One-Rate Bidirectional Clos Networks

• One-rate bidirectional Clos networks are multistage switching fabrics with uniform capacity links and symmetrical bidirectional operation in circuit, packet, and optical applications.
• They achieve full throughput and nonblocking performance through careful middle-stage dimensioning and deterministic scheduling, as seen in models like LBC and ROADM.
• Advanced control techniques, including LP duality and replacement chains, minimize rearrangement costs while ensuring in-order forwarding and scalable DCN operations.

One-rate bidirectional Clos networks are multistage switching fabrics in which all external and internal links operate at the same rate, while connectivity is supported in both directions under symmetric line capacities, full-duplex interpretations, or explicitly shared bidirectional capacities. In the classical setting they are three-stage Clos networks used for circuit switching; in later work they appear as packet-switch fabrics without speedup, optical ROADMs and all-optical datacenter networks, and centrally scheduled bidirectional Clos fabrics for optical DCNs. Across these settings, the central technical questions are nonblocking dimensioning, rate-1 throughput, path diversity, rearrangement cost, and the control mechanisms needed to exploit the fabric without introducing excessive scheduling complexity (Ngo et al., 2012, Sule et al., 2018, Lin et al., 2023, Zhu et al., 16 Jul 2025).

1. Formal definition and modeling conventions

In its general form, a three-stage Clos network is denoted $C(n_1,r_1,m,n_2,r_2)$, with $r_1$ input crossbars of size $n_1\times m$, $m$ middle crossbars of size $r_1\times r_2$, and $r_2$ output crossbars of size $m\times n_2$. The symmetric case $C(n,m,r)$ sets $n_1=n_2=n$ and $r_1=r_2=r$ (Ngo et al., 2012). In one-rate circuit models, all calls use one unit of capacity, and all internal links have the same capacity as the external links. In packet-switch realizations, the same condition appears as rate-1 operation with no internal speedup and no central-stage expansion (Sule et al., 2018).

Bidirectionality is represented in more than one way. In the standard circuit interpretation, a physical full-duplex line can be modeled as two directed links of capacity $r_1$0, one in each direction; under symmetric traffic, one-direction results apply identically to the other direction (Ngo et al., 2012). In optical ROADM models, a fiber degree is explicitly a pair of bidirectional fibers on a directional degree, and the line side is described as a pair of ingress and egress modules (Lin et al., 2023). In recent two-level datacenter formulations, bidirectionality is native: low-level switches $r_1$1 connect to top-level switches $r_1$2 through bidirectional links $r_1$3 with shared integer capacity $r_1$4, and a unit connection between $r_1$5 and $r_1$6 via $r_1$7 is represented as $r_1$8 (Zhu et al., 16 Jul 2025).

The one-rate condition also varies by domain while preserving the same structural idea. In packet switching it means that all links $r_1$9, $n_1\times m$0, $n_1\times m$1, $n_1\times m$2, and $n_1\times m$3 operate at the same line rate and that neither memory speedup nor central-stage expansion is used (Sule et al., 2018). In optical Clos-ROADMs it means homogeneous per-fiber capacity, equal numbers of wavelengths on line-side ports, and uniform traffic assumptions such as “the offered traffic load between any input-output port pair is the same” (Lin et al., 2023). In the bidirectional DCN model, it means that a connection is an atomic transfer unit with rate $n_1\times m$4, and each link carries as many unit connections as its capacity allows (Zhu et al., 16 Jul 2025).

A key special case in the bidirectional DCN literature is the proportional one-rate bidirectional Clos network, defined by the existence of nonnegative integers $n_1\times m$5 and $n_1\times m$6 such that

$n_1\times m$7

The uniform model, in which all $n_1\times m$8 are equal and even, is a special case of this proportional model (Zhu et al., 16 Jul 2025). This proportionality is what enables a reduction to a classical three-stage symmetric Clos network and thereby links modern bidirectional scheduling results back to classical nonblocking theory.

2. Classical nonblocking theory and middle-stage dimensioning

The classical theory distinguishes strict-sense nonblocking (SNB), wide-sense nonblocking (WSNB), and rearrangeably nonblocking (RNB) behavior. SNB means that a new request between any free input-output pair can always be routed regardless of how existing connections are arranged. WSNB means that a new request can always be routed provided that past and new requests are routed according to a specified algorithm. RNB means that any permutation can be realized, possibly after rearranging existing connections (Ngo et al., 2012).

For a symmetric three-stage Clos $n_1\times m$9 under one-rate circuit switching, the classical S-Clos condition is

$m$0

A standard LP-duality derivation bounds the number of unavailable middle crossbars for a new request by $m$1, so one additional middle module guarantees an available route (Ngo et al., 2012). Under the bidirectional full-duplex interpretation, the same bound applies per direction, so the structural requirement on $m$2 is unchanged.

For $m$3, Benes’s “reuse busy middle crossbar whenever possible” algorithm yields the WSNB condition

$m$4

and the result is stated as necessary and sufficient for that routing discipline (Ngo et al., 2012). This bound is substantially smaller than the SNB requirement and illustrates the traditional trade-off between stronger online guarantees and algorithm-dependent routing.

The same LP framework extends to multirate settings. The paper on linear-programming duality introduces dynamic weighted edge coloring (DWEC) and obtains the multirate WSNB sufficient condition

$m$5

For one-rate calls this remains sufficient but is not tight relative to the classical circuit-switching bounds (Ngo et al., 2012). The significance is methodological: a single LP/duality template covers strict-sense, wide-sense, multirate, and photonic variants of Clos-type networks.

These results establish the canonical dimensioning question for one-rate bidirectional Clos networks: how many middle-stage resources are needed to ensure that uniform-capacity connections can be admitted without blocking, either unconditionally, algorithmically, or after rearrangement. Later packet-switch and optical results can be read as alternative ways of achieving or approximating the same objective under different control models.

3. Rate-1 packet-switch realizations without speedup

A concrete packet-switch realization of the one-rate Clos idea is the split-central-buffered Load-Balancing Clos (LBC) switch, an $m$6 cell-based switch with four stages: input, central-input, central-output, and output. It uses $m$7 input modules (IMs), $m$8 output modules (OMs), and a central stage split into $m$9 central-input modules (CIMs) and $r_1\times r_2$0 central-output modules (COMs), with the symmetric choice $r_1\times r_2$1 and hence $r_1\times r_2$2 (Sule et al., 2018). The split central stage separates load balancing from routing to outputs and inserts buffers between CIM and COM.

The LBC uses pre-determined, periodic, disjoint permutations. At time slot $r_1\times r_2$3, the IM configuration is

$r_1\times r_2$4

the CIM configuration is

$r_1\times r_2$5

and the COM configuration mirrors the CIM schedule in reverse: $r_1\times r_2$6 The combined IM-CIM connectivity is represented by an $r_1\times r_2$7 permutation matrix $r_1\times r_2$8, and the compound permutation over one period is

$r_1\times r_2$9

An analogous compound COM permutation is

$r_2$0

These cyclic permutations evenly spread each input’s offered load across the $r_2$1 VOMQs associated with its destination OM (Sule et al., 2018).

The traffic model assumes admissible i.i.d. arrivals with rate matrix $r_2$2 satisfying

$r_2$3

After IM-CIM load balancing,

$r_2$4

so each row of $r_2$5 has exactly $r_2$6 nonzero entries, each equal to $r_2$7 of the row sum of $r_2$8. The analysis further shows

$r_2$9

which means that the traffic queued at the crosspoint buffers for output $m\times n_2$0 reproduces the original external load (Sule et al., 2018).

The main throughput result is explicit: LBC achieves $m\times n_2$1 throughput under admissible i.i.d. traffic without memory speedup or central-stage expansion. Stability is proved separately for VOQs, VOMQs, and CBs, with queue occupancies $m\times n_2$2, $m\times n_2$3, and $m\times n_2$4 shown to have bounded drift under admissible load (Sule et al., 2018). Simulations for $m\times n_2$5 and $m\times n_2$6 show $m\times n_2$7 throughput under Bernoulli uniform, bursty ON/OFF, unbalanced, hot-spot, and two CB-stressing traffic patterns; delay is reported as close to ideal output-queued behavior in several cases (Sule et al., 2018).

A distinct contribution is in-order forwarding. Because load balancing sprays cells across multiple VOMQs and COMs, the switch applies a hold-back mechanism at the inputs. If a cell $m\times n_2$8 is sent to a VOMQ with occupancy $m\times n_2$9, then no later cell of flow $C(n,m,r)$0 leaves the VOQ for the next $C(n,m,r)$1 time slots. Practically, each input keeps an input port counter per reachable VOMQ and a hold-down timer per VOQ; when the current VOMQ occupancy is $C(n,m,r)$2, the timer is set to $C(n,m,r)$3 (Sule et al., 2018). The resulting theorem states that for any two cells $C(n,m,r)$4 and $C(n,m,r)$5 of the same flow with $C(n,m,r)$6, the earlier cell always departs the destination output port before the later one.

Although this switch is analyzed for unidirectional traffic, its building blocks and its rate-1 scheduling are symmetric. This suggests that the same deterministic scheduling and buffering principles can be applied per direction in a bidirectional Clos fabric without changing link rates (Sule et al., 2018).

4. Optical and all-optical interpretations

In optical switching, one-rate bidirectional Clos networks appear in three-stage Clos-ROADMs and folded Clos datacenter networks. The relevant parameters are directional degree $C(n,m,r)$7, fiber degree $C(n,m,r)$8, and middle-stage size $C(n,m,r)$9. A fiber degree is a pair of bidirectional fibers on a directional degree, and a Clos-ROADM is denoted $n_1=n_2=n$0, in contrast to a Spanke-ROADM $n_1=n_2=n$1 (Lin et al., 2023). The ingress and egress stages use arrays of $n_1=n_2=n$2 and $n_1=n_2=n$3 WSSs, the middle stage uses $n_1=n_2=n$4 switches of size $n_1=n_2=n$5, and neighboring stages are fully connected by short-reach fibers.

The nonblocking condition for a WSS-based Clos-ROADM is the spatially strictly non-blocking requirement

$n_1=n_2=n$6

For comparison with a strictly non-blocking Spanke-ROADM, the paper chooses $n_1=n_2=n$7 (Lin et al., 2023). With wavelength continuity, a further wavelength-dimensional requirement is stated: $n_1=n_2=n$8 where $n_1=n_2=n$9 is the total number of wavelengths supported by the Clos network and $r_1=r_2=r$0 is the number of wavelengths per line-side port (Lin et al., 2023). These conditions are optical analogues of classical one-rate Clos dimensioning, with wavelengths acting as unit-capacity circuits.

The paper also derives a blocking benchmark using the Erlang-B formula

$r_1=r_2=r$1

together with

$r_1=r_2=r$2

The underlying assumptions are Erlang traffic on each input-output pair, homogeneous ports, and the same number of wavelengths $r_1=r_2=r$3 per port (Lin et al., 2023). This is explicitly a one-rate regime.

The complexity comparison with Spanke is one of the main quantitative results. For Spanke-ROADM $r_1=r_2=r$4,

$r_1=r_2=r$5

For Clos-ROADM $r_1=r_2=r$6,

$r_1=r_2=r$7

If $r_1=r_2=r$8, the Clos fiber complexity becomes $r_1=r_2=r$9, while the Spanke fiber complexity remains $r_1$00 (Lin et al., 2023). In the case $r_1$01 with 5 wavelengths per fiber and 2 Erlang per fiber degree, the blocking probability of $r_1$02 becomes “very close” to that of $r_1$03 when $r_1$04; the internal fiber count is then 120 for Clos versus 9,000 for Spanke (Lin et al., 2023).

Different middle-stage technologies alter the blocking regime. For the base network $r_1$05, WSS Clos-ROADM matches CDC-ROADM performance, TWC-WSS reaches the theory limit defined by the Erlang benchmark, AWG performs worse because of limited routing flexibility, TWC-AWG provides slight improvement, and TWC-AWG-TWC approaches the theory limit (Lin et al., 2023). In all-optical DCNs, the same three-stage Clos can be folded into a Spine-Leaf architecture, and the middle stage can also be replaced by optical splitters for multicast or by deterministic round-robin direct connections to form a Torus-like topology (Lin et al., 2023). A plausible implication is that one-rate bidirectional Clos theory extends naturally from electrical circuit interpretations to wavelength-level optical switching whenever capacities remain homogeneous.

5. Centralized scheduling and replacement chains in bidirectional DCNs

A recent formulation treats the one-rate bidirectional Clos network as a two-level complete bipartite fabric with low-level switches $r_1$06, top-level switches $r_1$07, and bidirectional links $r_1$08 of integer capacity $r_1$09 (Zhu et al., 16 Jul 2025). A connection between low-level switches $r_1$10 and $r_1$11 via $r_1$12 is $r_1$13, and the demand matrix $r_1$14 is symmetric with $r_1$15 and $r_1$16. For a new routing scheme $r_1$17, feasibility requires

$r_1$18

and

$r_1$19

The demand itself is feasible iff

$r_1$20

The associated optimization objective is minimal rearrangement. Given an original routing $r_1$21, the number of rearrangements needed to obtain $r_1$22 is

$r_1$23

FastReChain attacks this problem through replacement chains (Zhu et al., 16 Jul 2025). A connection is redundant when

$r_1$24

and a link is explicitly available if

$r_1$25

or implicitly available if it is full but carries some redundant connection. Replacement chains are alternating remove/add sequences that free capacity and end with a final add, while increasing the desired pairwise connectivity by 1 and keeping all other demanded connectivities unchanged (Zhu et al., 16 Jul 2025).

The central theorem is stated for proportional networks. In a two-level proportional one-rate bidirectional Clos network, a valid modification scheme for scheduling a single connection between $r_1$26 and $r_1$27 can be found using replacement chains, plus possibly removing a few redundant connections, iff there exists at least one available link incident to $r_1$28 and at least one available link incident to $r_1$29 (Zhu et al., 16 Jul 2025). The proof reduces the bidirectional proportional network to a three-stage proportional one-rate symmetric Clos network and then to a classical unit-capacity Clos setting, where Paull’s chain method applies. The throughput consequence is that, in proportional one-rate bidirectional Clos networks, FastReChain achieves the theoretical maximum throughput and is effectively rearrangeably non-blocking for one-rate calls (Zhu et al., 16 Jul 2025).

The algorithm’s practical contribution lies in responsiveness. It uses DFS with iterative deepening for chain search, randomizes the enumeration of candidate top-level switches and displaced connections, and accelerates filtering through bitsets. The maintained structures include countLT[j] [i], bitsetLT_j, countLL[j] [k], bitsetLL_j, and bitsetLLT[j] [k], all updated in $r_1$30 time per add or remove (Zhu et al., 16 Jul 2025). The reported speedup relative to a naive implementation is about $r_1$31, and running time becomes “almost independent of the number of network ports or switches” in practice.

The measured dynamic performance is unusually strong. For $r_1$32 and $r_1$33, the per-operation time is about 18 ns at load $r_1$34 with approximately 0.024 rearrangements per operation, 119 ns at $r_1$35 with 0.167 rearrangements, 419 ns at $r_1$36 with 0.694 rearrangements, and 1,237 ns at $r_1$37 with 1.29 rearrangements (Zhu et al., 16 Jul 2025). For static minimal rewiring, FastReChain is reported to run from a fraction to a few hundredths of the time of prior algorithms while producing steadily improved numbers of rearrangements (Zhu et al., 16 Jul 2025). A common misconception is that dynamic centralized scheduling is incompatible with one-rate bidirectional Clos fabrics; these results show that the obstacle is not the rate-1 model itself but the lack of an efficient rearrangement mechanism.

6. Related Clos-type multistage networks: the Benes case

Benes networks occupy a special position in the broader theory because they are recursively constructed Clos-type multistage networks built from $r_1$38 modules. A $r_1$39 Benes network has $r_1$40 columns, $r_1$41 rows, link capacity $r_1$42 packet/slot on every internal link, and $r_1$43 internal modules for $r_1$44 (Huang et al., 2012). As circuit switches, Benes networks are rearrangeably non-blocking, which implies that they are full-throughput as packet switches with suitable routing (Huang et al., 2012).

The packet-level capacity region is

$r_1$45

This is the same doubly sub-stochastic region that characterizes ideal one-rate nonblocking behavior: only input and output sums matter (Huang et al., 2012). The paper proposes a grouped backpressure scheme, G-BP, with end-to-end congestion control, and proves the utility bound

$r_1$46

Its distinctive implementation feature is that only four queues are required per internal module, independently of network size (Huang et al., 2012).

Routing in the Benes packet network exploits a partition between upper-division and lower-division flows and balances traffic across partition nodes. For any flow $r_1$47 and any partition node $r_1$48, the average rate through $r_1$49 is

$r_1$50

From the partition column onward, each destination has a unique path, which allows the second half of the network to operate in deterministic free-flow mode (Huang et al., 2012). The paper also states that it is straightforward to extend the results to include bi-directional traffic flows (Huang et al., 2012).

The Benes case matters for one-rate bidirectional Clos networks for two reasons. First, it shows that a uniform-capacity multistage fabric can achieve full throughput with distributed packet-level control rather than centralized permutation scheduling. Second, it demonstrates that low per-module state—four queues rather than per-destination queueing—can be enough when symmetry and recursive path structure are exploited. This suggests a broader design principle: one-rate bidirectional Clos networks need not choose between nonblocking structure and manageable control-state complexity.

7. Synthesis and design principles

Across circuit, packet, optical, and DCN contexts, one-rate bidirectional Clos networks are unified by three constraints: homogeneous internal and external rates, symmetric bidirectional operation, and a middle-stage resource whose dimensioning or control policy determines blocking, throughput, and rearrangement behavior. Classical circuit theory provides closed-form middle-stage requirements such as $r_1$51 for SNB and $r_1$52 for WSNB in $r_1$53 (Ngo et al., 2012). Packet-switch work shows that deterministic periodic scheduling plus split central buffering can yield $r_1$54 throughput and in-order delivery at rate 1 without speedup (Sule et al., 2018). Optical work shows that Clos structures can preserve near-nonblocking behavior while dramatically reducing element and fiber complexity relative to Spanke architectures, provided that parameters such as $r_1$55, $r_1$56, and wavelength budgets satisfy the appropriate one-rate conditions (Lin et al., 2023). Recent bidirectional DCN work shows that centralized dynamic scheduling becomes practical when the control primitive is a replacement chain rather than a global ILP or MCF computation (Zhu et al., 16 Jul 2025).

A persistent theme is that one-rate does not imply architectural weakness. In the packet-switch setting, the absence of speedup is compensated by deterministic load balancing and buffering (Sule et al., 2018). In the optical setting, homogeneous wavelength capacity supports tractable blocking analysis and scalable ROADM construction (Lin et al., 2023). In the bidirectional DCN setting, unit-rate connections and shared bidirectional capacities enable exact combinatorial rearrangement arguments in proportional networks (Zhu et al., 16 Jul 2025). In Clos-type Benes packet fabrics, unit-capacity links still support the full capacity region under suitable backpressure control (Huang et al., 2012).

The main open boundary, as reflected in these works, is not whether one-rate bidirectional Clos networks can be made high-performance, but under which traffic, technology, and structural assumptions the strongest guarantees hold. Some results are exact and classical, such as the SNB and WSNB bounds for symmetric Clos (Ngo et al., 2012). Some are exact but architecture-specific, such as the $r_1$57 throughput and in-order theorems for LBC under admissible i.i.d. traffic (Sule et al., 2018). Some are exact only in proportional subclasses, such as FastReChain’s full-throughput theorem for proportional one-rate bidirectional Clos networks (Zhu et al., 16 Jul 2025). Others are optical analogues whose validity depends on wavelength continuity, wavelength conversion, or homogeneous Erlang traffic assumptions (Lin et al., 2023). Taken together, these results define one-rate bidirectional Clos networks not as a single architecture, but as a research program centered on how far uniform-rate multistage fabrics can be pushed through structure, scheduling, and combinatorial control.