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Multi-Beam Multi-Hop Routing for IntelligentReflecting Surfaces Aided Massive MIMO

Weidong Mei and Rui Zhang, Fellow, IEEE

Abstract—Intelligent reflecting surface (IRS) is envisioned toplay a significant role in future wireless communication systemsas an effective means of reconfiguring the radio signal propa-gation environment. In this paper, we study a new multi-IRSaided massive multiple-input multiple-output (MIMO) system,where a multi-antenna BS transmits independent messages toa set of remote single-antenna users using orthogonal beamsthat are subsequently reflected by different groups of IRSs viatheir respective multi-hop passive beamforming over pairwiseline-of-sight (LoS) links. We aim to select optimal IRSs andtheir beam routing path for each of the users, along withthe active/passive beamforming at the BS/IRSs, such that theminimum received signal power among all users is maximized.This problem is particularly difficult to solve due to a new typeof path separation constraints for avoiding the IRS-reflectedsignal induced interference among different users. To tacklethis difficulty, we first derive the optimal BS/IRS active/passivebeamforming solutions based on their practical codebooks giventhe reflection paths. Then we show that the resultant multi-beam multi-hop routing problem can be recast as an equivalentgraph-optimization problem, which is however NP-complete. Tosolve this challenging problem, we propose an efficient recursivealgorithm to partially enumerate the feasible routing solutions,which is able to effectively balance the performance-complexitytrade-off. Numerical results demonstrate that the proposed al-gorithm achieves near-optimal performance with low complexityand outperforms other benchmark schemes. Useful insights intothe optimal multi-beam multi-hop routing design are also drawnunder different setups of the multi-IRS aided massive MIMOnetwork.

Index Terms—Intelligent reflecting surface, massive MIMO,passive beamforming, multi-beam multi-hop routing, graph the-ory.

I. INTRODUCTION

Wireless communication systems in the last decade haveundergone a remarkable progress with various advanced tech-nologies successfully implemented, such as adaptive modula-tion and coding, dynamic resource allocation, hybrid digitaland analog beamforming, etc., which significantly enhancedtheir throughput and efficiency. However, existing wirelesstechnologies were designed mainly to adapt to or compensatethe random and time-varying wireless channels only, but havevery limited control over them, thus leaving an ultimate barrieruncleared in achieving ultra-reliable and ultra-high-capacitywireless systems in the future. Recently, intelligent reflectingsurface (IRS) has emerged as an appealing solution to tackle

Part of this work has been presented in IEEE International Conference onCommunications, Montreal, Canada, 2021 [1].

The authors are with the Department of Electrical and Computer Engineer-ing, National University of Singapore, Singapore 117583 (e-mails: wmei,elezhang@nus.edu.sg).

this issue. By dynamically tuning its large number of reflectingelements (or so-called passive beamforming), IRS is able to“reconfigure” wireless channels and refine their realizationsand/or distributions [2]–[4], rather than adapting to them onlyin the traditional approach. In addition, IRS elements do notrequire transmit or receive radio frequency (RF) chains asthey simply reflect the incident signal as a passive array, thusdrastically reducing the hardware cost and energy consumptionas compared to traditional active transceivers and relays. Thus,by efficiently integrating IRSs into future wireless networks, aquantum-leap improvement in capacity and energy efficiencyis anticipated over today’s wireless systems.

Due to the great potential of IRS, its performance hasbeen recently studied in the literature under different wirelesssystem setups, such as IRS-aided multi-antenna/multiple-inputmultiple-output (MIMO) system [5], [6], massive MIMOsystem [7], [8], orthogonal frequency division multiplexing(OFDM) system [9], [10], non-orthogonal multiple access(NOMA) system [11], [12], multi-cell network [13], [14], si-multaneous wireless information and power transfer [15], [16],mobile edge computing [17], [18], physical-layer security [19],[20], unmanned aerial vehicle (UAV) communication [21],[22], and so on. However, all of these works consider one ormultiple distributed IRSs, which assist in the wireless commu-nication between the base station (BS) and users with only onesingle signal reflection by each IRS. This simplified approach,however, generally results in suboptimal performance. Thisis because by properly deploying IRSs, strong line-of-sight(LoS) channels can be achieved for inter-IRS links, which canprovide more pronounced cooperative passive beamforming(CPB) gains over the conventional single-IRS assisted system.In addition, leveraging the multiple signal reflections by IRSsprovides a higher path diversity to bypass the dense obstaclesin a complex environment and thereby establish a blockage-free end-to-end link between two communication nodes, whichgenerally has a stronger strength compared to other randomlyscattered links between them that suffer multi-path fading.

Inspired by the above, the authors in [23] first proposeda double-IRS system, where a single-antenna BS serves asingle-antenna user through a double-reflection link with twocooperative IRSs deployed near the BS and user, respectively.It was shown in [23] that this system provides a CPB gain thatincreases quartically with the total number of IRS reflectingelements, thus is significantly higher than the quadratic growthof the passive beamforming gain in the conventional single-IRS link. The authors in [24]–[27] further extended [23] toaddress the more practical Rician fading channel and multi-antenna/multi-user setups. Specifically, the authors in [24]

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…

…

BS

User 1

User 2

User k

User K

Used active beam Used passive beam

Unused beams

…

…

IRS iIRS j

x

zy

IRS back side

Reflection half-space IRS element

x

z

y⨂

Fig. 1. A multi-IRS aided massive MIMO system with the MBMH routingvia joint BS/IRS active/passive beamforming.

proposed two different channel estimation schemes for thedouble-IRS aided single-user system under arbitrary and LoS-dominant inter-IRS channels, respectively. In [25], the authorsstudied the channel estimation problem in a more challengingdouble-IRS aided multi-user MIMO system with coexistingsingle- and double-reflection links. Furthermore, the passivebeamforming optimization for the two IRSs under this systemwas studied in [26]. Finally, the authors in [27] considered asecure double-IRS aided system and optimized the two IRSs’passive beamforming to maximize the secrecy rate. Despiteof the above recent works, the general multi-IRS aided multi-user communication system with multi-hop (i.e., more thantwo hops) signal reflections has not been investigated in theliterature yet. Under this general setup with more availableIRSs in the network, different end-to-end LoS paths can beachieved between the BS and multiple remote users at the sametime via multi-hop signal reflections by different groups ofIRSs selected. This thus gives rise to a new cooperative multi-beam multi-hop (MBMH) routing design problem, where theselected IRSs and their beam-routing paths for different usersare jointly optimized with the active/passive beamforming atthe BS/IRSs to maximize the received signal power at all users.

In this paper, we study this new MBMH routing problemfor the downlink communication in a massive MIMO system,where a BS equipped with a large number of active antennastransmits independent messages to a set of remote single-antenna users simultaneously over the same frequency band,aided by multiple distributed IRSs as shown in Fig. 1. In [28],by considering only a single user in this system, we havederived the optimal single-beam multi-hop routing solution.However, different from [28], a new challenge arises in ourconsidered MBMH routing design in this paper, which is toavoid the inter-user/path interference due to undesired scatter-ing by the IRSs that serve for different users/paths, especiallywhen there exist LoS channels between them. This thus leadsto a new type of path separation constraints among differentusers, where the IRSs selected for different users/paths shouldavoid having LoS channels with each other. This stringentconstraint thus makes the MBMH routing problem in thispaper more challenging to solve, as compared to its single-beam special case in [28] without the inter-user/path inter-ference considered. Moreover, unlike [1] and [28] where the

continuous active/passive beamforming is assumed for the easeof exposition, in this paper we consider the more practicaldesign based on beamforming codebook, which consists ofonly a finite number active/passive beamforming directions atthe BS/IRSs, as shown in Fig. 1. This helps reduce the com-plexity of optimal beamforming design as well as hardwarecost in general, especially for frequency division duplex (FDD)systems.

To solve the proposed MBMH routing problem in a multi-IRS aided massive MIMO network, we first derive the op-timal BS/IRS active/passive beamforming solution in theirrespective codebooks for given beam-routing paths of theusers, by exploiting the high angular resolution of the massiveMIMO BS and the inter-IRS LoS channels, respectively. Next,we show that the resultant MBMH routing problem is NP-complete by recasting it into an equivalent neighbor-disjointpath optimization problem in graph theory. To deal with thischallenging problem, a recursive algorithm is proposed topartially enumerate the feasible MBMH routing solutions.By tuning its parameter, the proposed algorithm can strikea flexible balance between performance and complexity. Itis also shown that in the special case of continuous passivebeamforming at the IRSs, the MBMH routing problem can besolved in a more efficient manner by the proposed algorithm.Numerical results show that our proposed algorithm can findthe near-optimal MBMH routing solution with low computa-tional complexity and outperforms other benchmark schemes.It is also revealed that the optimal MBMH routing solutionvaries considerably with the number of reflecting elements aswell as the size of passive beamforming codebook at each IRS.

The rest of this paper is organized as follows. Section IIpresents the system model. Section III presents the optimalBS/IRS active/passive beamforming design and the problemformulation for our considered MBMH routing optimization.Section IV presents the proposed solution to this problembased on graph theory. Section V presents the simulationresults to show the performance of the proposed scheme ascompared to other benchmark schemes. Finally, Section VIIconcludes this paper and discusses future work.

The following notations are used in this paper. Bold symbolsin capital letter and small letter denote matrices and vectors,respectively. The conjugate, transpose and conjugate transposeof a vector or matrix are denoted as (·)∗, (·)T and (·)H ,respectively. Rn (Cn) denotes the set of real (complex) vectorsof length n. For a complex number s, s∗ and |s| denote itsconjugate and amplitude, respectively. For a vector a ∈ Cn,diag(a) denotes an n × n diagonal matrix whose entriesare given by the elements of a; while for a square matrixA ∈ Cn×n, diag(A) denotes an n×1 vector that contains then diagonal elements of A. ‖a‖ denotes the Euclidean normof the vector a. b·c denotes the greatest integer less than orequal to its argument. |A| denotes the cardinality of a set A. jdenotes the imaginary unit, i.e., j2 = −1. For two sets A andB, A∪B denotes the union of A and B. ∅ denotes an emptyset. and ⊗ denote the Hadamard product and Kroneckerproduct, respectively. O(·) denotes the order of complexity.For ease of reference, the main symbols used in this paper arelisted in Table I.

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TABLE ILIST OF MAIN SYMBOLS

Symbol Description Symbol DescriptionJ Number of IRSs K Number of usersNB Number of BS antennas M Number of IRS reflecting elementsWB BS beamforming codebook, |WB | = NB K Set of usersJ Set of IRSs Φj Reflection coefficient matrix of IRS jθj Passive beamforming vector of IRS j WI IRS beamforming codebookD Number of beam patterns in WI , D = |WI | b Number of controlling bits for WI , b = log2DH0,j Channel from the BS to IRS j Si,j Channel from IRS i to IRS j

gHj,J+k Channel from IRS j to user k dA/dI Antenna/element spacing at BS/IRS

M1/M2Number of IRS elements in horizontal/verticaldirection di,j Distance between nodes i and j

d0 Minimum distance for far-field propagation li,j LoS condition indicator between nodes i and je(φ,N) Steering vector function λ Carrier wavelengthaB Array response at the BS aI Array response at each IRSϑ0,j AoD from the BS to IRS j ϕa

j,i/ϕej,i Azimuth/elevation AoA at IRS j from node i

ϑai,j /ϑe

i,j Azimuth/elevation AoD from IRS i to node j β LoS path gain at 1 meterΩ(k) Reflection path from the BS to user k Nk Number of IRSs in Ω(k)

a(k)n Index of the n-th IRS in Ω(k) Nk Set of IRSs in Ω(k)

h0,J+k(Ω(k)) BS-user k effective channel κ(Ω(k)) End-to-end path gain between the BS and user k

wk BS active beamforming for serving user k W(1)I /W(1)

I

IRS codebook for horizontal/vertical passivebeamforming

b1/b2 Number of controlling bits for W(1)I /W(1)

I Q Number of candidate shortest paths for each userGp Constructed path graph Ωr Set of all cliques of size r in Gp

II. SYSTEM MODEL

As shown in Fig. 1, we consider a massive MIMO downlinksystem, where J distributed IRSs are deployed to assist inthe communications from a multi-antenna BS to K remotesingle-antenna users. Assume that the BS is equipped withNB K active antennas, while each IRS is equipped withM passive reflecting elements. Without loss of generality andfor ease of practical implementation, we assume that the BSserves the K users by selecting K beams from a predefinedcodebook, denoted as WB , which consists of NB orthogonaland unit-power beams, where NB can be arbitrarily large inmassive MIMO. For the purpose of exposition, we considerthe challenging scenario where the BS-user direct links areseverely blocked for all the K users considered in this paper.As such, the BS can only communicate with each user througha multi-reflection signal path that is formed by a set of IRSsassociated with the user. To mitigate the potential inter-userinterference during the multi-hop signal reflection, the signalpaths for all K users should be sufficiently separated and thuseach IRS is associated with at most one user at one time,while it can serve multiple users over different time slots viaproper user scheduling. As such, we focus on the MBMHrouting design for a set of users in one given time slot. Forconvenience, we denote the sets of users and IRSs as K ,1, 2, · · · ,K and J , 1, 2, · · · , J, respectively.

To maximize the reflected signal power by each selectedIRS and ease the hardware implementation, we set the re-flection amplitude of all its elements to the maximum valueof one. As such, the reflection coefficient matrix of eachIRS j, j ∈ J is given by Φj = diagejθj,1 , · · · , ejθj,M ∈CM×M , and its passive beamforming vector is denoted asθj = diag(Φj) ∈ CM×1. The passive beamforming vectorof each IRS is assumed to be selected from a codebook WI ,

i.e., θj ∈ WI ,∀j ∈ J , and WI consists of D = 2b beampatterns, where b denotes the number of controlling bits forWI . For convenience, we refer to the BS and user k, k ∈ Kas nodes 0 and J+k in the system, respectively. Accordingly,we define H0,j ∈ CM×NB , j ∈ J as the channel from theBS to IRS j, gHj,J+k ∈ C1×M , j ∈ J as that from IRS j

to user k, and Si,j ∈ CM×M , i, j ∈ J , i 6= j as that fromIRS i to IRS j. For ease of exposition, we assume that thepassive reflecting elements of each IRS in J are arrangedin a uniform rectangular array (URA) perpendicular to theground and facing a fixed direction, while the BS employsa uniform linear array (ULA). For convenience, we apply athree-dimensional (3D) coordinate system locally at each IRSand assume that its URA is parallel to the x-z plane, as shownin Fig. 1. The antenna and element spacing at the BS and eachIRS is assumed to be dA and dI , respectively. The numbers ofelements in each IRS’s horizontal and vertical directions areassumed to be M1 and M2, respectively, with M1M2 = M .

Let di,j , i 6= j denote the distance between nodes i and j,for which some reference transmitting/reflecting elements ofthe BS/IRSs are selected without loss of generality. To ensurethe far-field propagation between any two nodes, we assumethat di,j ≥ d0,∀i 6= j, where d0 denotes the minimum distanceto satisfy this condition. According to [23], it must hold thatd0

√Md2Iλ , where λ denotes the carrier wavelength. Then,

by carefully deploying the J IRSs, LoS dominant propagationmay be achieved between some pair of nodes i and j ifdi,j is practically small (but larger than d0). To simplify theactive and passive beamforming designs as well as enhance thestrength of the multi-reflection signal paths, we only exploitthe LoS links in the system for the multi-hop signal reflection.Then, to describe the LoS condition between any two nodes i(BS/IRS) and j (IRS/user) in the considered system, we define

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a binary LoS condition indicator li,j ∈ 0, 1. In particular,li,j = 1 indicates that the link between nodes i and j consistsof an LoS link; otherwise, li,j = 0. In addition, we setli,i = 0,∀i and thus, li,j = lj,i,∀i, j.

Furthermore, each IRS can only achieve 180 half-spacereflection, i.e., only the signal incident on its reflection sidecan be reflected, as shown in Fig. 1. Thus, for any two nodes iand j, if they are both IRSs, each of them needs to be locatedin the reflection half-space of the other to achieve effectivesignal reflection between them. For example, in Fig. 1, IRSj and IRS i cannot successively reflect the signal from theBS as they do not meet the above condition. Similarly, if oneof the two nodes (say, node i) is the BS/user and the othernode (say, node j) is an IRS, then node i should be located inthe reflection half-space of node j. Equivalently, if the aboveconditions cannot be satisfied for any two nodes i and j, wecan set li,j = 0. In this paper, to focus on the new MBMHrouting design, we assume that the LoS condition indicatorsli,j’s are known and constant after deploying the IRSs, whilehow to acquire such knowledge in practice is an interestingproblem to be addressed in our future work. Based on the LoScondition between any two nodes in the considered system,a multi-hop LoS link can be established between the BS andeach user k, k ∈ K by properly selecting a subset of associatedIRSs. For example, if l0,i = li,j = lj,J+k = 1, i, j ∈ J , wecan select IRSs i and j as the associated IRSs of user k, whichsuccessively reflect its intended signal from the BS toward itsreceiver. For all IRSs that are not associated with any userin K, the BS can inform their controllers can via the controllinks to turn them off based on its optimized MBMH routingsolution, so as to minimize the scattered interference in thesystem.

Next, we characterize the LoS channel between any twonodes in the system (if any), which is modeled as the productof array responses at their two sides. For convenience, wedefine the following steering vector function,

e(φ,N) = [1, e−jπφ, · · · , e−jπ(N−1)φ]T ∈ CN×1, (1)

where N denotes the number of elements in a ULA, and φdenotes the phase difference between the observations at twoadjacent elements. Obviously, e(φ,N) is a periodic functionsof φ and has a period of 2. Hence, we restrict φ ∈ [0, 2) in thesequel of this paper. If φ ≥ 2 or φ < 0, we set φ as φ−2bφ2 c.Then, the array response at the BS is expressed as

aB(ϑ) = e(2dAλ

sinϑ,NB

), (2)

where ϑ denotes the angle-of-departure (AoD) relative to theBS antenna boresight. For the URA at each IRS, its arrayresponse is expressed as the Kronecker product of two steeringvector functions in the horizontal and vertical directions,respectively, i.e.,

aI(ϑa, ϑe)=e

(2dIλ

sinϑe cosϑa,M1

)⊗e(2dIλ

cosϑe,M2

),

(3)where ϑe and ϑa denote its elevation angle-of-arrival(AoA)/AoD and azimuth AoA/AoD, respectively. Then, wedefine ϑ0,j as the AoD from the BS to IRS j, ϕaj,i/ϕ

ej,i as the

azimuth/elevation AoA at IRS j from node i (BS or IRS), andϑai,j /ϑ

ei,j as the azimuth/elevation AoD from IRS i to node j

(IRS or user). The above AoAs and AoDs can be estimatedby exploiting the geometric relationship of the BS, IRSs andusers in the system [23] or by integrating sensors to the IRSs[3].

Based on the above, we define hj,1 = aB(ϑ0,j) andhj,2 = aI(ϕ

aj,0, ϕ

ej,0) for the LoS channel from the BS to IRS

j, j ∈ J , si,j,1 = aI(ϑai,j , ϑ

ei,j) and si,j,2 = aI(ϕ

aj,i, ϕ

ej,i)

for that from IRS i to IRS j, i, j ∈ J , as well as gj,J+k =aI(ϑ

aj,J+k, ϑ

ej,J+k) for that from IRS j to user k, j ∈ J , k ∈

K. Then, if l0,j = 1, the BS-IRS j channel is expressed as

H0,j =

√β

d0,je−

j2πd0,jλ hj,2h

H

j,1, j ∈ J , (4)

where β (< 1) denotes the LoS path gain at the referencedistance of 1 meter (m), and the exponential term capturesthe transmission delay over the LoS link. Similarly, if li,j =1, i, j ∈ J , the IRS i-IRS j channel is given by

Si,j =

√β

di,je−

j2πdi,jλ si,j,2s

Hi,j,1, i, j ∈ J , i 6= j. (5)

Finally, if lj,J+k = 1, the IRS j-user k channel is expressedas

gHj,J+k=

√β

dj,J+ke−

j2πdj,J+kλ gHj,J+k, j∈J , k∈K. (6)

Based on (4)-(6), we can characterize the multi-hop LoSchannel between the BS and each user k, k ∈ K, with thegiven reflection path and BS/IRS active/passive beamform-ing. Specifically, let Ω(k) = a(k)

1 , a(k)2 , · · · , a(k)

Nk, k ∈ K

denote the reflection path from the BS to user k, whereNk (≥ 1) and a(k)

n ∈ J denote the number of associated IRSsfor user k and the index of the n-th associated IRS, withn ∈ Nk , 1, 2, · · · , Nk, respectively. For convenience, wedefine a(k)

0 = 0 and a(k)Nk+1 = J + k, k ∈ K, corresponding to

the BS and user k, respectively. Then, to ensure that each IRSin Nk only reflects user k’s information signal at most once,the following constraints should be met:

a(k)n ∈ J , a(k)

n 6= a(k)n′ ,∀n, n

′ ∈ Nk, n 6= n′, k ∈ K. (7)

Moreover, each constituent link of Ω(k), along with the BS-IRS a

(k)1 link and the IRS a

(k)Nk

-user k link, should consist ofan LoS link, i.e.,

la(k)n ,a

(k)n+1

= 1,∀n ∈ Nk ∪ 0, k ∈ K. (8)

Furthermore, to avoid the scattered inter-user interference,we consider that there is no direct LoS link1 between anytwo nodes belonging to different reflection paths (except thecommon node 0 or the BS). Thus, we have

la(k)n ,a

(k′)n′

=0, a(k)n 6=a

(k′)n′ ,∀n, n

′ 6= 0, k, k′ ∈ K, k 6= k′. (9)

Note that the condition a(k)n 6= a

(k′)n′ ensures that there is

1The methods and results in this paper are extendible to the more generalpath separation constraints, e.g., without q-hop LoS link between any tworeflection paths, with q ≥ 1, by utilizing a similar approach as in SectionIV-B.

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no common node (except the BS) between any two differentreflection paths Ω(k) and Ω(k′). As will be shown in Section V,the inter-user interference can be mitigated to a considerablylower level as compared to the information signal at each userk’s receiver thanks to the constraint (9).

Thus, each Ω(k) is a feasible path if and only if theconstraints in (7)-(9) are satisfied. Given K feasible pathsΩ(k), k ∈ K, we define wk ∈ CN×1, k ∈ K as the BS activebeamforming design for user k, with wk ∈ WB . Then, theBS-user k effective channel is expressed as h0,J+k(Ω(k)) =

gHa(k)Nk,J+k

Φa(k)Nk

( ∏n∈Nk,n6=Nk

Sa(k)n ,a

(k)n+1

Φa(k)n

)H

0,a(k)1wk, k ∈ K,

(10)which depends on both the CPB design for the Nk selectedIRSs and the active beamforming design wk for the BS. Bysubstituting (4)-(6) into (10) and rearranging the terms in it,we obtain

h0,J+k(Ω(k)) = e−j$kκ(Ω(k))( Nk∏n=1

A(k)n

)hH

a(k)1 ,1wk, k ∈ K,

(11)where

A(k)n =

sHa(k)1 ,a

(k)2 ,1

Φa(k)1ha(k)1 ,2

if n = 1

sHa(k)n ,a

(k)n+1,1

Φa(k)nsa(k)n−1,a

(k)n ,2

if 2 ≤ n ≤ Nk − 1

gHa(k)Nk,J+k

Φa(k)Nk

sa(k)Nk−1,a

(k)Nk,2

if n = Nk,

(12)$k = 2π

λ

∑Nkn=0 da(k)n ,a

(k)n+1

is proportional to the end-to-endtransmission distance, and

κ(Ω(k)) =(√β)Nk+1

Nk∏n=0

da(k)n ,a

(k)n+1

(13)

denotes the cascaded LoS path gain between the BS and userk under the path Ω(k), which turns out to be the productof the LoS path gains of all constituent links in Ω(k). Notethat the end-to-end delay under the path Ω(k) is equal to∑Nkn=0 da(k)n ,a

(k)n+1

divided by the speed of light, which is thusnegligible.

Thus, the equivalent channel gain between the BS and userk, |h0,J+k(Ω(k))|2, is expressed as

|h0,J+k(Ω(k))|2 =

βNk+1Nk∏n=1|A(k)n |2 · |h

H

a(k)1 ,1wk|2

Nk∏n=0

d2

a(k)n ,a

(k)n+1

, k ∈ K.

(14)

Based on (14), we can obtain the optimal active/passivebeamforming design at the BS/IRSs with a given MBMHrouting solution, whereby the MBMH routing problem canbe formulated, as detailed in the next section.

III. OPTIMAL BEAMFORMING DESIGN AND PROBLEMFORMULATION

In this section, we first derive the optimal active and passivebeamforming design to maximize each |h0,J+k(Ω(k))|2, k ∈ K

in (14) under a given reflection path Ω(k). With the optimalbeamforming design, we then formulate the MBMH routingproblem where only the reflection paths Ω(k), k ∈ K, need tobe optimized.

A. Optimal Active and Passive Beamforming Design

First, it is observed from (14) that for any given reflectionpath for user k, to maximize |h0,J+k(Ω(k))|2, the magnitudeof each A(k)

n and hH

a(k)1 ,1wk should be maximized, subject to

the codebook constraints at each IRS and the BS, respectively.First, given the codebook WI at each IRS, consider that anIRS j reflects the signal from its last node i to the next noder. We denote by θI(i, j, r) its corresponding optimal passivebeamforming vector, which can be obtained by enumeratingall beam patterns in WI , i.e., ∀i, j, r,2

θI(i, j, r) =

arg max

θ∈WI

|sHj,r,1diag(θ)hj,2| if i = 0

arg maxθ∈WI

|gHj,J+kdiag(θ)si,j,2| if r = J + k

arg maxθ∈WI

|sHj,r,1diag(θ)si,j,2| otherwise.

(15)In particular, if the continuous passive beamforming with

b → ∞ is applied at each IRS, as all array responses haveunit-modulus entries, (15) can be simplified as

θI(i, j, r) =

sj,r,1 h

∗j,2 if i = 0

gj,J+k s∗i,j,2 if r = J + k

sj,r,1 s∗i,j,2 otherwise.∀i, j, r (16)

Accordingly, in the reflection path of user k, the passivebeamforming of each IRS a

(k)n , n ∈ Nk should be set as

θa(k)n

= diag(Φa(k)n

) = θI(a(k)n−1, a

(k)n , a

(k)n+1). (17)

Note that in the special case of continuous IRS beamformingwith b→∞, by substituting (16) and (17) into (12), we haveA

(k)n = M,∀n ∈ Nk, k ∈ K.3

To gain more useful insights into the optimal passive beam-forming solutions at each IRS in (15) and (16), we considerthat both nodes i and r are IRSs, which corresponds to thethird case in (15). Then, according to (3), it can be shown that[30]

sHj,r,1diag(θj)si,j,2 = aHI (ϑaj,r, ϑej,r)diag(θj)aI(ϕ

aj,i, ϕ

ej,i)

=(aHI (ϑaj,r, ϑej,r) aTI (ϕaj,i, ϕ

ej,i))θj

=(eH(2dIλφ

(1)i,j,r,M1

)⊗ eH

(2dIλφ

(2)i,j,r,M2

))θj , (18)

where φ(1)i,j,r , sinϑej,r cosϑaj,r − sinϕej,i cosϕaj,i and φ(2)

i,j,r ,cosϑej,r − cosϕej,i. Similarly, it can be verified that (18) alsoholds if node i is the BS (the first case in (15)) or node r is auser (the second case in (15)). It follows from (18) that if b→∞, the optimal passive beamforming in (16) can be rewritten

2Similar passive beam search can also be performed in the case with mutualcoupling among IRS elements, where the reflection coefficient matrix of eachIRS is non-diagonal [29].

3For ease of exposition, we assume that a full amplitude gain of M canbe obtained in this paper, while it may be dependent on the incident andreflection angles at each IRS in practice.

6 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS

as θI(i, j, r) = e(

2dIλ φ

(1)i,j,r,M1

)⊗ e(

2dIλ φ

(2)i,j,r,M2

), which

perfectly aligns the horizontal and vertical directions at thesame time.

Motivated by this observation, we set θj = θ(1)I (i, j, r) ⊗

θ(2)I (i, j, r) in (18), where θ(1)

I (i, j, r) and θ(2)I (i, j, r) denote

the horizontal and vertical passive beamforming vectors forIRS j, respectively. Then, we can obtain

sHj,r,1diag(θj)si,j,2

=(eH(2dIλφ

(1)i,j,r,M1

)⊗ eH

(2dIλφ

(2)i,j,r,M2

))·(θ

(1)I (i, j, r)

⊗ θ(2)I (i, j, r)

)=(eH(2dIλφ

(1)i,j,r,M1

)θ

(1)I (i, j, r)

)·(eH(2dIλφ

(2)i,j,r,M2

)θ

(2)I (i, j, r)

). (19)

It is noted from (19) that the passive beamforming ofIRS j can be decoupled into horizontal and vertical IRSpassive beamforming. Accordingly, we define W(1)

I and W(2)I

as the codebooks for the horizontal and vertical IRS passivebeamforming, respectively.4 The numbers of controlling bitsfor W(1)

I and W(2)I are denoted as b1 and b2, respectively,

which satisfy b1 + b2 = b. Hence, the IRS codebook WI canbe decomposed as

WI = θ|θ = θ(1) ⊗ θ(2),θ(1) ∈ W(1)I ,θ(2) ∈ W(2)

I , (20)

while the optimal IRS passive beamforming in (15) can becomputed as θI(i, j, r) = θ

(1)I (i, j, r)⊗ θ(2)

I (i, j, r), where

θ(1)I (i, j, r) = arg max

θ1∈W(1)I

∣∣∣∣eH(2dIλφ

(1)i,j,r,M1

)θ1

∣∣∣∣ , (21)

θ(2)I (i, j, r) = arg max

θ2∈W(2)I

∣∣∣∣eH(2dIλφ

(2)i,j,r,M2

)θ2

∣∣∣∣ . (22)

Note that compared to the joint 3D beam search in (15),the complexity of beam search can be greatly reduced fromO(2b) to O(2b1 + 2b2) by separately solving (21) and (22). Inparticular, if node r is not a user, i.e., r ∈ J , the above beamsearch for any triple nodes (i, j, r) can be conducted offline,since the BS and all IRSs are fixed and their channels can beassumed to be constant over a long period.

Accordingly, in the reflection path of user k, k ∈ K, thepassive beamforming of each IRS a(k)

n , n ∈ Nk in (17) can berewritten as

θI(a(k)n−1, a

(k)n , a

(k)n+1) =

θ(1)I (a

(k)n−1, a

(k)n , a

(k)n+1)⊗ θ(2)

I (a(k)n−1, a

(k)n , a

(k)n+1), (23)

which can be simplified as

θI(a(k)n−1, a

(k)n , a

(k)n+1) =

e(2dIλφ

(1)

a(k)n−1,a

(k)n ,a

(k)n+1

,M1

)⊗ e(2dIλφ

(2)

a(k)n−1,a

(k)n ,a

(k)n+1

,M2

),

(24)

4In the general case of multi-path channel model, a more sophisticatedcodebook structure may be needed. For example, in [31], an additionalwavefront phase codebook is utilized to enable constructive or destructivesuperposition of the waves from different elements at the receivers.

in the case of continuous passive beamforming at each IRSwith b1, b2 →∞.

Next, we focus on the optimal active beamforming de-sign for the BS, which should maximize the amplitude ofhH

a(k)1 ,1wk, k ∈ K in (14). To this end, we define

wB(j) = arg maxw∈WB

|hH

j,1w|, j ∈ J , (25)

as the optimal active beamforming solution for the BS totransmit the beam to IRS j, which is obtained by enumeratingall beam patterns in the codebook at the BS, WB , thusincurring the complexity of O(NB). Similar to the beamsearch in (21) and (22), the beam search in (25) can alsobe performed offline. In particular, if the continuous activebeamforming is applied at the BS, (25) becomes equivalent tothe maximum-ratio transmission (MRT) based on hj,1, i.e.,

wB(j) = hj,1/‖hj,1‖, k ∈ K. (26)

With this definition, the BS active beamforming in the reflec-tion path of user k should be set as

wk = wB(a(k)1 )ej$k , k ∈ K, (27)

where the effective phases $k, k ∈ K of the K reflection pathsare compensated at the BS.

It is worth noting that if NB is sufficiently large, the MRT-based beamforming in (26) ensures that the power of theinformation signal for each user k, k ∈ K overwhelms thatof the inter-user interference in the BS-IRS a

(k)1 link, i.e.,

the first link in Ω(k). This is because with a large NB , theBS antenna array has a practically high angular resolution.If all first-hop IRSs in the reflection paths for the K users,i.e., IRS a

(k)1 , k ∈ K, are sufficiently separated in the angular

domain, the following asymptotically favorable propagationfor massive MIMO [32] can be achieved:

1

NB|hH

a(k)1 ,1wk|2 = 1, k ∈ K,

1

NB|hH

a(k)1 ,1wk′ |2 ≈ 0, k, k′ ∈ K, k 6= k′.

(28)

The asymptotically favorable propagation in (28) may alsobe achieved with practical finite-size codebooks, e.g., thediscrete Fourier transform (DFT)-based codebook (see SectionV for details). This is because when the codebook size NBis sufficiently large, the codebook will have a high resolution,such that the selected beam patterns are close to the MRT-based beamforming in (26). Hence, the inter-user interferencecan be approximately nulled in the first link of each reflectionpath Ω(k) by properly selecting WB . Furthermore, since thepath separation constraints in (9) ensure that the scatteredinter-user interference in the subsequent links of Ω(k) is wellmitigated, user k is approximately free of inter-user interfer-ence, while achieving the maximum end-to-end channel gainwith the BS via IRSs’ passive beamforming in (23) and theBS’s active beamforming in (27).

Under the above optimal beamforming designs, we defineA

(k)n as the maximum value of A(k)

n in (12) by following(23). It is worth noting that A(k)

n depends on the AoAs/AoDsbetween nodes a(k)

n−1 and a(k)n , as well as those between nodes

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 7

a(k)n and a

(k)n+1. Besides, it also depends on the numbers of

controlling bits for the IRS codebooks, i.e., b1 and b2. Inparticular, with increasing b1 or b2, the resolution of IRScodebook can be improved, thus resulting in a larger A(k)

n .In the special case of continuous IRS beamforming withb1, b2 → ∞, we have A

(k)n = M , which is regardless of

the AoAs/AoDs. It follows that the effect of AoAs and AoDsdiminishes when the resolution of IRS codebooks becomeshigher. By substituting (23) and (28) into (14), the maximumBS-user k equivalent channel gain is given by

|h0,J+k(Ω(k))|2 =

βNk+1NBNk∏n=1|A(k)n |2

Nk∏n=0

d2

a(k)n ,a

(k)n+1

, k ∈ K. (29)

It is observed from (29) that besides the conventional activeBS beamforming gain of NB , a new multiplicative CPBgain of

∏Nkn=1|A

(k)n |2 is also achieved for each BS-user k

equivalent channel. As previously discussed, if b1 or b2 issmall, this multiplicative CPB gain will depend heavily on theAoAs and AoDs between any two consecutive nodes in Ω(k).However, if both b1 and b2 are sufficiently large, this CPBgain can be greatly enhanced and approaches its maximumvalue, M2Nk . In general, there exists a fundamental trade-off between maximizing the CPB gain versus the end-to-endpath gain, i.e., κ2(Ω(k)) in (13) (or minimizing the end-to-endpath loss κ−2(Ω(k))), as the former monotonically increaseswith Nk, while the latter generally decreases with Nk. Besidesthis trade-off, there exists another trade-off in balancing all|h0,J+k(Ω(k))|’s for different users in K. Specifically, dueto the practically finite number of IRSs and LoS paths inthe system as well as the path separation constraints in (9),maximizing the channel gain for one user generally reducesthe number of feasible paths for the other users. Particularly, ifthe number of users is large, some users may be denied accessdue to the lack of feasible paths. As such, the optimal MBMHrouting design should reconcile the above trade-offs and takeinto account the resolution of practical IRS codebooks, so asto achieve the optimum performance of all K users in a fairmanner.

It should be mentioned that in addition to the reflection pathΩ(k), there may also exist some other signal paths between theBS and user k due to the random scattering of all active IRSsin the system. Nonetheless, as will be shown in Section V,the strength of these scattered links is practically much lowerthan that of Ω(k) in (29), due to the lack of joint active andCPB gains over these scattered links. As such, we only focuson the multi-reflection path Ωk in this paper.

Numerical Example: To better manifest the benefits of theproposed multi-IRS aided system, we provide the followingtwo numerical examples, as shown in Fig. 2. In Example 1,as shown in Fig. 2(a), there are two IRSs (labelled as 1 and2) deployed near the user and BS, respectively, such thatLoS-dominant channels can be achieved between the BS andIRS 2 as well as between the user and IRS 1. However, dueto the scattered obstacles in the environment, IRS 1 cannotestablish an LoS-dominant channel with IRS 2. Accordingly,

Rayleigh fading

BS

User

IRS 2 IRS 1

BS

User

IRS 2 IRS 1

IRS 3

(a) Example 1

LoS LoS

LoS LoS LoS LoS

(0,0,3)

(2,2,3)(5,2,3)

(6,0,1.5)

(0,0,3)

(2,2,3)(5,2,3)

(6,0,1.5)

(b) Example 2

(3.5,-0.5,3)

z yx

Fig. 2. Simulation setup of numerical examples.

2.5 3 3.5 4 4.5 5

Path-Loss Exponent of Rayleigh-Fading Channel,

-105

-95

-85

-75

-65

-55

-45

Eff

ective

Ch

an

ne

l G

ain

(d

B)

Example 2

Example 1M=400

M=100

Fig. 3. Effective channel gain versus path-loss exponent of inter-IRS Rayleigh-fading channels in Example 1, α.

we assume Rayleigh fading for the channel between them, witha path-loss exponent denoted by α. In Example 2, as shown inFig. 2(b), an additional IRS 3 is properly deployed such thatLoS-dominant channels can be established between it and bothIRSs 1 and 2. For convenience, in both examples, we assumethat each IRS and the BS employ continuous passive and activebeamforming, respectively. Moreover, we follow the notationsin the previous sections and refer to the BS and user as nodes0 and 4, respectively. As such, based on (26), the optimal BSbeamforming is given by the MRT wB(1) = h1,1/‖h1,1‖ inboth examples. Furthermore, in Example 2, based on (16), theoptimal passive beamforming vectors of IRSs 1, 2, and 3 aregiven by θI(0, 1, 3), θI(1, 3, 2), and θI(3, 2, 4), respectively.However, it is generally difficult to derive the optimal passivebeamforming vectors of IRSs 1 and 2 in Example 1 due tothe arbitrary-rank channel between them. In this paper, weapply a similar alternating optimization approach as in [26] toalternately optimize the passive beamforming of one IRS withthat of the other being fixed, until convergence is reached.

In Fig. 3, we plot the effective BS-user channel gain inExample 1 versus the path-loss exponent of the Rayleigh-

8 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS

fading channel between IRSs 1 and 2, α, and compare itwith that in Example 2. The BS is equipped with NB = 32antennas, while each IRS is equipped with M = 100 or 400reflecting elements. The carrier frequency is set to 5 GHz.All results are averaged over 100 random channel realizations.It is observed from Fig. 3 that the effective BS-user channelgain in Example 1 monotonically decreases with α and islower than that in Example 2. In particular, when M = 400,it is around 20 dB lower than that in Example 2 even forα = 2.5. This is expected since no CPB gain can be achievedin Example 1 while a significant CPB gain of M6 can beachieved in Example 2 despite its generally higher end-to-end path loss. Thus, the proposed multi-IRS aided systemis practically useful in enhancing the communication linkstrength in a complex environment with dense obstacles.

B. Problem Formulation

In this paper, we aim to maximize the minimum signal-to-noise-plus-interference ratio (SINR) achievable by the Kusers, by optimizing the reflection paths Ω(k), k ∈ K, sub-ject to the feasibility constraints in (7)-(9). Due to the wellmitigated inter-user interference at each user’s receiver, thisis equivalent to maximizing the minimum BS-user effectivechannel gain, i.e., mink∈K|h0,J+k(Ω(k))|2. The optimizationproblem is thus formulated as

(P1) maxΩ(k)k∈K

mink∈K

|h0,J+k(Ω(k))|2 s.t. (7)-(9). (30)

However, (P1) is a combinatorial optimization problem dueto its integer and coupled variables. In addition, A(k)

n , k ∈ Kin h0,J+k(Ω(k)) are functions of the corresponding AoAs andAoDs if b1 or b2 is finite, while they become a constant Min the case of continuous IRS beamforming, as consideredin [1] and [28]. Thus, it is challenging to obtain the optimalsolution to (P1) via standard optimization methods in general,especially in the case with a small b1 or b2. To tackle thischallenging problem, we reformulate it as an equivalent graph-optimization problem which is then solved, as detailed in thenext section.

IV. PROPOSED SOLUTION TO (P1)

In this section, we first reformulate (P1) as an equivalentproblem in graph theory under the general case with finiteb1 and b2, and thereby show that it is NP-complete. Then,a parametrized recursive algorithm is proposed to efficientlysolve this problem sub-optimally in general. Finally, we showthat (P1) can be more efficiently solved by the proposedalgorithm in the special case of continuous IRS beamformingwith b1 and b2 →∞.

A. Problem Reformulation via Graph Theory

Obviously, in (P1), it is equivalent to minimizing the max-imum |h0,J+k(Ω(k))|−2 among all k ∈ K. Based on (14), wehave

|h0,J+k(Ω(k))|−2 =d2

0,a(k)1

βNB·Nk∏n=1

d2

a(k)n ,a

(k)n+1

β|A(k)n |2

, k ∈ K. (31)

Then, by taking the logarithm of (31), (P1) becomes equiv-alent to

minΩ(k)k∈K

maxk∈K

F (Ω(k)), s.t. (7)-(9), (32)

where

F (Ω(k)) = lnd2

0,a(k)1

βNB+

Nk∑n=1

lnd2

a(k)n ,a

(k)n+1

β|A(k)n |2

. (33)

Next, we recast problem (32) as an equivalent problem ingraph theory subject to the constraints (7)-(9). Following thesimilar procedures in [1] and [28], we construct a directed andunweighted graph G0 = (V0, E0). The vertex set V0 consistsof all nodes in the system, i.e., V0 = 0, 1, 2, · · · , J + K.Furthermore, we consider that each of the K beams can onlybe routed outwards from one IRS i to a farther IRS j fromthe BS with dj,0 > di,0, i, j ∈ J , so as to reach its intendeduser as quickly as possible. Hence, the edge set E is definedas

E0 =(0, j)|l0,j = 1, j ∈ J ∪ (i, j)|li,j = 1, dj,0 > di,0, i, j ∈ J ∪ (j, J + k)| lj,J+k = 1, j ∈ J , k ∈ K, (34)

i.e., there exists an edge from vertex i to vertex j if and only ifan LoS path exists between them and dj,0 > di,0, except thatvertex j corresponds to a user, i.e., j = J+k, k ∈ K. Thus, wehave |E0| = 1

2

∑J+Ki=0

∑J+Kj=0 li,j . Note that (34) ensures that

there is no circle in G, i.e., G is a direct acyclic graph (DAG).Given the constructed graph G, any reflection path from theBS to user k corresponds to a path from node 0 to node J+kin G. However, different from the beam routing problems in[1] and [28] with continuous IRS beamforming with b1 andb2 → ∞, it is difficult to assign a weight to each edge inG0 to recast problem (32) as an equivalent graph-optimizationproblem. This is because each A(k)

n in (33) is associated withthree vertices, i.e., vertices a(k)

n−1, a(k)n and a(k)

n+1, but each edgein G0 is only associated with two vertices. However, in [1] and[28], we have A(k)

n = M , which greatly simplifies the weightassignment in G0.

To resolve the above issues, a new DAG of higher dimen-sion, denoted as G = (V,E), should be constructed from G0.Specifically, besides vertex 0 and vertices J + k, k ∈ K, wecreate a vertex in G for each edge in G0; while for everytwo edges in G0 that share a common vertex, we create anedge between their corresponding vertices in G. The resultinggraph G is known as the line graph of G0 in graph theory.Mathematically, for G, its vertex set V is given by

V = vi,j | (i, j) ∈ E0∪0, J + 1, J + 2, · · · , J +K. (35)

Obviously, we have |V | = |E0| + K + 1. The edge set E isgiven by

E =(0, v0,j)| j ∈ J ∪ (vj,J+k, J + k)| j ∈ J , k ∈ K∪ (vi,j , vj,r)| i, j, r ∈ V . (36)

It follows from (35) and (36) that the edge (0, v0,j)((vj,J+k, J + k)) indicates that there exists an LoS path

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 9

from the BS (IRS j) to IRS j (user k). Moreover, the edge(vi,j , vj,r) indicates that there exist two pairwise LoS pathsfrom node i and node r via IRS j. In this new graph G, someedges in E involve three vertices, thus making the weightassignment possible. To determine the edge weights in G, wefirst rewrite F (Ω(k)), k ∈ K in (33) as

F (Ω(k))=lnd0,a

(k)1√

βNB

+ln

da(k)Nk

,J+k

√β

+

Nk∑n=1

lnda(k)n−1,a

(k)nda(k)n ,a

(k)n+1

β|A(k)n |2

,

(37)by rearranging the terms in it. Accordingly, the weight of eachedge in E is set as follows:

W (0, v0,j) = lnd0,j√βNB

, W (vj,J+k, J + k) = lndj,J+k√

β,

W (vi,j , vj,r)=

ln

di,jdj,r

β|sHj,r,1diag(θI(0,j,r))˜hj,2|2

if i = 0

lndi,jdj,r

β|gHj,J+kdiag(θI(i,j,J+k))si,j,2|2if r = J+k

lndi,jdj,r

β|sHj,r,1diag(θI(i,j,r))si,j,2|2otherwise.

(38)

Note that the above weights may be negative, e.g., when M ,b1 and b2 are practically large, such that the argument of thelogarithm in (38) is smaller than one.

With the constructed line graph G, we can establish a one-to-one correspondence between each path from vertex 0 tovertex J + k, k ∈ K in G and that in G0. For example, if apath in G is given by 0 → v0,1 → v1,3 → v3,J+1 → J + 1,then it corresponds to the path 0 → 1 → 3 → J + 1 in G0

and thus a reflection path from the BS to user 1 via IRSs 1and 3. In particular, the sum of edge weights of any path fromvertex 0 to vertex J + k, k ∈ K in G is equal to F (Ω(k)),if its corresponding path in G0 is Ω(k). Since G0 is a DAG,it is easy to verify that G is also a DAG. Thus, for any pathin G, its corresponding path in G0 can automatically satisfythe constraints in (7)-(8). To handle the more challengingconstraint (9), we present the following definitions.

Definition 1: Neighbor-disjoint paths refer to the paths in agraph which do not have any common or neighboring verticesexcept their starting points.

According to Definition 1, the constraints in (9) can be sat-isfied if the K paths from vertex 0 to vertices J +k, k ∈ K inG0 are neighbor-disjoint. As such, problem (32) is equivalentto the following graph-optimization problem, denoted as (P2).

(P2) Find K paths from vertex 0 to vertices J+k, k ∈K in G, respectively, such that the length of the longestpath (i.e., the path with the maximum sum of edgeweights) is minimized and their corresponding pathsin G0 are neighbor-disjoint.

Note that neighbor-disjoint routing design has been previ-ously studied in various multi-hop wireless networks, suchas ad-hoc networks and wireless sensor networks, for thepurpose of load balancing or interference mitigation [33], [34].However, most of these works only focus on discovering aset of neighbor-disjoint paths through different medium accesscontrol (MAC) layer protocols, but not from an optimal routing

design perspective. A common routing design is by utilizingthe shortest path algorithm to sequentially update the pathsfor the K users [33]. Specifically, after deriving the shortestpath for a user in G, the nodes in its corresponding path inG0 (except node 0) and their neighbors are removed. Then,a new line graph G is constructed to determine the shortestpath for the next user, so as to satisfy (9). However, as will beshown in Section V, this sequential update design generallyyields suboptimal paths and even fails to return feasible paths.This is because the set of feasible paths for the current usercritically depends on the previously optimized paths for theother users. In fact, it has been proved in [34] that findingK neighbor-disjoint paths in G0 is NP-complete even in thecase of K = 2. As such, (P2) remains a challenging problem,which will be solved next.

B. Proposed Solution to (P2)

The basic idea of the proposed solution to (P2) is byfirst finding Q (≥ 1) candidate shortest paths from node 0to each node J + k, k ∈ K in G (thus G0). Given thesecandidate shortest paths, we further construct a new pathgraph, based on which a recursive algorithm is performed topartially enumerate the feasible paths and select the best oneas the solution to (P2), as specified below.

1) Step 1: Find the candidate shortest paths. First, for thenodes 0 and J+k, k ∈ K in G, we invoke the Yen’s algorithm[35] to find Q candidate shortest paths between them. If thetotal number of paths between the two nodes is less than Q, weassume that there exist additional virtual paths between themwith infinite sum of edge weights. For convenience, we denoteby p(q)

k and c(q)k , k ∈ K, q ≤ Q the q-th candidate shortest pathbetween vertices 0 and J + k and its sum of edge weights,respectively. Let P = p(q)

k , k ∈ K, q ≤ Q be the set of allcandidate shortest paths. The time complexity for this step isO(KQ|V |(|E|+ |V | log|V |)) [35].

2) Step 2: Construct the path graph. Next, we constructa new undirected graph Gp = (Vp, Ep), where each vertexin Vp corresponds to one candidate shortest path obtained inStep 1 (thus termed as path graph), i.e., Vp = v(p

(q)k ) | k ∈

K, q ≤ Q. Hence, we have |Vp| = KQ. By this means, wecan establish a one-to-one mapping between any path in P andone vertex in Gp. Moreover, since there also exists a one-to-one mapping between any path in G (thus in P) and a path inG0, each vertex in Gp also corresponds to a unique path in G0.In particular, the vertex v(p

(q)k ) in Gp corresponds to a path

from vertex 0 to vertex J+k in G0. For example, as shown inFig. 4, if the path 0→ v0,1 → v1,3 → v3,J+1 → J + 1 is thesecond candidate shortest path from node 0 to node J + 1 inG and also included in P (e.g., Q = 2), then it corresponds tothe vertex v(p

(2)1 ) in Gp, which thus corresponds to the path

0 → 1 → 3 → J + 1 in G0. Based on this fact, for any twovertices in Gp, we add an edge between them if and only iftheir corresponding paths in G0 are neighbor-disjoint or satisfythe considered path separation constraints. As |Vp| = KQ,we need to execute this procedure KQ(KQ − 1)/2 times;while in each execution, we need to check the connectivitybetween any two vertices in the two corresponding paths in

10 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS

0

J+1J+2

J+3

J+4

(1)

G

v( )p1

(2)

p1(2) v( )p1

(1)

v( )p2(1)

v( )p3(1)

v( )p4(1)

(1)p2

p3(1)

p4(1)

Vp,1

Vp,2

Vp,3

Vp,4

Gp with Q=2

v0,1 v1,3 v3,J+1

v0,2 p1v2,J+1

v0,4

v0,5

v0,6

v4,J+2

v5,J+3

v6,J+4

01

3 J+1

24

J+25

J+3

6

J+4

G0

Fig. 4. An example of the constructed graphs with J = 6 and K = 4, wherethe corresponding paths and vertices are marked by the same color.

G0, respectively, so as to determine whether they are neighbor-disjoint or not. Since the number of vertices in any path in G0

should not exceed |V0| = J+K+1, the worst-case complexityof this step is given by O(K2Q2(J+K)2). Finally, we assigneach vertex v(p

(q)k ) in Gp with a weight, which is equal to the

sum of edge weights of its corresponding path in G, i.e., c(q)k ,obtained in Step 1.

To relate the path graph Gp to (P2), we first introduce thefollowing definitions.

Definition 2: A K-partite graph refers to a graph whosevertices can be partitioned into K disjoint sets, such that thereis no edge between any two vertices within the same set.

Definition 3: A clique is a subset of vertices of an undirectedgraph, such that every two distinct vertices in the clique areadjacent.

Based on Definitions 2 and 3, we can verify the followingfacts, which specify the relationship among Gp, G and G0.

Fact 1: Gp is a K-partite graph, with the k-th disjoint setgiven by Vp,k = v(p

(q)k ) | q ≤ Q, k ∈ K.

Fact 2: For K neighbor-disjoint paths from vertex 0 tovertices J + k, k ∈ K in G0, if their corresponding pathsin G are included in P , they correspond to a clique of size Kin Gp.

Fact 1 can be proved by noting that for any two ver-tices in each Vp,k, k ∈ K, their corresponding paths in G0

should not be neighbor-disjoint, since they share the sameend vertex J + k. For Fact 2, it can be easily verifiedbased on Definition 3 and the definition of Gp. For ex-ample, in Fig. 4, Gp is a 4-partite graph and consists oftwo cliques of size 4, i.e., (v(p

(1)1 ), v(p

(1)2 ), v(p

(1)3 ), v(p

(1)4 ))

and (v(p(2)1 ), v(p

(1)2 ), v(p

(1)3 ), v(p

(1)4 )), each corresponding to

4 neighbor-disjoint paths in G0. The two vertices v(p(1)1 ) and

v(p(2)1 ) in Vp,1 are not connected as their corresponding paths

in G0 share the same end vertex J + 1.According to Facts 1 and 2, we aim to solve the following

clique search problem, denoted as (P3).

(P3) Find a clique of size K in a K-partite graph Gp,whose maximum vertex weight is minimized.

The optimal clique for (P3) corresponds to the best solutionto (P2) among the paths in P . Thus, if Q is set to besufficiently large, such that the optimal paths from node 0to each node J + k, k ∈ K are included in P , the proposedalgorithm ensures to find an optimal solution to (P2) (andhence (P1)), if (P3) is optimally solved. Accordingly, by tuning

the value of its parameter Q, the proposed algorithm canflexibly balance between its performance and complexity.

3) Step 3: Clique enumeration. To find the optimal solutionto (P3), we can enumerate all cliques of size K in Gp and thencompare their respective maximum vertex weights. However,finding all cliques of size K in a graph is also an NP-completeproblem in general when K > 2 [35]. As such, we propose arecursive algorithm to achieve this purpose by leveraging theK-partite property of Gp, thereby optimally solving (P3).

Specifically, we will show that each clique of size K inGp can be recursively constructed based on the cliques ofsmaller sizes. Note that its K vertices must be selected fromthe K disjoint sets Vp,k, k ∈ K, respectively. Without loss ofoptimality, we assume that its k-th vertex is selected from Vp,k.Accordingly, let Ωr, r ≤ K denote the set of all cliques of sizer in Gp, with the s-th vertex of each clique in Ωr selectedfrom Vp,s, s = 1, 2, · · · , r. Obviously, we have Ω1 = Vp,1.Moreover, for each clique (of size r) in Ωr, r ≤ K − 1, ifthere exists a vertex in Vp,r+1 which is adjacent to all verticesin this clique, then a new clique (of size r+1) in Ωr+1 can beconstructed by appending the vertex to this clique. As such,based on the initial condition for Ω1 and the recursion forΩr, r ≤ K−1, all cliques of size K in Gp can be enumeratedin the set ΩK , which requires the worst-case complexity ofO(QK). To further reduce complexity, it is noted that when aclique of size K−1 is constructed, among all feasible verticesin Vp,K , we only need to append the vertex with the lowestweight to it. This is because the cliques obtained by appendingother feasible vertices cannot yield a lower maximum vertexweight. Thus, the worst-case complexity of the above recursivealgorithm can be reduced to O(QK−1). In fact, since thenumber of feasible vertices may significantly decrease whenincreasingly larger cliques are constructed (owing to the morestringent adjacency constraint), the actual complexity of theproposed recursive enumeration is much lower thanO(QK−1),as will be shown in Section V.

Denote by Ci the i-th clique (of size K) in ΩK after theenumeration. For each Ci ∈ ΩK , we can obtain the maximumvertex weight among all of its K vertices, denoted as

ci = maxv(p

(q)k )∈Ci

c(q)k .

Thus, the best clique in ΩK can be obtained as Ci? , withi? , arg min

ici. The main procedures of the proposed clique

enumeration method for solving (P3) are summarized inAlgorithm 1, where a function “RECENUM” is defined andrecursively called to achieve the recursive enumeration.

4) Step 4: Map and output. Finally, a generally suboptimalMBMH routing solution with a finite value of Q can be ob-tained by mapping the K vertices in Ci? to K neighbor-disjointpaths in G0. The process of solving (P2) is summarized inAlgorithm 2. The worst-case complexity of Algorithm 2 isgiven by the sum of the complexity of the first three steps, i.e.,O(KQ|V ||E|+KQ|V |2 log|V |+K2Q2(J +K)2 +QK−1).Since the active beam search in (25) and the passive beamsearch in (21) and (22) with r ∈ J can be conducted offline,the online complexity of the proposed MBMH routing designis given by the sum of the complexity of Algorithm 2 and that

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 11

Algorithm 1 Proposed Clique Enumeration Method for Solv-ing (P3)

1: Initiate r = 1 and a clique C = ∅.2: Execute RECENUM (r, C) and obtain ΩK .3: Compare the maximum vertex weights for all obtained

cliques in ΩK , i.e., ci’s, and determine the best cliqueCi? .

4: function RECENUM (r, C)5: if r = K then6: Among all vertices in Vp,K which are adjacent

to every vertex in C, append the vertex with thelowest weight to C and obtain a new clique of sizeK, C′.

7: Add C′ to the set ΩK .8: else9: Initialize s = 1.

10: while s ≤ Q do11: if C = ∅ or the s-th vertex in Vp,r is adjacent

to every vertex in C then12: Append this vertex to C and obtain a clique

of size r, C′.13: Add C′ to the set Ωr and execute

RECENUM (r + 1, C′).14: end if15: Update s = s+ 1.16: end while17: end if18: end function

of the passive beam search for the K users at their associatedfinal-hop IRSs.

It is worth noting that if ΩK = ∅ with a given Q afterperforming Algorithm 1, this indicates that (P3) is infeasible.To obtain a feasible clique of size K, the value of Q can beincreased to enlarge the solution set of (P3). However, if (P3)is still infeasible even with the maximum allowable Q, thenit can be claimed that (P2) (thus (P1)) is infeasible. As such,some users would be denied access to the considered system.This may happen if the number of users is large (e.g., K >J) or some users are close to each other, such that the pathseparation constraints cannot be met. Besides, if the numberof IRSs is small or they are not properly deployed, (P2) mayalso be infeasible due to the limited number of reflection paths.In this case, the proposed algorithms can help determine theoptimal user selection and the reflection paths for the selectedusers. Specifically, let K ′ be the maximum number of usersthat can be granted access to the considered system, which isgiven by the maximum value of k such that Ωk 6= ∅. Then,the selected users and their reflection paths can be obtained bymapping the best clique (of size K ′) in ΩK′ to K ′ neighbor-disjoint paths in G0.

C. Special Case with Continuous IRS Beamforming

If the continuous beamforming with b1 and b2 → ∞ isapplied at each IRS, (P1) can be more efficiently solved basedon G0, without the need of constructing its line graph G.

Algorithm 2 Proposed Algorithm for Solving (P2)1: Input the line graph G and the number of candidate

shortest paths for each user, Q.2: Find Q candidate shortest paths from vertex 0 to each

vertex J + k, k ∈ K by invoking the Yen’s algorithm anddetermine the path set P = p(q)

k , k ∈ K, q ≤ Q.3: Construct the path graph Gp = (Vp, Ep) with the follow-

ing steps based on P .4: a) Make a vertex for each path in P , i.e., Vp =

v(p(q)k ) | k ∈ K, q ≤ Q.

5: b) Add an edge between any two vertices in Gp if theircorresponding paths in G0 are neighbor-disjoint todetermine Ep.

6: c) Assign each vertex v(p(q)k ) in Gp with the weight

c(q)k .

7: Obtain Ci? by performing Algorithm 1.8: Map the K vertices in Ci? to K neighbor-disjoint paths

in G0 and output them.

Specifically, since we have A(k)n = M in this case, (31)

becomes

|h0,J+k(Ω(k))|−2 =M2

NB

Nk∏n=0

d2

a(k)n ,a

(k)n+1

M2β, k ∈ K. (39)

By taking the logarithm of (39) and discarding irrelevantconstant terms therein, (P1) becomes equivalent to

minΩ(k)k∈K

maxk∈K

Nk∑n=0

lnda(k)n ,a

(k)n+1

M√β

, s.t. (7)-(9). (40)

Similarly as in Section IV-A, we construct the DAG G0 =(V0, E0). However, according to (40), we can directly assigna weight to each edge (i, j) in E0, denoted as W (i, j) =ln

di,jM√β

. As a result, (P2) reduces to finding K neighbor-disjoint paths from vertex 0 to vertices J + k, k ∈ K inG0, respectively, such that the length of the longest path isminimized. To solve this simplified problem, Algorithm 2 canbe similarly applied. The only difference is that the input graphis G0 instead of G.

V. NUMERICAL RESULTS

In this section, we provide numerical results to evaluateour proposed MBMH routing design. We focus on an indoormulti-IRS aided system (e.g., in a smart factory) with K = 4users and J = 13 IRSs. The 3D coordinates of all nodes,their available LoS links, as well as the facing directions ofall IRSs are shown in Fig. 5(a). The system is assumed tooperate at a carrier frequency of 5 GHz. Thus, the carrierwavelength is λ = 0.06 m and the LoS path gain at thereference distance 1 m is β = (λ/4π)2 = −46.4 dB. Basedon the LoS probability specified in [36], for any two nodes iand j that satisfy the facing condition, we consider that thereis an LoS-dominant link between them, i.e., li,j = 1, i, j ∈ V ,if its occurrence probability is equal to one, or di,j ≤ 5m. Whereas if li,j = 0, we assume that there exist richscatterers between nodes i and j and model their channel as

12 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS

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Rayleigh fading with a path-loss exponent of α. The antennaand element spacing at the BS and each IRS are set todA = λ/2 and dI = λ/4, respectively. Moreover, we set theminimum distance for far-field propagation as d0 = 2.5 m.Accordingly, the graph representation of the considered multi-IRS aided system, i.e., G0, is shown in Fig. 5(b). The numbersof elements in each IRS’s horizontal and vertical dimensionsare set to be identical as M0 ,

√M = M1 = M2. The BS is

equipped with NB = 32 antennas. We use the NB-point DFT-based codebook as the BS’s codebook WB , which equallydivides the spatial domain [0, 2) into NB sectors. Specifically,let wB,i ∈ CNB×1 denote the i-th beam pattern in WB . Wehave

wB,i =1√NB

e(2(i− 1)

NB, NB

), i = 1, 2, · · · , NB . (41)

It is verified via simulation that with the deployment of IRSs inFig. 5 and the codebook in (41), the asymptotically favorablepropagation in (28) can be achieved for all links between theBS (node 0) and the possible first-hop IRSs (the neighborsof node 0 in G0). The numbers of controlling bits for eachIRS’s codebooks in the horizontal and vertical dimensions,i.e., W(1)

I and W(2)I , are assumed to be identical as b0 ,

b/2 = b1 = b2. Thus, the number of beam patterns in W(1)I

and W(2)I is identical to D0 , 2b0 =

√D. Similarly as the

BS, the D0-point DFT codebook is used for W(1)I and W(2)

I ,while the proposed MBMH routing design is applicable to anyIRS beamforming codebook. Let θ(1)

I,i and θ(2)I,i denote the i-th

beam patterns in W(1)I and W(2)

I , respectively. Then, we have

θ(1)I,i = θ

(2)I,i = e

(2(i− 1)

D0,M0

), i = 1, 2, · · · , D0. (42)

In the proposed recursive algorithm, the number of candidateshortest paths for each node J + k or user k, k ∈ K is set toQ = 5.

First, Fig. 6 shows the optimized reflection paths for allusers under different numbers of IRS reflecting elements andcontrolling bits for the IRS codebook in each dimension,i.e., M0 and b0. In Fig. 6(a), by utilizing the Bellman-Fordalgorithm [35] for the shortest path problem on G, we plotthe optimal reflection path for each user without the pathseparation constraints in (9) under M0 = 24 (i.e., M = 576)and b0 = 7 (i.e., b = 14) bits. It is observed that IRS 13exists in the reflection paths for both users 3 and 4. Thus,the proposed algorithm is needed to obtain a feasible MBMHrouting solution to (P1) that meets (9). In Figs. 6(b)- 6(d), weplot the optimized MBMH routing solutions by the proposedalgorithm subject to (9). By comparing Fig. 6(b) with Fig. 6(a),it is observed that the paths for users 2 and 4 are changeddue to the path separation constraints in (9). As a result, theireffective channel gains with the BS are sacrificed in order toyield K = 4 neighbor-disjoint paths between the BS and allusers. On the other hand, by comparing Fig. 6(b) with Fig. 6(c),it is observed that for a given M0, increasing the resolutionof the IRS codebook may lead to different optimized paths.This is expected since with a larger b0, each IRS has a higherdegree of freedom in controlling the direction of the reflectedsignal, which may result in different reflection paths. Next, bycomparing Fig. 6(b) with Fig. 6(d), it is observed that whenb0 = 7, the optimized paths for some users, e.g., user 1,may go through more IRSs under M0 = 28 than those underM0 = 24. This is due to the different dominating effects ofthe end-to-end path loss and the CPB gain in maximizing theusers’ effective channel gains with the BS as b0 becomes large.In particular, as M0 = 24, minimizing the end-to-end pathloss is dominant over maximizing the CPB gain. However,as M0 increases to 28, maximizing the CPB gain becomesmore dominant. Since the CPB gain monotonically increaseswith the hop count of the reflection path when b0 is large, theoptimized reflection paths generally go through more IRSs.

To measure the strength of the scattered links in the system,under the optimized MBMH routing solution in Fig. 6(b), weplot in Fig. 7 the power of the overall channel between theBS and user 4, i.e., the cascaded LoS channel via IRSs 8 and11 (or Ω(4)) plus other scattered channels by active IRSs inthe system, versus the path-loss exponent of Rayleigh-fadingchannels, α. As there are two hops in Ω(4), we only considerall single-hop and double-hop scattered links between theBS and user 4, due to the more severe multiplicative path-loss and lack of CPB gain over other multi-hop scatteredlinks. The results are averaged over 100 channel realizations.It is observed from Fig. 7 that the overall channel gain is

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Fig. 6. Optimized reflection paths under different setups.

comparable to the cascaded LoS channel gain over the wholerange of α considered. This indicates that the strength ofthe scattered links in the system is practically much lowerthan that of Ω(4) and thus can be ignored. Furthermore, toevaluate the performance of the proposed algorithm in termsof mitigating the inter-path interference, we plot in Fig. 8 thetotal interference channel gain between the BS and user 4 forserving the other three users. By comparing Fig. 8 with Fig. 7,it is observed that even with α = 2.5, the strength of theinterfering links is about 35.5 dB lower than that of Ω(4). Asα increases, the former further decreases and becomes around57.5 dB lower than the latter with α = 5. Given a commonsignal-to-noise ratio (SNR) level between 20 and 30 dB inmodern data transmission, the inter-user interference has beenwell suppressed below the receiver noise (including other non-IRS reflected interference) in the considered system.

Next, Fig. 9 shows the max-min BS-user channel gainamong all users by different schemes versus the number ofIRS reflecting elements in each dimension, M0, under b0 = 5and b0 = 6. For performance comparison, we consider the

following three benchmark schemes. The first benchmark isthe sequential update scheme, as mentioned at the end ofSection IV-A, where we sequentially update the reflectionpaths from user 1 to user 4. The second benchmark minimizesthe maximum path loss among all BS-user LoS links, while thethird benchmark maximizes the minimum CPB gain among allBS-user LoS links. Their corresponding reflection paths canbe obtained by assuming unit CPB gain and unit end-to-endpath loss, i.e., |A(k)

n | = 1,∀n, k and κ2(Ω(k)) = 1,∀k, in ourproposed algorithm, respectively.

First, it is observed from Fig. 9(a) that when b0 = 5 orthe resolution of the IRS codebook is low, all benchmarks areobserved to achieve a much worse performance as comparedto the proposed algorithm. The sequential update schemeeven fails to output feasible paths when M0 < 28. This isbecause its performance critically depends on the order ofthe update for the users. In addition, the second benchmarkfails to take into account the effect of AoAs and AoDs inthe network when b0 is small, as discussed at the end ofSection III-A; while the third benchmark overestimates the

14 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS

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effect of CPB gain and overlooks that of end-to-end pathloss. On the other hand, as b0 increases to 7, the secondand third benchmarks are observed to achieve the comparableperformance as the proposed algorithm when M0 ≤ 26 andM0 ≥ 28, respectively. This is because the CPB gain is greatlyimproved with increasing b0 and the effect of AoAs and AoDsdiminishes. As such, the CPB gain and end-to-end path losscan dominate the BS-user effective channel gain when M0 islarge and small, respectively. However, when M0 = 27, thesetwo schemes are observed to yield worse performance than theproposed algorithm, which strikes a better trade-off betweenmaximizing the CPB gain and minimizing the end-to-end pathloss.

Finally, in Fig. 10, we plot the max-min BS-user chan-nel gain by the proposed algorithm and the above threebenchmark schemes versus the number of controlling bits forIRS codebook in each dimension, b0, under M0 = 24. Inaddition, we also show the performance by the continuousIRS beamforming with b0 → ∞. It is observed that thecontinuous IRS beamforming yields the largest max-min BS-user channel gain among all schemes considered. This is be-

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cause the maximum passive beamforming gain can be achievedat each selected IRS for any given reflection paths, i.e.,A

(k)n = M,∀n, k. Nonetheless, as b0 increases, it is observed

that the performance of the proposed algorithm improves andeventually achieves a performance very close to the continuousIRS beamforming as b0 ≥ 7. In contrast, the sequentialupdate scheme is observed to achieve a worse performancethan our proposed algorithm as b0 ≤ 4 and fails to outputfeasible paths when b0 = 6. Although the second benchmarkyields the same performance as our proposed algorithm whenb0 ≥ 6, its performance becomes worse than ours when b0decreases. The reason is that it fails to consider the effectof AoAs and AoDs in the network as b0 is small. The thirdbenchmark is observed to achieve a worse performance thanour proposed algorithm, since it overlooks the end-to-end pathloss. However, it outperforms the second benchmark whenb0 ≤ 3. This indicates that the AoAs and AoDs in the networkor the placement of IRSs may be more dominant than the end-to-end path-loss when b0 is extremely small. All the aboveobservations are consonant with our analysis provided at the

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end of Section III-A.

VI. CONCLUSIONS

This papers studies a new MBMH routing problem fora multi-IRS aided massive MIMO system, where cascadedLoS links are established between the multi-antenna BS andmultiple users by exploiting the cooperative signal reflectionsof selected IRSs. We present the optimal active and passivebeamforming solutions at the BS and each selected IRS,respectively. However, under the stringent path separationconstraints for avoiding the inter-user interference, the MBMHrouting problem is NP-complete and challenging to solve. Toderive a high-quality suboptimal solution without incurringprohibitive complexity, we propose a parameterized recursivealgorithm for this problem by leveraging graph theory. It isshown that both the number of IRS reflecting elements andsize of IRS beamforming codebook can greatly impact theoptimal MBMH routing solution as well as the achievablemax-min BS-user channel gain. In particular, the optimalMBMH routing design should take into account the AoAs andAoDs in the system if the size/resolution of IRS beamformingcodebook is not large. Besides, there exists a fundamentaltrade-off between minimizing the end-to-end path loss andmaximizing the CPB gain, which have different dominatingeffects under different numbers of IRS reflecting elements.

This paper can be extended in several promising directionsfor future work, some of which are listed as follows to motivatefuture works.• First, it is interesting to study the MBMH routing prob-

lem under the general multi-path channel model. Inthis case, the MBMH routing problem becomes morechallenging to be solved as the beamforming designcannot be simplified by assuming the LoS inter-IRSchannels. In a parallel work [37], the authors applieda deep reinforcement learning (DRL) approach to op-timize the BS/IRS active/passive beamforming, under agiven reflection path. As such, it is worthy of furtherinvestigating new approaches for the joint beamformingand MBMH routing design under the general multi-path channel model. Moreover, how to find an efficient

approach without assuming any prior channel knowledgeis challenging.

• Second, the considered MBMH routing problem maybecome infeasible as the number of users is large orsome users are close to each other in location. In thiscase, each IRS may be associated with more than oneuser to aid their transmission over orthogonal time slotsor simultaneous transmission over orthogonal frequencyresource blocks (RBs). In the latter case, each IRS cansplit its elements into multiple sub-surfaces (or co-locatedsmaller IRSs equivalently), each associated with one userby reflecting the signal in its corresponding frequencyband. As such, under any RB allocation scheme, ourproposed MBMH routing design can be extended to thissetup by treating the random scattering by co-locatedIRSs as part of environment scattering and applyingour proposed path separation constraint to those IRSsreflecting user signals over the same frequency band.However, in both the cases above, user scheduling/RBallocation and IRS beam routing/passive reflection needto be jointly designed, which is interesting to study infuture work.

• Third, as our main focus is on the new MBMH routingdesign, we consider a simplified IRS model in this paper,while a more accurate model may be needed in practicalrouting design to account for other aspects pertaining toelectromagnetic propagation and device hardware imper-fections, such as mutual coupling among IRS elements[29], angle-dependent passive beamforming gain [31],near-field effects [38], etc. Nonetheless, the proposedalgorithm provides a theoretical bound for the perfor-mance of practical MBMH routing designs, which canbe calibrated by properly introducing correction factorsto account for hardware effects by adjusting the weightsin the constructed graphs accordingly.

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