rected paths, irrespective of their lengths, is already NP-complete4 [10]. On the oth-
er hand, as shown in [15], if the paths exist and the objective is to minimize the

4
It is essential that the terminals of each path are specified. If we just look for two edge-disjoint paths to connect
the sets {s1, s2} and {t1, t2}, then it can be solved by network flow techniques in polynomial time.
433
16.3 A UNIFIED REPRESENTATION OF PATH METRICS


length of the longer path, rather than the sum, then it can be approximated in poly-
nomial time within a factor of 2. This also implies an approximation factor of at
most 2 for the summed length, due to max{|P1|, |P2|} < |P1| + |P2| 2 max{|P1|,
|P2|}.
Let us express the problem with the F-measure, for the |P1| + |P2| min objec-
tive. First, if the graph does not contain the edge (t1, s2), then let us add this edge.
Now construct F as follows. Add each edge, except (t1, s2), as a singleton set to F.
Further, let H be the set of s1“t2 paths that contain the edge (t1, s2). For each path P
H let us add the set P (the complement of P) to F with multiplicity m, where m is
the number of edges in the original graph. Every simple path P H is of the form P
= P1(t1, s2)P2, that is, the concatenation of three paths: P1 from s1 to t1, the edge (t1,
s2), as well as another disjoint path P2 from s2 to t2. It is not difficult to see (exer-
cise!) that for such a P

F(P) = |P1| + |P2| + m(|H | “ 1) m|H |

H we have F (P) > m|H |. Thus, the F-shortest path
holds, whereas in case of P
will correspond to the required pair of disjoint paths with the minimum sum of hop
counts.

Having seen some examples, the reader may start wondering: what are the limits of
the expressive power of the F-measure? Can every path metric be expressed in this uni-
fied way by picking the appropriate family of sets for F ? Below, we show that essen-
tially every path metric, no matter how complicated, can be represented this way. Thus,
the F-measure can serve as a canonical representation of all path metrics and, therefore,
the concept of F-shortest path is a canonical representation of any path-optimization
task.
Before going into the representation, let us note that the F-measure, by definition, can
only be a nonnegative integer. As remarked after Definition 1, the model can be directly
extended to the real-valued case. On the other hand, when we search for a u“v path that is
optimal for some path metric h(P) in a graph, the order relationship between the paths
does not change if the metric h(P) is replaced by a + bh(P) for any constants a, b, with b >
0. This implies that, from the path-optimization point of view, it is enough to restrict our-
selves to positive integer valued metrics. That is why we stay with integer multiplicities in
this introductory exposition.
Now we can state the main representation theorem that shows that the F-measure pro-
vides a realization of any integer-valued metric, apart from a constant translation.

Theorem 1 (Path Metric Representation Theorem). Let G be a (directed or undirected)
graph with two distinguished vertices u and v that serve as endpoints of the considered
paths. Let h be an arbitrary path metric that assigns a positive integer value h(P) to every
u“v path P in G. Then there exists a family F of sets of edges in G and a constant M, such
that the F-measure represents the path metric h(P) for every u“v path in G, up to constant
translation, that is,

F (P) = h(P) + M

holds for every u“v path in G.
434 ALGORITHMIC CHALLENGES IN AD HOC NETWORKS


Proof. Let Pu,v be the set of u“v paths in G and let R = |Pu,v|. Set

A= h(P)
P Pu,v

Construct the family F of edge sets as follows. For each P Pu,v include the edge set P
(the complement of P) in F with multiplicity A “ h(P). Define the constant M as

M = (R “ 2)A

We claim that with this choice of F and M the relationship F (P) = h(P) + M holds for
every u“v path in G.
First we show that a simple u“v path P be can never be a proper subset of another sim-
ple u“v path Q. Assume indirectly that P Q, but P Q. Let us traverse Q from u to v.
(Note that if G is undirected, then it is not a priori necessary that the common edges of P,
Q are traversed in the same direction by both paths.) In the traversal of Q, let e be the first
edge in Q that is not contained in P (such an edge must exist, since P is assumed to be a
proper subset of Q). Let w be the endpoint of e that is reached first in the traversal. Then w
v, since Q is not fully traversed yet when we reach w. Now we have two possibilites. (1)
If w = u, then P must leave u on an edge f e. But then, since Q does not visit u again, f
Q must hold, a contradiction to P Q. (2) If w u, then let f1 be the edge on which P
arrives at w and f2 be the edge on which it leaves. Since e P, therefore, f1, f2 e. But
then at least one of f1, f2 cannot be in Q, since Q cannot contain three edges that are inci-
dent to w, so we again contradict P Q. Thus, in either case we arrive at a contradiction,
proving that an u“v path cannot be a proper subset of another one.
Now let us compute F (P) for an arbitrary P Pu,v. By the construction, the sets in F
are all of the form Q with Q Pu,v. Observe that whenever P Q, we have P Q 0, as /
we have shown that in case of P Q the relationship P Q cannot hold, since then P
would be a proper subset of Q. Therefore, if P Q holds, then P must intersect the com-
plement of Q. On the other hand, naturally, P P = 0. Thus, P intersects all sets in F, ex-
/
cept the ones that correspond to its own complement. This implies, by the construction of
F and the multiplicities,

F (P) = [A “ h(Q)] = (R “ 1)A “ h(Q)
Q Pu,v“{P} Q Pu,v“{P}

Observing that the second sum is equal to A “ h(P), we obtain

F (P) = (R “ 1)A “ [A “ h(P)] = h(P) + (R “ 2)A = h(P) + M

í
which proves the theorem.

One can observe at this point that the representation provided in the proof of Theorem 1
generally uses exponentially many different sets for F. This is the case even if a much small-
er F would do, such as for the hop metric in Example 1. Clearly, it would be desirable to
have a “small” representation whenever possible. This points to the first open question.

Open Problem 1. Characterize the path metrics that can be represented by F-measures
that have only polynomially many different sets in F. Find a general method for construct-
ing this “small” representation, whenever it exists.
435
16.4 FINDING THE F-SHORTEST PATH


FINDING THE F -SHORTEST PATH
16.4

Some of the path metrics generate NP-hard path-optimization tasks. For example, as men-
tioned in Section 16.2, the shortest weight constrained path problem has this property.
Since, as we have seen in Section 16.3, the F -measure can represent any path metric,
therefore, we cannot expect that a polynomial-time algorithm exists to find an F -shortest
path for an arbitrary F. On the other hand, we show that it is possible to find an approxi-
mately F -shortest path efficiently, if F is “small.”

Theorem 2. Let F be a family of sets, given by explicit listing, in a (directed or undirect-
ed) graph with n vertices. Assume there are k sets in F (counted with multiplicity), each
with cardinality at most r. Then there is an algorithm of complexity O(max{n2, kr}) that
finds a path P between any two given vertices such that P is an r-approximation of the F -
shortest path, that is,

F (P) rF (P0)

holds, where P0 is an F -shortest path between the same end nodes.

Proof. Let e1, e2 . . . be the edges of the graph and u, v be the end vertices of the sought
path. Let us define edge weights, as follows. To edge ei assign the weight

wi = m(H)
ei H F


where m(H) is the multiplicity of the set H in F. In other words, wi is the number of sets in
F, counted with multiplicity, that contain the edge ei. Each wi can be found and recorded
in altogether O(max{n2, kr}) time, by simply scanning all sets in F and recording for each
edge in how many sets it occured. Now find a minimum weight u“v path P according to
this weighting. Since the path can be found in O(n2) time by Dijsktra™s algorithm, there-
fore, the overall complexity remains O(max{n2, kr}).
Now we claim that the found path P has the desired property of being an r-approxi-
mation of the F-shortest path. Let P0 be an F -shortest u“v path. Let us count for each
edge ei P0 the number of sets H F, counted with multiplicity, that contain ei. Then
we get precisely wi. If we sum up the wi weights along the path P0, then we can get
F can be counted at most r times, as it contains at
at most rF (P0), since each set H
most r edges. On the other hand, this sum is exactly the weight w(P0) under the weight-
ing wi. Thus, we have w(P0) rF (P0). Now observe that the path P, found as a mini-
mum weight path under the weighting wi, cannot have larger weight than any other u“v
path. Therefore, w(P) w(P0) must hold, which implies the desired relationship F (P)
í
rF (P0).

Note that in the above proof we did not really require that the sets of F were explicitely
listed; this was only assumed for simplicity. In fact, it is enough if we have a subroutine
that can count for each edge the number of sets in F that contain the edge.
Regarding the algorithmic issues on F-shortest paths, two open problems are definitely
worth mentioning.
436 ALGORITHMIC CHALLENGES IN AD HOC NETWORKS


Open Problem 2. Characterize the families (or find special families) F of edge sets for
which it is possible to find an F -shortest path in polynomial time.
Open Problem 3. In the case when no polynomial-time algorithm is available to com-
pute an exact solution, find efficient approximations with provable bounds on the
approximation error. An example in this direction is Theorem 2.


16.5 MOBILE PATHS

A proposed method for the assessment and comparison of routing protocols in mobile ad
hoc networks is the MERIT framework [6, 7]. This method compares the route sequences
found by a given routing protocol with an ideal route sequence that could be found if we
knew the node movement in advance. We do not discuss the details of the entire protocol
assessment framework here, as it is covered by [6, 7]. We only consider the interesting
graph algorithmic issues from this framework and point out that they also fit as a special
case in the F -shortest path model.
To model mobility, the MERIT framework introduces the concepts of mobile graph and
mobile path. A mobile graph G is defined as a sequence Gi, i = 1, . . . , T, of graphs on the
same node set, where the successive graphs represent a history of the network topology
changes over some time horizon T:

G = G1G2 . . . GT

In a mobile graph, a mobile path between a source“destination s“t pair is defined as a se-
quence of paths

P = P1P2 . . . PT

where Pi is a static path in Gi between the same source“destination s“t pair.
The weight of a mobile path w(P) includes two basic components:

1. The weights wi(Pi) of the individual static paths, according to arbitrary individual
weightings in the graphs Gi
2. Some transition cost ctrans(Pi, Pi+1) incurred by the protocol whenever there is a
change in the path sequence

Thus, the weight (cost) of a mobile path P is defined as

T T“1
w(P) = wi(Pi) + ctrans(Pi, Pi+1) (16.1)
i=1 i=1


The transition cost between paths is a cost function associated with having to update from
one path to another in the routing protocol. In general, it is the overhead associated with
updating the routing state to reflect the change in the path.
Now we can define the shortest mobile path problem as follows.

<<

. 82
( 87 .)



>>