Expander Routing: Mechanisms and Applications

• Expander routing is a paradigm that uses graph expansion measures (like conductance and edge expansion) to enable low-congestion, short-path routing.
• It encompasses diverse mechanisms such as electrical flows, tree-based hierarchies, and sparse overlays, and is applied in static, dynamic, and distributed network models.
• Its practical impact spans multicommodity routing, robust network design in datacenters, and fault-tolerant algorithms that adapt to length constraints and edge deletions.

Expander routing is a family of routing paradigms that exploit graph expansion—expressed through edge expansion, conductance, spectral gap, or related linkedness notions—to route many point-to-point demands with low congestion over short paths. In the literature, the term refers both to routing on an expander and to algorithmic frameworks that reduce general routing problems to expander-like substructures. It appears in at least five distinct settings: multicommodity routing on static expanders, oblivious and dynamic routing in robust network design, distributed routing primitives in the $\mathsf{CONGEST}$ model, practical routing over expander-based datacenter fabrics, and length-constrained or deletion-resilient routing structures (Haeupler et al., 17 Jul 2025, Chang et al., 2020, Goyal et al., 2013, Haeupler et al., 2024).

1. Formal models and core notions

A standard starting point is the expander condition itself. For an undirected graph $G=(V,E)$, one common formulation uses conductance:

$$|\partial S| \;\ge\; \Phi \cdot \mathrm{vol}(S)\quad\text{for all }S\subseteq V\text{ with }\mathrm{vol}(S)\le \mathrm{vol}(V)/2,$$

where $\mathrm{vol}(S)$ is the sum of weighted degrees in $S$ and $\partial S$ is the boundary of $S$. Another formulation uses edge expansion

$$h(G) \;=\; \min_{\emptyset \ne S \subset V,\, |S| \le |V|/2} \frac{|E(S,V\setminus S)|}{|S|}.$$

These equivalent perspectives support different routing analyses: conductance is natural for electrical, spectral, and distributed arguments, while edge expansion is prominent in expander constructions and lower bounds (Florescu et al., 2024, Goyal et al., 2013).

The demand model also varies by context. In routing on expanders as combinatorial objects, a demand is often required to be unit or degree-respecting:

$$\sum_{u \in V} D(u,v) \le \deg(v)\quad\text{and}\quad \sum_{v \in V} D(u,v) \le \deg(u).$$

In this regime, a $\phi$-expander is a graph on which every unit demand can be routed with congestion $G=(V,E)$0 over paths of length at most $G=(V,E)$1. In distributed computing, the analogous promise is that each vertex is the source and destination of at most $G=(V,E)$2 tokens, or more generally $G=(V,E)$3 tokens, and the objective is to deliver them rapidly under $G=(V,E)$4 bandwidth limits (Haeupler et al., 17 Jul 2025, Chang et al., 2020).

A different formalization arises in robust network design. There, one is given a network $G=(V,E)$5 with edge costs $G=(V,E)$6, capacity variables $G=(V,E)$7, and a demand polytope $G=(V,E)$8. The goal is to minimize $G=(V,E)$9 subject to supporting every demand matrix $$|\partial S| \;\ge\; \Phi \cdot \mathrm{vol}(S)\quad\text{for all }S\subseteq V\text{ with }\mathrm{vol}(S)\le \mathrm{vol}(V)/2,$$0. The critical distinction is between dynamic routing, where the flow $$|\partial S| \;\ge\; \Phi \cdot \mathrm{vol}(S)\quad\text{for all }S\subseteq V\text{ with }\mathrm{vol}(S)\le \mathrm{vol}(V)/2,$$1 may depend arbitrarily on the realized demand $$|\partial S| \;\ge\; \Phi \cdot \mathrm{vol}(S)\quad\text{for all }S\subseteq V\text{ with }\mathrm{vol}(S)\le \mathrm{vol}(V)/2,$$2, and oblivious routing templates, where a unit flow $$|\partial S| \;\ge\; \Phi \cdot \mathrm{vol}(S)\quad\text{for all }S\subseteq V\text{ with }\mathrm{vol}(S)\le \mathrm{vol}(V)/2,$$3 or a single path $$|\partial S| \;\ge\; \Phi \cdot \mathrm{vol}(S)\quad\text{for all }S\subseteq V\text{ with }\mathrm{vol}(S)\le \mathrm{vol}(V)/2,$$4 is fixed in advance for each pair. In the oblivious fractional model,

$$|\partial S| \;\ge\; \Phi \cdot \mathrm{vol}(S)\quad\text{for all }S\subseteq V\text{ with }\mathrm{vol}(S)\le \mathrm{vol}(V)/2,$$5

whereas the dynamic model is a semi-infinite LP over all $$|\partial S| \;\ge\; \Phi \cdot \mathrm{vol}(S)\quad\text{for all }S\subseteq V\text{ with }\mathrm{vol}(S)\le \mathrm{vol}(V)/2,$$6 and corresponding feasible multicommodity flows (Goyal et al., 2013).

These definitions already indicate that “expander routing” is not a single algorithm. It names a class of guarantees whose exact meaning depends on whether the focus is congestion minimization, fixed-template robustness, distributed token routing, or bounded-length multicommodity flow. This suggests that the unifying object is not the routing rule itself but the way expansion suppresses bottlenecks across cuts.

2. Routing mechanisms: electrical, tree-based, spectral, and sparse-overlay constructions

One prominent oblivious mechanism is electrical routing. For a connected undirected graph with incidence matrix $$|\partial S| \;\ge\; \Phi \cdot \mathrm{vol}(S)\quad\text{for all }S\subseteq V\text{ with }\mathrm{vol}(S)\le \mathrm{vol}(V)/2,$$7, diagonal conductance matrix $$|\partial S| \;\ge\; \Phi \cdot \mathrm{vol}(S)\quad\text{for all }S\subseteq V\text{ with }\mathrm{vol}(S)\le \mathrm{vol}(V)/2,$$8, and Laplacian

$$|\partial S| \;\ge\; \Phi \cdot \mathrm{vol}(S)\quad\text{for all }S\subseteq V\text{ with }\mathrm{vol}(S)\le \mathrm{vol}(V)/2,$$9

electrical routing sends a demand vector $\mathrm{vol}(S)$0 by solving

$\mathrm{vol}(S)$1

This defines a fixed linear oblivious routing operator

$\mathrm{vol}(S)$2

For a $\mathrm{vol}(S)$3-expander with $\mathrm{vol}(S)$4 edges, the 2024 analysis proves that electrical routing is $\mathrm{vol}(S)$5-competitive in the $\mathrm{vol}(S)$6- and $\mathrm{vol}(S)$7-norms, with the explicit bound

$\mathrm{vol}(S)$8

It further gives $\mathrm{vol}(S)$9 unconditionally, interpolation bounds for all $S$0, and matching lower bounds

$S$1

In $S$2 and $S$3, the bound is unconditionally tight up to constants (Florescu et al., 2024).

A second mechanism represents routing structure by a tree. The expander hierarchy is a tree representation built from boundary-linked expander decompositions. If $S$4 is an $S$5-expander hierarchy of depth $S$6 and slack $S$7, then $S$8 is a tree flow sparsifier for $S$9 with quality

$\partial S$0

Static and fully dynamic instantiations yield $\partial S$1, and the hierarchy supports $\partial S$2-approximate conductance, $\partial S$3-$\partial S$4 maximum flows, and $\partial S$5-$\partial S$6 minimum cuts with subpolynomial update time and $\partial S$7 query time. In this picture, expander routing becomes routing on a dynamically maintained tree flow sparsifier whose capacities encode cut structure across contracted boundary-linked expanders (Goranci et al., 2020).

A third mechanism builds sparse overlays that preserve expansion approximately. A $\partial S$8-splicer is the union of $\partial S$9 independent uniformly random spanning trees. For bounded-degree $S$0-regular graphs, if $S$1 and $S$2, then with probability $S$3,

$S$4

For two trees, the overlay has exactly $S$5 edges. On $S$6, the union of two random spanning trees has constant vertex expansion with probability $S$7; for $S$8 with $S$9, two trees again form an expander with probability $$h(G) \;=\; \min_{\emptyset \ne S \subset V,\, |S| \le |V|/2} \frac{|E(S,V\setminus S)|}{|S|}.$$0. The paper explicitly frames this as enabling classical expander routing methods such as Valiant’s scheme and Leighton–Maggs–Rao on an $$h(G) \;=\; \min_{\emptyset \ne S \subset V,\, |S| \le |V|/2} \frac{|E(S,V\setminus S)|}{|S|}.$$1-edge overlay (0807.1496).

Spectral regularization extends these ideas to irregular graphs. The Perron eigenvector $$h(G) \;=\; \min_{\emptyset \ne S \subset V,\, |S| \le |V|/2} \frac{|E(S,V\setminus S)|}{|S|}.$$2 of the adjacency matrix makes the weighted degree

$$h(G) \;=\; \min_{\emptyset \ne S \subset V,\, |S| \le |V|/2} \frac{|E(S,V\setminus S)|}{|S|}.$$3

constant across vertices, which “regularizes” the graph in a precise sense. The Perron-weighted Expander Mixing Lemma gives, for all $$h(G) \;=\; \min_{\emptyset \ne S \subset V,\, |S| \le |V|/2} \frac{|E(S,V\setminus S)|}{|S|}.$$4,

$$h(G) \;=\; \min_{\emptyset \ne S \subset V,\, |S| \le |V|/2} \frac{|E(S,V\setminus S)|}{|S|}.$$5

where $$h(G) \;=\; \min_{\emptyset \ne S \subset V,\, |S| \le |V|/2} \frac{|E(S,V\setminus S)|}{|S|}.$$6. Applied to routing via matchings, this yields a dense-graph routing-number bound

$$h(G) \;=\; \min_{\emptyset \ne S \subset V,\, |S| \le |V|/2} \frac{|E(S,V\setminus S)|}{|S|}.$$7

whenever $$h(G) \;=\; \min_{\emptyset \ne S \subset V,\, |S| \le |V|/2} \frac{|E(S,V\setminus S)|}{|S|}.$$8 and $$h(G) \;=\; \min_{\emptyset \ne S \subset V,\, |S| \le |V|/2} \frac{|E(S,V\setminus S)|}{|S|}.$$9 (Abiad et al., 2024).

Across these mechanisms, the common theme is that expansion can be encoded algebraically, combinatorially, or hierarchically. Electrical flows exploit voltage-threshold cuts; tree sparsifiers encode recursive linkedness; splicers preserve cuts in a sparse overlay; Perron weighting regularizes irregular structure. The resulting routing algorithms differ substantially, but their guarantees are driven by the same anti-bottleneck phenomenon.

3. Robust network design and the expander separation between dynamic and oblivious routing

In robust network design, expanders yield a sharp separation between routing models. The canonical instance starts from a constant-degree edge-expander $$\sum_{u \in V} D(u,v) \le \deg(v)\quad\text{and}\quad \sum_{v \in V} D(u,v) \le \deg(u).$$0 with $$\sum_{u \in V} D(u,v) \le \deg(v)\quad\text{and}\quad \sum_{v \in V} D(u,v) \le \deg(u).$$1, adds a special sink $$\sum_{u \in V} D(u,v) \le \deg(v)\quad\text{and}\quad \sum_{v \in V} D(u,v) \le \deg(u).$$2, and connects every $$\sum_{u \in V} D(u,v) \le \deg(v)\quad\text{and}\quad \sum_{v \in V} D(u,v) \le \deg(u).$$3 to $$\sum_{u \in V} D(u,v) \le \deg(v)\quad\text{and}\quad \sum_{v \in V} D(u,v) \le \deg(u).$$4 by a port edge. The resulting graph $$\sum_{u \in V} D(u,v) \le \deg(v)\quad\text{and}\quad \sum_{v \in V} D(u,v) \le \deg(u).$$5 carries costs $$\sum_{u \in V} D(u,v) \le \deg(v)\quad\text{and}\quad \sum_{v \in V} D(u,v) \le \deg(u).$$6 on expander edges and $$\sum_{u \in V} D(u,v) \le \deg(v)\quad\text{and}\quad \sum_{v \in V} D(u,v) \le \deg(u).$$7 on ports. The demand polytope is a single-sink hose model with

$$\sum_{u \in V} D(u,v) \le \deg(v)\quad\text{and}\quad \sum_{v \in V} D(u,v) \le \deg(u).$$8

and only demands incident to $$\sum_{u \in V} D(u,v) \le \deg(v)\quad\text{and}\quad \sum_{v \in V} D(u,v) \le \deg(u).$$9. For $\phi$0, the main theorem gives

$\phi$1

and therefore, for infinitely many $\phi$2,

$\phi$3

for an absolute constant $\phi$4. The separation holds for both fractional multipath and integral single-path oblivious routing, and already in the asymmetric hose model (Goyal et al., 2013).

The dynamic upper bound is obtained by installing capacities

$\phi$5

The total cost is

$\phi$6

Feasibility is proved by a max-flow/min-cut argument in an augmented graph $\phi$7 with a super-source linked to an arbitrary active set $\phi$8 of size $\phi$9. For every cut $G=(V,E)$00,

$G=(V,E)$01

and edge expansion guarantees that every $G=(V,E)$02–$G=(V,E)$03 cut has capacity at least $G=(V,E)$04.

The oblivious lower bounds use different reductions for the fractional and integral cases. In the fractional case, if $G=(V,E)$05, then the capacity required on an edge is the sum of the $G=(V,E)$06 largest template values:

$G=(V,E)$07

which is exactly the buy-and-rent single-sink problem with parameter $G=(V,E)$08. The analysis partitions each terminal’s unit flow into a rented-port part $G=(V,E)$09, a locally bought-port part $G=(V,E)$10, and a far-traveling part $G=(V,E)$11, then sums rental costs over nested expanding shells $G=(V,E)$12 to obtain

$G=(V,E)$13

for $G=(V,E)$14. In the integral case, oblivious routing reduces to single-sink rent-or-buy with parameter $G=(V,E)$15, and bounded degree forces either $G=(V,E)$16 path lengths or heavy-edge bottlenecks, again giving $G=(V,E)$17.

The significance of this construction is negative and precise. It shows that even on a graph whose expansion makes dynamic routing easy, fixed templates can be inherently too rigid. In the single-sink hose model, oblivious routing cannot anticipate which $G=(V,E)$18 terminals will be active, and the expander geometry converts this uncertainty into either repeated cut-crossings or expensive port usage. The paper also states that the $G=(V,E)$19 gap is tight up to constant factors for general instances.

4. Distributed expander routing and expander decompositions

In distributed graph algorithms, expander routing is a communication primitive in the $G=(V,E)$20 model. The basic input is a $G=(V,E)$21-expander $G=(V,E)$22 together with a multiset of $G=(V,E)$23-bit messages. In the classical formulation, each vertex $G=(V,E)$24 is the source and destination of at most $G=(V,E)$25 messages; the deterministic generalization allows $G=(V,E)$26 messages and also permits destination ranges rather than fixed destination identifiers. The objective is to route all messages in a bounded number of rounds, with each edge carrying only $G=(V,E)$27 bits per round (Chang et al., 2020).

The 2020 deterministic framework gives two central guarantees. First, an $G=(V,E)$28-expander decomposition can be deterministically computed in $G=(V,E)$29 rounds for $G=(V,E)$30, and randomized polylogarithmic-round decomposition is also obtained for $G=(V,E)$31. Second, deterministic expander routing on a bounded-degree expander with conductance $G=(V,E)$32 and per-node load parameter $G=(V,E)$33 can be performed in

$G=(V,E)$34

rounds, with final per-node load

$G=(V,E)$35

General graphs are reduced to bounded-degree expanders through the expander split graph $G=(V,E)$36, which preserves conductance up to $G=(V,E)$37 factors (Chang et al., 2020).

The 2024 deterministic improvement matches the randomized single-instance round complexity of Ghaffari–Kuhn–Su for a single query and restores preprocessing/query tradeoffs. A routing instance in a $G=(V,E)$38-expander can be solved deterministically in

$G=(V,E)$39

rounds. For every $G=(V,E)$40, there is a deterministic preprocessing procedure taking

$G=(V,E)$41

rounds, after which every routing query is answered in

$G=(V,E)$42

rounds. The technical ingredients are a one-shot hierarchical decomposition, deterministic shufflers derived from a cut–matching game on a cluster graph, and a deterministic equivalence between expander routing and expander sorting (Chang et al., 2024).

Expander routing is also a core subroutine in decomposition-based algorithms on non-expanders. The 2019 decomposition work constructs an $G=(V,E)$43-expander decomposition with

$G=(V,E)$44

in

$G=(V,E)$45

rounds, and then uses a slight modification of the Ghaffari–Kuhn–Su routing framework to obtain triangle enumeration in $G=(V,E)$46 rounds, matching the known lower bound up to polylogarithmic factors (Chang et al., 2019).

A common misconception is that expander routing in distributed settings is synonymous with random-walk routing. The deterministic line of work shows otherwise. Random walks motivated the original algorithms, but the later deterministic constructions replace them with expander embeddings, matching embeddings, deterministic load balancing, shufflers, and sorting networks, while preserving nearly the same dependence on conductance.

5. Practical network architectures: datacenter expanders and dynamic expander snapshots

In datacenter systems, expander routing has been studied as a practical alternative to Clos fabrics. The 2018 experimental study evaluates expander-based topologies using only traditional protocols: standard link-state shortest-path forwarding, ECMP, and TCP. It reports that expanders can realize 3x more throughput than an equivalent fat tree and 1.5x more throughput than an equivalent leaf-spine topology, while also achieving lower flow completion times, lower queue occupancy under bursty conditions, and graceful degradation with increasing load. For network incast and outcast, the paper reports >2x median FCT improvement and shows that expander queues build and drain faster under the same droptail and TCP settings (Harsh et al., 2018).

The routing mechanisms in that evaluation remain deliberately conventional. The baseline is shortest-path routing plus ECMP with per-flow hashing. The paper also studies $G=(V,E)$47-shortest paths and $G=(V,E)$48 edge-disjoint paths, implemented by a segment-routing construction based on an expressible path: a path $G=(V,E)$49 from $G=(V,E)$50 to $G=(V,E)$51 is expressible if there exists an intermediate hop $G=(V,E)$52 such that $G=(V,E)$53 and $G=(V,E)$54 are both shortest paths. It further analyzes Valiant Load Balancing for comparison with earlier expander proposals. The operational interpretation is that expanders need not rely on exotic transport or routing machinery to obtain measurable performance gains (Harsh et al., 2018).

The same work gives an analytic explanation through the Uplink-to-Downlink Factor:

$G=(V,E)$55

For a fat tree of degree $G=(V,E)$56,

$G=(V,E)$57

and for a leaf–spine topology with oversubscription $G=(V,E)$58,

$G=(V,E)$59

This quantifies the flatness advantage of expander rewiring: with the same equipment, an expander fabric can expose more effective rack uplink capacity.

Opera extends the idea to a time-varying network. At any instant, every circuit switch realizes a predetermined random matching, and the inter-ToR fabric is the union of $G=(V,E)$60 active matchings. For $G=(V,E)$61, the union of $G=(V,E)$62 random matchings is an expander with high probability; even with one switch reconfiguring, $G=(V,E)$63 suffices to preserve expander properties with high probability. Across time, however, the schedule factors the complete graph into disjoint matchings so that every rack pair eventually sees a direct one-hop circuit. Opera therefore routes low-latency traffic over an expander snapshot and bulk traffic over time-varying direct circuits (Mellette et al., 2019).

The implementation details are explicit. Opera uses topology slices of duration

$G=(V,E)$64

where $G=(V,E)$65 is the reconfiguration delay and $G=(V,E)$66 is the worst-case traversal time for a low-latency packet. In the example with $G=(V,E)$67 and $G=(V,E)$68, using shallow queues and 10 Gb/s links, the paper gives $G=(V,E)$69, worst-case ToR-to-ToR path length $G=(V,E)$70, duty cycle $G=(V,E)$71, and overall cycle time $G=(V,E)$72 ms. In a 648-host example, Opera delivers up to 4x the bandwidth for all-to-all shuffle and supports 60% higher load for published datacenter workloads than cost-equivalent static designs (Mellette et al., 2019).

Taken together, these system papers broaden the meaning of expander routing. In theory papers, expansion typically certifies cut robustness or congestion bounds. In these network designs, the same property is used as an engineering substrate: shortest-path ECMP and priority queuing operate over a topology whose path diversity and mixing characteristics are inherited from expansion, while time variation can be layered on top without discarding the expander viewpoint.

6. Length-constrained, deletion-resilient, and self-pruning variants

Classical expander routing is insensitive to path-length budgets except through polylogarithmic bounds. Length-constrained expander decompositions refine this by requiring routes to stay close to the original metric. An $G=(V,E)$73-length $G=(V,E)$74-expander decomposition is a collection of edge-length increases such that nodes within distance $G=(V,E)$75 can route flow over paths of length $G=(V,E)$76 with congestion at most $G=(V,E)$77. The 2024 framework strengthens this to a witnessed $G=(V,E)$78-length guarantee: after modifying lengths, any demand between pairs at distance $G=(V,E)$79 is routed along paths of length at most $G=(V,E)$80, with congestion $G=(V,E)$81. Its main algorithm computes such a decomposition with cut slack

$G=(V,E)$82

and length slack

$G=(V,E)$83

in close-to-linear work, while also providing sparse length-bounded flow routines whose support is

$G=(V,E)$84

independent of $G=(V,E)$85 (Haeupler et al., 2024).

A different line treats routing and pruning simultaneously. The 2025 paper introduces semi-hypercubes, a recursively defined family generalizing hypercubes and designed to remain easy to route in after adversarial edge deletions. In a noncritical $G=(V,E)$86-semi-hypercube, there exists an oblivious routing of length $G=(V,E)$87 that routes any unit demand with congestion $G=(V,E)$88, and a path can be sampled in expected time $G=(V,E)$89. In a general $G=(V,E)$90-semi-hypercube with $G=(V,E)$91 and $G=(V,E)$92, there exists an oblivious routing of length $G=(V,E)$93 that routes any unit demand with congestion

$G=(V,E)$94

and a path can be sampled in expected time $G=(V,E)$95. The same work gives a deterministic dynamic routing algorithm with congestion

$G=(V,E)$96

and path length

$G=(V,E)$97

for integral $G=(V,E)$98-load demands (Haeupler et al., 17 Jul 2025).

The structural novelty is self-pruning. Under online adversarial edge deletions, a deterministic algorithm prunes a small number of vertices so that the remaining graph stays within the semi-hypercube family. For $G=(V,E)$99 and $$|\partial S| \;\ge\; \Phi \cdot \mathrm{vol}(S)\quad\text{for all }S\subseteq V\text{ with }\mathrm{vol}(S)\le \mathrm{vol}(V)/2,$$00, every deletion causes worst-case pruning ratio

$$|\partial S| \;\ge\; \Phi \cdot \mathrm{vol}(S)\quad\text{for all }S\subseteq V\text{ with }\mathrm{vol}(S)\le \mathrm{vol}(V)/2,$$01

Through embeddings, these guarantees transfer to general graphs: if a graph initially contains an embedded router of congestion $$|\partial S| \;\ge\; \Phi \cdot \mathrm{vol}(S)\quad\text{for all }S\subseteq V\text{ with }\mathrm{vol}(S)\le \mathrm{vol}(V)/2,$$02 and path length $$|\partial S| \;\ge\; \Phi \cdot \mathrm{vol}(S)\quad\text{for all }S\subseteq V\text{ with }\mathrm{vol}(S)\le \mathrm{vol}(V)/2,$$03, then after pruning one maintains a subgraph that remains a $$|\partial S| \;\ge\; \Phi \cdot \mathrm{vol}(S)\quad\text{for all }S\subseteq V\text{ with }\mathrm{vol}(S)\le \mathrm{vol}(V)/2,$$04-congestion router for $$|\partial S| \;\ge\; \Phi \cdot \mathrm{vol}(S)\quad\text{for all }S\subseteq V\text{ with }\mathrm{vol}(S)\le \mathrm{vol}(V)/2,$$05-load demands, with path length $$|\partial S| \;\ge\; \Phi \cdot \mathrm{vol}(S)\quad\text{for all }S\subseteq V\text{ with }\mathrm{vol}(S)\le \mathrm{vol}(V)/2,$$06 in the oblivious case or $$|\partial S| \;\ge\; \Phi \cdot \mathrm{vol}(S)\quad\text{for all }S\subseteq V\text{ with }\mathrm{vol}(S)\le \mathrm{vol}(V)/2,$$07 in the explicit deterministic case (Haeupler et al., 17 Jul 2025).

These results point to a broader evolution of expander routing. The original guarantee—route degree-respecting demands with low congestion over short paths—has been refined along two axes. One axis is geometric, replacing unrestricted short paths by explicit length-competitive paths. The other is dynamic, replacing static expansion by routing structures that survive deletions, pruning, or online maintenance. A plausible implication is that future uses of expansion in routing will increasingly combine these constraints rather than treating them separately.