COMBINATORICA
Bolyai Society – SpringerVerlag
C
OMBINATORICA
19
(3) (1999) 375–
401
FAST ALGORITHMS FOR FINDING O(Congestion + Dilation)
PACKET ROUTING SCHEDULES
TOM LEIGHTON, BRUCE MAGGS
and
ANDR
´
EA W. RICHA
Received July 8, 1996
In 1988, Leighton, Maggs, and Rao showed that for any network and any set of packets whose
paths through the network are ﬁxed and edgesimple, there exists a schedule for routing the packets
to their destinations in
O
(
c
+
d
) steps using constantsize queues, where
c
is the congestion of the
paths in the network, and
d
is the length of the longest path. The proof, however, used the Lov´asz
Local Lemma and was not constructive. In this paper, we show how to ﬁnd such a schedule in
O
(
m
(
c
+
d
)(log
P
)
4
(loglog
P
)) time, with probability 1

1
/
P
β
, for any positive constant
β
,where
P
is the sum of the lengths of the paths taken by the packets in the network, and
m
is the number
of edges used by some packet in the network. We also show how to parallelize the algorithm so
that it runs in
NC
. The method that we use to construct the schedules is based on the algorithmic
form of the Lov´asz Local Lemma discovered by Beck.
1.
Introduction
I n this paper, we consider the problem of scheduling the movements of packets
whose paths through a network have already been determined. The problem is
formalized as follows. We are given a network with
n
nodes (switches) and
m
edges
(communication channels). Each node can serve as the source or destination of an
arbitrary number of
packets
(or
cells
or
ﬂits
, as they are sometimes referred to).
Let
N
denote the total number of packets to be routed. The goal is to route the
N
packets from their origins to their destinations via a series of synchronized time
steps, where at each step at most one packet can traverse each edge and each packet
can traverse at most one edge. Without loss of generality, we assume that all edges
in the network are used by the path of some packet, and thus that
m
gives the
number of such edges (all the other edges are irrelevant to our problem).
Mathematics Subject Classiﬁcation (1991): 68M20, 68M10, 68M07, 60C05
Tom Leighton
is supported in part by ARPA Contracts N0001491J1698 and N0001492
J1799. Bruce Maggs is supported in part by an NSF National Young Investigator Award under
Grant No. CCR–94–57766, with matching funds provided by NEC Research Institute, and by
ARPA Contract F33615–93–1–1330. Part of this research was conducted while Andr´
ea Richa
was at Carnegie Mellon University, supported by NSF National Young Investigator Award under
Grant No. CCR–94–57766, with matching funds provided by NEC Research Institute, and ARPA
Contract F33615–93–1–1330.
0209–9683/99/$6.00 c
±
1999 J´anos Bolyai Mathematical Society