Finding Best Swap Edges Minimizing the Routing Cost of a Spanning Tree
Given an n -node, undirected and 2-edge-connected graph G =( V , E ) with positive real weights on its m edges, given a set of k source nodes S ⊆ V , and given a spanning tree T of G , the routing cost from S of T is the sum of the distances in T from every source s ∈ S to all the other nodes of G ....
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| Vydáno v: | Algorithmica Ročník 68; číslo 2; s. 337 - 357 |
|---|---|
| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Boston
Springer US
01.02.2014
Springer |
| Témata: | |
| ISSN: | 0178-4617, 1432-0541 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | Given an
n
-node, undirected and 2-edge-connected graph
G
=(
V
,
E
) with positive real weights on its
m
edges, given a set of
k
source
nodes
S
⊆
V
, and given a spanning tree
T
of
G
, the
routing cost from
S
of
T
is the sum of the distances in
T
from every source
s
∈
S
to all the other nodes of
G
. If an edge
e
of
T
undergoes a
transient
failure, and one needs to promptly reestablish the connectivity, then to reduce set-up and rerouting costs it makes sense to temporarily replace
e
by means of a
swap edge
, i.e., an edge in
G
reconnecting the two subtrees of
T
induced by the removal of
e
. Then, a
best swap edge
for
e
is a swap edge which minimizes the routing cost from
S
of the tree obtained after the swapping. As a natural extension, the
all-best swap edges
problem is that of finding a best swap edge for
every
edge of
T
, and this has been recently solved in
O
(
mn
) time and linear space. In this paper, we focus our attention on the relevant cases in which
k
=
O
(1) and
k
=
n
, which model realistic communication paradigms. For these cases, we improve the above result by presenting an
time and linear space algorithm. Moreover, for the case
k
=
n
, we also provide an accurate analysis showing that the obtained swap tree is effective in terms of routing cost. Indeed, if the input tree
T
has a routing cost from
V
which is a constant-factor away from that of a
minimum routing-cost spanning tree
(whose computation is a problem known to be in APX), and if in addition nodes in
T
enjoys a suitable distance stretching property from a tree centroid (which can be constructively induced, as we show), then the tree obtained after the swapping has a routing cost from
V
which is still a constant-ratio approximation of that of a new (i.e., in the graph deprived of the failed edge) minimum routing-cost spanning tree. |
|---|---|
| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-012-9674-y |