![Figure 4: Modified Edmonds-Karp algorithm [2]. is a flow network or graph constructed as described in Figure 3. is the set of all edges in ; or is a vertex in representing an input or output; is an edge from to ; is the total flow through the network; denotes a flow from to .](/figures/figure-4-modified-edmonds-karp-algorithm-2-is-a-flow-network-2q7u31ou.webp)
Abstract
Input queueing is becoming increasingly used for high-
bandwidth switches and routers. In previous work, it was
proved that it is possible to achieve 100% throughput for
input-queued switches using a combination of virtual out-
put queueing and a scheduling algorithm called LQF.
However, this is only a theoretical result: LQF is too com-
plex to implement in hardware. In this paper we introduce
a new algorithm called Longest Port First (LPF), which is
designed to overcome the complexity problems of LQF,
and can be implemented in hardware at high speed. By
giving preferential service based on queue lengths, we
prove that LPF can achieve 100% throughput.
1 Introduction
Traditionally, switches and routers have been most
often designed as a collection of line-cards connected to a
single shared bus. Packets waiting to be transmitted on
outgoing links are stored in a centralized, shared pool of
memory. If the aggregate bandwidths of the bus and mem-
ory are high enough, the system is able to keep all of the
outgoing links continuously busy, making the system
highly efficient. Furthermore, the system is able to control
packet departure times and hence provides guaranteed
qualities-of-service (QoS) [3][15][20][21]. However,
switch and router designers are finding that the continued
growth in bandwidth is making it increasingly difficult to
design a shared bus and centralized memory that run fast
enough. The data rate of a shared bus is limited by electri-
cal considerations, such as the loading on the bus, and
reflections from connectors. And the data rate of a central-
ized shared memory is limited because it requires buffer
memories that run times faster than the line rate, where
is the number of switch ports.
N
N
Increasingly, a passive shared bus is being replaced by
an active non-blocking switch fabric — most often a
crossbar switch. Each line card is connected by a dedi-
cated point-to-point link to the central switch fabric, and
therefore has fewer electrical limitations due to loading
and reflections. More importantly, each connection to the
switch need run only as fast as the line rate, rather than at
the aggregate bandwidth of the switch. Centralized shared
memory is also being replaced—by separate queues at
each input of the switching fabric. Input queues need only
run at the line rate, and therefore allow a faster overall sys-
tem to be built [6][11].
The very fastest switches and routers usually transfer
packets across the switching fabric in fixed size units, that
we shall refer to as “cells.” Variable length packets are
segmented into cells upon arrival, transferred across the
switch fabric and then reassembled again before they
depart. At the beginning of each cell time, a (usually cen-
tralized) scheduler selects a configuration for the switch-
ing fabric and then transfers cells from inputs to outputs.
Using fixed sized cells simplifies the switch design, and
makes it easier for the scheduler to configure the switch
fabric for high throughput.
But systems that use input queues have two potential
problems: low throughput due to head-of-line (HOL)
blocking and the difficulty of controlling cell delay. In this
paper, we focus on the first problem: achieving high
throughput.
It is well known that if an input-queued switch
employs a single FIFO queue at each input, HOL blocking
limits the throughput to just 58.6% of the maximum [7].
But HOL blocking can be eliminated entirely using a
queueing technique known as virtual output queueing
(VOQ) in which each input maintains a separate queue for
each output [1][10][12][13][17]. It has been shown that
with a suitable centralized scheduling algorithm, the
throughput can be increased from 58.6% to 100% [12].
A Practical Scheduling Algorithm to Achieve
100% Throughput in Input-Queued Switches.
Adisak Mekkittikul Nick McKeown
Computer Systems Laboratory
Stanford University, Stanford, CA 94305-9030
{adisak, nickm}@stanford.edu
This work was funded by a fellowship from National Semicon-
ductor and also by Texas Instruments, Cisco Systems, the Alfred
P. Sloan Foundation and a Robert N. Noyce faculty fellowship.
Unfortunately, the algorithms known to-date (LQF [12]
and OCF [13]) are too complex to implement in hardware,
and are therefore unsuitable for switches operating at high
speed. Instead, most switches and routers use a much sim-
pler scheduling algorithm to configure the switch fabric
[1][10][18]. Typically, a configuration is selected in an
attempt to maximize the number of connections made dur-
ing each cell time. Such an algorithm is called a maximum
size bipartite matching algorithm, and is found to perform
well when the arriving traffic is uniformly distributed over
all the switch outputs.
But real traffic is not uniform: traffic tends to be
focused on a relatively small number of active ports. And
unfortunately, a maximum size matching algorithm is
known to perform poorly when traffic is non-uniform [12].
The algorithm performs poorly in two (albeit related)
ways: increased buffer overflows, and reduced throughput.
Increased overflows occur because a maximum size
matching algorithm does not consider queue lengths when
deciding which input queues to service. When traffic is
non-uniform, the occupancies of the various input queues
can differ greatly, and queues with heavy traffic can over-
flow while ones with light traffic remain empty most of the
time. The reason for reduced throughput is a little more
complex. For a given number of cells in the system, if the
traffic is non-uniform, the cells are concentrated on a rela-
tively small number of VOQs. This reduces the number of
configurations available to the scheduler, and therefore
reduces the size of the maximum size match. If instead the
traffic was uniform, the cells in the system would be dis-
tributed uniformly over a relatively large number of
VOQs, making available a larger number of configurations
for the scheduler to choose from.
In earlier work [12][13], it was found that LQF (long-
est queue first) can achieve 100% for both uniform and
non-uniform traffic by considering the occupancies of the
queues. LQF gives preferential service to long queues by
using a maximum weight matching algorithm, where each
weight is set to the corresponding queue length. But LQF
is very difficult to implement in hardware at high speed.
First of all, it takes too long to run—the most efficient
algorithm known to-date has a running-time complexity
. Second, an implementation requires a large
number of multi-bit comparators to perform many weight
comparisons in parallel. Attempts to implement LQF (and
even heuristic approximations [10]) have been limited by
the design of a single-chip scheduler that: (i) has fast
enough comparators, (ii) can support a sufficient number
of comparators, and (iii) can interconnect them in a rich
enough pattern.
ON
3
Nlog()
Motivated by the desire to overcome the impracticali-
ties of LQF, yet achieve its high performance, we propose
a new algorithm: LPF (longest port first). With LPF our
goal is to combine the benefits of a maximum size match-
ing algorithm, with those of a maximum weight algorithm,
while lending itself to simple implementation in hardware.
LPF effectively finds the set of maximum size matches,
and from among this set chooses the match with the largest
total weight. In LPF each weight is a function of queue
lengths (we shall see later that the weights in LPF are not
exactly equal to the queue lengths, but are similar). This
enables LPF to take advantage of both the high instanta-
neous throughput of a maximum size matching algorithm,
and the ability of a maximum weight matching algorithm
to achieve high throughput, and a small number of over-
flows even when the arriving traffic is non-uniform. We
find that LPF—like LQF—can achieve 100% throughput
for both uniform and non-uniform traffic.
LPF has a running-time complexity of ; lower
than LQF. Furthermore, the comparators that limit the per-
formance of LQF are removed from the critical path of the
LPF algorithm. In fact, the heart of the LPF algorithm uses
a slightly modified maximum size matching algorithm, for
which there are a variety of existing, heuristic approxima-
tions [1][9][10][17].
The paper is organized as follows. In Section 2, we
provide some definitions. In Section 3, we describe LPF
and its properties before presenting our performance anal-
ysis.
2 Our Switch Model
We follow the general definitions used in [12]. Figure
1 shows an input-queued switch consisting of
input and output ports, a non-blocking switching fabric
and a scheduler. To eliminate head-of-line (HOL) block-
ing, each input maintains FIFO virtual output queues,
one for each output. denotes the VOQ at input con-
taining cells destined to output . Arrivals are fixed size
packets or cells, allowing us to split time into discrete cell
times, or slots. During any given slot, there is at most one
arrival to and departure from each input, and similarly for
each output. is the arrival process of cells to input
destined to output at rate . Consequently, is
the aggregate process of all arrivals to input at rate
.
ON
2.5
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Definition 1: An arrival process is said to be admissible
when no input or output is oversubscribed, i.e., when
.
Definition 2: The traffic is uniform if all arrival processes
have the same arrival rate, and if the destinations of cells
are uniformly distributed over all outputs. Otherwise the
traffic is non-uniform.
The scheduler determines which inputs and outputs
are connected during each slot. The scheduling problem
can be viewed as a bipartite graph matching problem
Figure 1: A Simple Model of VOQ Switches.
Input 1
Q
1,1
Q
1,N
A
1
(t)
Input M
Q
M,1
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M,N
A
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(t)
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Output 1
Output N
Crossbar
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Scheduler
switch
λ
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Inputs, I
Outputs, J
1
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Request graph, G
Matching, M
a) Example of G for
b) Example of matching
Inputs, I
Outputs, J
1
2
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1
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w
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w
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I = M and J = N.
M on G.
w
2,1
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1,3
w
3,2
F
igure 2: A request graph and a matching graph of an
s
witch. Define G = [V,E] as an undirected graph connecting the
s
et of vertices V with the set of edges E. The edge connecting ver-
t
ices i, 1≤i≤M and j, 1≤j≤N has an associated weight denoted w
i,j
.
G
raph G is bipartite if the set of inputs I = {i: 1≤i≤M} and outputs
J
= {i: 1≤j≤N} partition V such that every edge has one end in I
a
nd one end in J. Matching M on G is any subset of E such that no
t
wo edges in M have a common vertex.
MN×
[2][19], an example of which is shown in Figure 2. Each
input makes a request to every output for which it has cells
queued. An edge in the graph represents a request from
with weight (denoted in Figure 2 as ).
Let be a service indicator such that
and ; a value of one indicates
that input is matched to output , i.e., is allowed to
forward one cell to its output.
Definition 3: A maximum size match is one that maximizes
, i.e., the number of connections.
Definition 4: A maximum weight match is one that maxi-
mizes , i.e., the total weight.
Alternatively, a bipartite graph matching problem can
be easily solved and understood by transforming it into a
flow network [2][19], as illustrated in Figure 3.
3 The LPF Algorithm
Although in practice LPF can be thought of as a spe-
cial maximum size matching algorithm, in theory it is eas-
ier to consider LPF as a maximum weight matching
algorithm. Each LPF request weight, , for a request
from input to output is defined as follows:
(1)
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ij,
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ij,
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ij,
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ij,
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i 1=
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∑
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j 1=
N
∑
1≤
1
2
3
N
a) Weighted request graph.
b) A corresponding flow network
.
1
2
3
M
Figure 3: Transformation of a request graph into a flow network.
(a) A weighted request graph. (b) The corresponding flow net-
work, G, whose all edges are of unity capacity. A source and a
target are added. The cost of every edge from and to is set
to zero. The cost of all other edges are equal to the negated value
of the corresponding weight.
s
t s t
1
2
3
N
1
2
3
M
-w
1,1
-w
2,1
-w
1,3
-w
3,2
-w
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t
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-w
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i
j
Q
ij,
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ij,
n()
ij,
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ij,
n()w
ij,
n()
ij,
∑
w
ij,
n()
i
j
w
ij,
n()
R
i
n() C
j
n()+ L
ij,
n() 0>,
0 otherwise,,
=
where is the occupancy of at slot ,
and .
, which we call the input occupancy, is the total
number of cells that are currently waiting at input to be
forwarded to their respective outputs. Similarly, , the
output occupancy, is the total number of cells at all inputs
waiting to be forwarded to output . Together, the sum of
the input and output occupancies represents the work load
or congestion that a cell faces as it competes for transmis-
sion to its output. We call this sum the port occupancy;
LPF favors queues with high port occupancy.
Property 1: The total weight of an LPF match is equal to
the occupancy sum of all matched inputs and outputs, i.e,
, where and are the
set of matched inputs and matched outputs respectively.
We now show that LPF is a special case of a maxi-
mum size matching algorithm.
Theorem 1: LPF finds a match that is both maximum size
and maximum weight.
Proof of Main Theorem: see Appendix A. ❒
Since an LPF match is a maximum size match, we can
use a maximum size matching algorithm to find an LPF
L
ij,
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ij,
n
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n()
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∑
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ij,
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i
Figure 4: Modified Edmonds-Karp algorithm [2]. is a flow
network or graph constructed as described in Figure 3. is
the set of all edges in ; or is a vertex in representing
an input or output; is an edge from to ; is the total
flow through the network; denotes a flow from to .
is a residual network [2] [19], also called a residual graph.
LPFS is a largest unmatched port first search.
G
EG[]
Gu v G
uv,() uv
f
f
uv,[] uv
G
f
Modified Edmonds-Karp algorithm
1 for each edge
2 do
3
4 while LPFS finds a path from to in the residual
network
5 for each edge in
6 do if then
7 else
8
uv,()EG[]∈
f
uv,[]0=
f
vu,[]0=
p st
G
f
uv,()p
f
uv,[]0=
f
uv,[]cuv,[]=
f
vu,[]0←
f
uv,[] f–vu,[]=
C
j
n()
j
S
ij,
n()w
ij,
n()
ij,
∑
R
i
iI∈
∑
C
j
jJ∈
∑
+= I J
match. But we need to make sure that among all possible
maximum size matches we choose one with the largest
total weight.
3.1 Finding an LPF Match Using a Maximum
Size Matching Algorithm
Existing maximum size matching algorithms cannot
be used to implement LPF because they are unable to
select the maximum size match with the largest weight. A
simple modification is called for. First, in order to keep the
algorithm free of complex magnitude comparisons, all
inputs and outputs are pre-ordered according to their LPF
weights prior to running the maximum size matching algo-
rithm. Then we use a modified Edmonds-Karp maximum
size matching algorithm [2][19] to find the LPF match (see
Figure 4). A breadth-first search (BFS) in the Edmonds-
Karp algorithm is replaced by a largest-unmatched-port
first search (LPFS) described in Figure 5. LPFS enables
the modified algorithm to search for a maximum weight
match while performing path augmentation [19] to find a
maximum size match. As a result, line 2 of the LPFS-Visit
does not involve any magnitude comparison. It is proved
in [14] that the modified algorithm finds an LPF match.
Figure 5: A largest-unmatched-port first search (LPFS). First,
LPFS builds a tree with as its root. Initially every input and out-
put is colored white — undiscovered, then is grayed when it is
discovered, and finally is blackened when it is finished. is
the predecessor of . From the tree, an augmenting path from
to which must go through an unmatched input can be found by
walking the predecessor list which begins at a selected un-
matched input.
t
π v[]
v s
t
LPFS(G)
1 for each vertex
2 do white
3 nil
4 LPFS-Visit(t)
LPFS-Visit(u)
1 gray
2 for each , starting the largest to
the smallest.
3 do if white
4 then
5 LPFS-Visit(v)
6 black
uVG[]∈
color u[]←
πu[]←
color u[]←
v Adjacent u[]∈
color v[]=
πv[] u←
color u[]=
Theorem 2: The maximum size match found by the modi-
fied Edmonds-Karp algorithm is also a maximum weight
match with weights as defined in Equation 1.
Proof of Main Theorem: see reference [14]. ❒
3.2 A Practical Approximation to LPF
LPF can be adapted to run at higher speed using sim-
ple heuristic approximations. Shown in Figure 6 is an iter-
ative algorithm called iLPF that approximates LPF. All
weight processing is done in step 1 prior to the iterative
steps. The second step consists of a double for-loop used
to find a maximal size match. Since the requests have
already been ordered in the first step, the maximal size
matching in the second step does not need to compare
request weights. Figure 7 shows the schematic of a hard-
ware implementation of iLPF. Our exploratory design
work suggests that the second step can be implemented
using simple hardware; for a switch, our synthe-
sized design can make a scheduling decision in just 10ns
using a commercial 0.25 CMOS ASIC technology.
The first step, which requires simple integer arithmetic,
can also run in 10ns, allowing the switch to run at a line
rate of 20 Gb/s.
1
3.3 Stability
We now prove that LPF can achieve 100% throughput
for all traffic patterns with independent arrivals, using the
1. Calculated based on the size of an ATM cell.
Figure 6: An iterative LPF algorithm. First, the algorithm builds
a sorted list of all inputs and outputs based on their occupancies.
Then, starting from the largest output and input, the algorithm
finds a maximal size match.
Iterative LPF algorithm
Step 1.
1 Sort inputs&outputs based on their occupancies
2 Reorder requests according to their input and output
occupancies
Step 2. Maximal size matching
1 for each output from largest->smallest
2 for each input from largest->smallest
3 if (there is a request) and (both input and output
unmatched)
4 then match them
32 32×
µm
notion of stability [8]. We define a switch to be stable for a
particular arrival process if the expected length of the
input queues does not grow without bound, i.e.,
. (2)
Definition 5: A switch can achieve 100% throughput if it
is stable for all independent and admissible arrivals.
Theorem 3: The LPF algorithm is stable for all admissi-
ble independent arrival processes.
Proof of Main Theorem: see Appendix B. ❒
3.4 Stability With a Finite Pipeline Delay
Because the modified maximum size matching algo-
rithm requires the input and outputs to be pre-ordered,
LPF and iLPF need sorting networks to sort all inputs and
outputs. Due to the relatively high complexity of the sort-
ing networks, they could dominate the running time of the
algorithm. Alternatively, we can pipeline the design to
reduce its running time; the sorting networks can operate
in one slot, and the maximum size matching algorithm in
the next. This means that the maximum size matching
algorithm is operating on weights that are now one slot out
of date — it is possible for the algorithm to favor the
Figure 7: A block diagram of iLPF. Referring to the algorithm in
Figure 6, inputs and outputs are pre-sorted by the two sorter net-
works. Raw requests (requests with weights removed) is given in
a matrix form. Request reordering is done by the two crossbars
which are configured by the sorting results. The maximal size
matching block, which implements the double for-loop, finds a
maximal size match that approximates an LPF match. The match
needs to be permuted back to its natural order.
Input Occupancies
Output Occupancies
Maximal size
Raw Requests
Sorter
Sorter
X Bar
X Bar
1
1
0
1
1
0
0
0
1
Matching
Permuted Requests
Match
1
0
0
0
1
0
0
0
1
{10, 20, 30}
{20, 25, 15}
{3, 2, 1}
{2, 1, 3}
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