Jesse Rivest was born in 1977.

Jean-Claude Rivest was born on 1943-01-27.

Jayson Rivest was born on August 9, 1975, in Holyoke, Massachusetts, USA.

Marcel Rivest has written:

'Les droits des travailleurs' -- subject(s): Labor laws and legislation

There are number of encryption techniques one such technique is RSA. RSA stands for rivest shamir algorithm.

Gilles Rivest has written:

'Saint-Ignace du Lac, histoire, exil et inondation' -- subject(s): History

RSA is a data-encryption technology utilizing prime factorization. Its name is derived from the developers who created it: Rivest, Shamir and Adelman.

RSA (Rivest, Shamir, and Adelman) is the best public key algorithm.

can i get solutions to core man introduction to algorithms

RSA is a data-encryption technology utilizing prime factorization. Its name is derived from the developers who created it: Rivest, Shamir and Adelman.

Ronald L. Rivest, Adi Shamir and Leonard M. Adleman was the RSA Code inventors, and code's name is the first letter of the last name of each inventor.

The cast of The One Way Door - 2013 includes: Robert Rivest as Brian Thompson Kyle Van Elliot as The Devil

The cast of Free Delivery - 2008 includes: Omar Alexis Ramos Rebecca Barnstaple Dominic Campos Phil Minas Nancy Rivest Toccara Willis

Some popular textbooks for learning computer programming include "Introduction to Algorithms" by Cormen, Leiserson, Rivest, and Stein, "The C Programming Language" by Kernighan and Ritchie, and "Python Crash Course" by Eric Matthes.

Rivest, Shamir and Adleman, the inventors of the algorithm. Rivest Shamir Adleman (RSA) Authentication Mechanism is used to simplify the security environment for the Flexible Management Topology. It supports the ability to securely and easily register new servers to the Flexible Management topology. With the Flexible Management topology, you can submit and manage administrative jobs, locally or remotely, by using a job manager that manages applications, performs product maintenance, modifies configurations, and controls the application server runtime. The RSA authentication mechanism is only used for server-to-server administrative authentication, such as admin connector and file transfer requests. The

This can be easily found in any book which deals with design and analysis of algorithms .Refer to the topic "map colouring problem". One of the best books is the one by Cormen/Lieserson/Rivest

The cast of Les colombes - 1972 includes: Paul Berval as Philippe Carmen Champagne Jean Coutu as M. Ferland Jacques Garand Ernest Guimond Michel Jasmin Armand Labelle Willie Lamothe Pierre Letourneau Danielle Lord Manda Parent as Armande Ginette Rivest Lise Thouin as Josianne Boucher

In cryptography, RSA (which stands for Rivest, Shamir and Adleman who first publicly described it) is an algorithm for public-key cryptography.[1] It is the first algorithm known to be suitable for signing as well as encryption, and was one of the first great advances in public key cryptography. RSA is widely used in electronic commerce protocols, and is believed to be secure given sufficiently long keys and the use of up-to-date implementations.

A text book about algorithm could be found in your school's library, or campus book shop. Your teacher may also have access to them. If you are off campus; try your local library or bookstore. Or perhaps ask a student.

I have a 2003 FLHTC, You can put a 150 tire, make sure it's not a dunlap cause of the weght capasity rateing, Michelin have a 800 plus weight capasity, also there are things you have to do, chainge the rear sproket for a 2004 model, it has an ofset that allows more room, you also have to change the belt to a 2004 model, it more narrow then 2003, you also have to change the front spraket this is the same one, but you want the belt to ware even or it will brake prematurelly, the chain guard has a flesible rubble that touches the tire, simply pot the rivest of the rubber and your good to go, Now all of this is not as simple as you may think, You have to take the entire side of the bike off to get to the belt pulley, you have to buy a special too for the trans nut it's about 130 bucks, you also have to take out the exaust and the swing arm and the starter bolts, it is a lot of work, but if you have a service manual, it will show you how to do it, I took my time did mine in 4 days, and I am so glad I did, it handles great no more zig zag in the rain groves of the road. Good luck look at utube videos it helped me a lot as well.

• William the Conqueror
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• Alexandre Dumas (1799-1850)
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• Jeanne d'Arc Joan of Arc (1412-1431 approx.)
• Napoléon Bonaparte (1769-1821)
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The cast of J.A. Martin photographe - 1977 includes: Charlie Beauchamp as The Old Man Colette Berthiaume as A Dancer Jacques Bilodeau as Hormidas Lambert Christiane Breton Yvan Canuel as Uncle Joseph Paul Cormier as The Fieddler Colette Cortois as Mrs. Lambert Pierre Daigneault as The Dance Caller Colette Dorsay as An OldMaid Denis Drouin as Beaupre Louise Dubogue as The Bride Francine Dufault as A Dancer Mariette Duval as Neighbor Gaetan Girard as The Married Man Pierre Gobeil as Mr. Tremblay Luce Guilbeault as Mrs. Beaupre Denis Hamel as Mathieu Martin Bobby Lalonde as The Fieddler Jean Lapointe as Adhemar Germaine Lemyre as Aunt Demerise Yvon Leroux as Hector Mireille Machado as A Dancer Walter Massy as Mr. Wilson Jean Mathieu as A Workman Monique Mercure as Rose-Aimee Martin Marthe Nadeau as Aunt Aline Madeleine Pageau as A Client Henry Ramer as Scott Rene Rivest as A Dancer Denis Robinson as Julien Tremblay Marcel Sabourin as Joseph-Albert Martin Marthe Thierry as Granma Catherine Tremblay as Dolores Martin

Gilbert Sicotte has: Played Louis Pelletier in "Les vautours" in 1975. Played Martin in "Ti-Cul Tougas, ou, Le bout de la vie" in 1976. Played Julien in "Fantastica" in 1980. Played Narrator in "Les illusions tranquilles" in 1984. Performed in "Le million tout-puissant" in 1985. Performed in "Anne Trister" in 1986. Played Jean-Paul Belleau in "Des dames de coeur" in 1986. Played Albert Laberge in "Lamento pour un homme de lettres" in 1988. Played Jean-Paul Belleau in "Un signe de feu" in 1989. Played Bouchard in "Les noces de papier" in 1990. Played Paul Lachance in "Le choix" in 1991. Played Narrator in "Un homme de parole" in 1991. Played Joseph-Armand Bombardier in "Bombardier" in 1992. Played Jean-Louis McKenzie in "Cap Tourmente" in 1993. Performed in "Shabbot Shalom" in 1994. Played Claude Rivest in "Urgence" in 1996. Played Claude Volant in "Marguerite Volant" in 1996. Played Antoine Beauchemin in "Bouscotte" in 1998. Played Gabriel Johnson in "Fortier" in 2001. Played Gerry elliot in "Les soeurs Elliot" in 2007. Played Marcel Beaudoin in "Continental, un film sans fusil" in 2007. Played Gustave Lambert in "Louis Cyr" in 2013. Played Mari de Louise in "Miraculum" in 2014.

In the famous paper "New Directions in Cryptography", Whitfield Diffie and Martin Hellman first described the notion of a digital signature scheme, although they only conjectured that such schemes existed.[5][6] Soon afterwards, Ronald Rivest, Adi Shamir, and Len Adleman invented the RSA algorithm that could be used for primitive digital signatures[7]. (Note that this just serves as a proof-of-concept, and "plain" RSA signatures are not secure.) The first widely marketed software package to offer digital signature was Lotus Notes 1.0, released in 1989, which used the RSA algorithm. Basic RSA signatures are computed as follows. To generate RSA signature keys, one simply generates an RSA key pair containing a modulus N that is the product of two large primes, along with integers e and d such that e d = 1 mod φ(N), where φ is the Euler phi-function. The signer's public key consists of N and e, and the signer's secret key contains d. To sign a message m, the signer computes σ=md mod N. To verify, the receiver checks that σe = m mod N. As noted earlier, this basic scheme is not very secure. To prevent attacks, one can first apply a cryptographic hash function to the message m and then apply the RSA algorithm described above to the result. This approach can be proven secure in the so-called random oracle model. Other digital signature schemes were soon developed after RSA, the earliest being Lamport signatures[8], Merkle signatures (also known as "Merkle trees" or simply "Hash trees")[9], and Rabin signatures[10]. In 1984, Shafi Goldwasser, Silvio Micali, and Ronald Rivest became the first to rigorously define the security requirements of digital signature schemes[11]. They described a hierarchy of attack models: # In a key-onlyattack, the attacker is only given the public verification key. # In a known message attack, the attacker is given valid signatures for a variety of messages known by the attacker but not chosen by the attacker. # In a chosen message attack, the attacker first learns signatures on arbitrary messages of the attacker's choice. They also describe a hierarchy of attack results: # A total break results in the recovery of the signing key. # A universal forgery attack results in the ability to forge signatures for any message. # A selective forgery attack results in a signature on a message of the adversary's choice. # An existential forgery merely results in some valid message/signature pair not already known to the adversary. They also present the GMR signature scheme, the first that can be proven to prevent even an existential forgery against a chosen message attack.[11] Most early signature schemes were of a similar type: they involve the use of a trapdoor permutation, such as the RSA function, or in the case of the Rabin signature scheme, computing square modulo composite n. A trapdoor permutation family is a family of permutations, specified by a parameter, that is easy to compute in the forward direction, but is difficult to compute in the reverse direction. However, for every parameter there is a "trapdoor" that enables easy computation of the reverse direction. Trapdoor permutations can be viewed as public-key encryption systems, where the parameter is the public key and the trapdoor is the secret key, and where encrypting corresponds to computing the forward direction of the permutation, while decrypting corresponds to the reverse direction. Trapdoor permutations can also be viewed as digital signature schemes, where computing the reverse direction with the secret key is thought of as signing, and computing the forward direction is done to verify signatures. Because of this correspondence, digital signatures are often described as based on public-key cryptosystems, where signing is equivalent to decryption and verification is equivalent to encryption, but this is not the only way digital signatures are computed. Used directly, this type of signature scheme is vulnerable to a key-only existential forgery attack. To create a forgery, the attacker picks a random signature σ and uses the verification procedure to determine the message mcorresponding to that signature.[12] In practice, however, this type of signature is not used directly, but rather, the message to be signed is first hashed to produce a short digest that is then signed. This forgery attack, then, only produces the hash function output that corresponds to σ, but not a message that leads to that value, which does not lead to an attack. In the random oracle model, this hash-and-decrypt form of signature is existentially unforgeable, even against a chosen-message attack.[6] There are several reasons to sign such a hash (or message digest) instead of the whole document. * For efficiency: The signature will be much shorter and thus save time since hashing is generally much faster than signing in practice. * For compatibility:Messages are typically bit strings, but some signature schemes operate on other domains (such as, in the case of RSA, numbers modulo a composite number N). A hash function can be used to convert an arbitrary input into the proper format. * For integrity: Without the hash function, the text "to be signed" may have to be split (separated) in blocks small enough for the signature scheme to act on them directly. However, the receiver of the signed blocks is not able to recognize if all the blocks are present and in the appropriate order.

Gerard Butler ... King Leonidas

Lena Headey ... Queen Gorgo

Dominic West ... Theron

David Wenham ... Dilios

Vincent Regan ... Captain

Michael Fassbender ... Stelios

Tom Wisdom ... Astinos

Andrew Pleavin ... Daxos

Andrew Tiernan ... Ephialtes

Rodrigo Santoro ... Xerxes

Giovani Cimmino ... Pleistarchos (as Giovani Antonio Cimmino)

Stephen McHattie ... Loyalist

Greg Kramer ... Ephor #1

Alex Ivanovici ... Ephor #2

Kelly Craig ... Oracle Girl

Eli Snyder ... Leonidas at 7 / 8 yrs

Tyler Neitzel ... Leonidas at 15 yrs

Tim Connolly ... Leonidas' Father

Marie-Julie Rivest ... Leonidas' Mother

Sebastian St. Germain ... Fighting Boy (12 years old)

Peter Mensah ... Messenger

Dennis St John ... Spartan Baby Inspector

Neil Napier ... Spartan with Stick

Dylan Smith ... Sentry #1 (as Dylan Scott Smith)

Maurizio Terrazzano ... Sentry #2

Robert Paradis ... Spartan General

Kwasi Songui ... Persian

Alexandra Beaton ... Burned Village Child

Frédéric Smith ... Statesman

Loucas Minchillo ... Spartan Baby A

Nicholas Minchillo ... Spartan Baby B

Tom Rack ... Ephor #3

David Francis ... Ephor #4

James Bradford ... Ephor #5

Andrew Shaver ... Free Greek-Potter

Robin Wilcock ... Free Greek-Sculptor

Kent McQuaid ... Free Greek-Blacksmith

Marcel Jeannin ... Free Greek-Baker

Jere Gillis ... Spartan General #2

Jeremy Thibodeau ... Spartan Boy

Tyrone Benskin ... Persian Emissary

Robert Maillet ... Uber Immortal (Giant)

Patrick Sabongui ... Persian General

Leon Laderach ... Executioner

Dave Lapommeray ... Persian General Slaughtered

Vervi Mauricio ... Armless Concubine

Charles Papasoff ... Blacksmith

Isabelle Champeau ... Mother at Market

Veronique-Natale Szalankiewicz ... Daughter at Market (3 / 5 years old)

Maéva Nadon ... Girl at Market

David Thibodeau ... Boy #1 at Market

David Schaap ... Potter

Jean Michel Paré ... Other Council Guard

Stewart Myiow ... Persian General

Andreanne Ross ... Concubine #1

Sara Giacalone ... Concubine #2

Ariadne Bourbonnière ... Kissing Concubine #1

Isabelle Fournel ... Kissing Concubine #2

Sandrine Merette-Attiow ... Contortionist

Elisabeth Etienne ... Dancer

Danielle Hubbard ... Dancer

Ruan Vibegaard ... Dancer

Genevieve Guilbault ... Slave Girl

Bonnie Mak ... Slave Girl

Amélie Sorel ... Slavegirl

Caroline Aspirot ... Slave Girl

Gina Gagnon ... Slave Girl

Tania Trudell ... Slave Girl

Stéphanie Aubry ... Slave Girl

Mercedes Leggett ... Slave Girl

Atif Y. Siddiqi ... Transsexual (Arabian) #3 (as Atif Siddiqi)

Stephania Gambarova ... Slave Girl

Chanelle Lamothe ... Slave Girl

Sabrina-Jasmine Guilbault ... Slave Girl

Manny Cortez Tuazon ... Transsexual (Asian) #1

Camille Rizkallah ... Giant with Arrow

Trudi Hanley ... Long Neck Woman

Neon Cobran ... Litter Bearer / Slave

Gary A. Hecker ... Ubermortal Vocals (voice)

rest of cast listed alphabetically:

Arthur Holden ... Partisan (uncredited)

Deke Richards ... Spartan Soldier (uncredited)

Darren Shahlavi ... Persian (uncredited)

Michael Sinelnikoff ... Senator (uncredited)

Marc Trottier ... Spartan Warrior (uncredited)

Lexicographic Max-Min Fairness in a Wireless Ad

Hoc Network with Random Access

Xin Wang, Koushik Kar, and Jong-Shi Pang

¤

Abstract

We consider the lexicographic max-min fair rate control problem at the link layer in a

random access wireless network. In lexicographic max-min fair rate allocation, the min-

imum link rates are maximized in a lexicographic order. For the Aloha multiple access

model, we propose iterative approaches that attain the optimal rates under very general

assumptions on the network topology and communication pattern; the approaches are also

amenable to distributed implementation. The algorithms and results in this paper gener-

alize those in our previous work [7] on maximizing the minimum rates in a random access

network, and nicely connects to the \bottleneck-based" lexicographic rate optimization

algorithm popularly used in wired networks [1].

1 Introduction

In a wireless network, the Medium Access Control (MAC) protocol de¯nes rules by which

nodes regulate their transmission onto the shared broadcast channel. An e±cient MAC proto-

col should ensure high system throughput, and distribute the available bandwidth fairly among

the competing nodes. In this paper, we consider the problem of optimizing a random access

MAC protocol with the goal of attaining lexicographic max-min fair rate allocations at the

link layer. Fairness is a key consideration in designing MAC protocols, and the lexicographic

max-min fairness metric [1] is one of the most widely used notions of fairness. The objective,

stated simply, is to maximize the minimum rates in a lexicographic manner. More speci¯cally,

a lexicographic max-min fair rate allocation algorithm should maximize the minimum rate,

then maximize the second minimum rate, then maximize the third minimum rate, and so on.

In our previous work [7], we have proposed algorithms that maximizes the minimum rate in

a wireless ad hoc network in a distributed manner, and showed that the proposed algorithms

can achieve lexicographic max-min fairness under very restrictive \symmetric" communication

patterns. However, the question of achieving lexicographic max-min fair rate allocation in a

more general wireless ad hoc network, preferably in a distributed manner, remained an open

question. In this paper, we propose algorithms that solve this question.

¤

The ¯rst two authors are with the Electrical, Computer, and Systems Engineering Department, and the

third author is with the Mathematical Sciences Department. All authors are at Rensselaer Polytechnic Institute,

Troy, NY 12180, USA. fwangx5,kark,pangjg@rpi.edu

1Speci¯cally, we propose two multi-step approaches that can achieve lexicographic max-

min fair allocation. Intuitively, both algorithms attain lexicographic max-min fairness in the

network by solving a sequence of max-min rate optimization problem and identifying bottleneck

links at each step. Loosely speaking, a bottleneck link is a link that has the minimum rate in

the network, in all possible optimal allocations. We will prove rigorously the convergence of

the two algorithms, and discuss the possibility of implementing the algorithms in a distributed

manner.

The paper is organized as follows. Section 2 describes the system model and problem

formulation. Section 3 provides a few important de¯nitions which are used later in describing

the solution approach. In Section 4, we propose an approach for providing lexicographic max-

min fair rate, which is based on identifying a subset of the bottleneck links. In Section 5, we

discuss how we can identify all bottleneck links so as to improve the e±ciency of the algorithm.

Section 5 concludes the work, and all proofs are presented in the appendix.

2 Problem Formulation

2.1 System Model

A wireless network can be modelled as an undirected graph G = (N; E), where N and E

respectively denote the set of nodes and the set of undirected edges. An edge exists between

two nodes if and only if they can receive each other's signals (we assume a symmetric hearing

matrix). Note that there are 2jEj possible communication pairs, but only a subset of these

may be actively communicating. The set of active communication pairs is represented by the

set of links, L. Each link (i; j) 2 L is always backlogged. Without loss of generality we assume

that all the nodes share a single wireless channel of unit capacity.

For any node i, the set of i's neighbors, Ki = fj : (i; j) 2 Eg, represents the set of nodes that

can receive i's signals. For any node i, the set of out-neighbors of i, Oi = fj : (i; j) 2 Lg µ Ki

,

represents the set of neighbors to which i is sending tra±c. Also, for any node i, the set of

in-neighbors of i, Ii = fj : (j; i) 2 Lg µ Ki

, represents the set of neighbors from which i

is receiving tra±c. A transmission from node i reaches all of i's neighbors. Each node has

a single transceiver. Thus, a node can not transmit and receive simultaneously. We do not

assume any capture, i.e., node j can not receive any packet successfully if more than one of

its neighbors are transmitting simultaneously. Therefore, a transmission in link (i; j) 2 L is

successful if and only if no node in Kj [ fjg n fig, transmits during the transmission on (i; j).

We focus on random access wireless networks, and use the slotted Aloha model [1] for

modeling interference and throughput. In this model, i transmits a packet with probability Pi

in a slot. If i does not have an outgoing edge, i.e., Oi = Á, then Pi = 0. Once i decides to

transmit in a slot, it selects a destination j 2 Oi with probability pij=Pi

, where

P

j2Oi

pij = Pi

.

Therefore, in each slot, a packet is transmitted on link (i; j) with probability pij . Let p =

(pij ;(i; j) 2 L) be the vector of transmission probabilities on all edges, and let Pf denote the

feasible region for p, i.e. Pf = fp : 0 · pij · 1; 8(i; j) 2 L; Pi =

P

j2Oi

pij ; 0 · Pi · 1; 8i 2

2Ng. Then, the rate or throughput on link (i; j), xij , is given by

xij (p) = pij (1 ¡ Pj )

Y

k2Kjnfig

(1 ¡ Pk); p 2 Pf : (1)

Note that (1 ¡ Pj )

Q

k2Kjnfig

(1 ¡ Pk) is the probability that a packet transmitted on link (i; j)

is successfully received at j.

2.2 Lexicographic Max-Min Fair Rate

Let x = (xij ;(i; j) 2 L) denote the vector of rates for all links in the active communication

set E (also referred to as the allocation vector), and ~x be the allocation vector x sorted

in nondecreasing order. An allocation vector x1 is said to be lexicographically greater than

another allocation vector x2, denoted by x1 Â x2, if the ¯rst non-zero component of ~x1 ¡ ~x2

is positive. Consequently, an allocation vector x1 is said to be lexicographically no less than

than another allocation vector x2, denoted by x1 º x2, if ~x1 ¡ ~x2 = 0, or the ¯rst non-zero

component of ~x1 ¡ ~x2 is positive.

A rate allocation is said to be lexicographic max-min fair if the corresponding rate alloca-

tion vector is lexicographically no less than any other feasible rate allocation vector. In the

lexicographic max-min fair rate allocation vector, therefore, a rate component can be increased

only at the cost of decreasing a rate component of equal or lesser value, or by making the vector

infeasible.

3 Preliminaries

In this section we introduce a few de¯nitions which are used later in the paper in describing

our solution approach, and stating and proving our results. We also de¯ne the max-min fair

rate allocation problem, and the concept of bottleneck link, which are two main components

of our solution approach in solving the lexicographic max-min fair rate allocation problem.

3.1 Directed Link Graph and Its Component Graph

3.1.1 Directed Link Graph

Recall that the transmission on link (i; j) is successful if and only if node j, as well as

all neighbors of node j (except node i), are silent. From this, it is straightforward to see

that the interference relationship between two links (i; j) and (s; t), may not be symmetric.

As an example, consider two links (i; j) and (j; k), i =6 k. Obviously, transmission on (i; j)

is successful only if (j; k) is silent. However, if i is not in the neighborhood of k, then the

successful transmission on link (j; k) does not require that link (i; j) be silent.

We de¯ne a directed graph, called directed link graph GL = (VL; EL), where each vertex

stands for a link in the original network. There is an edge from link (i; j) to link (s; t) in the

directed link graph if and only if a successful transmission on link (s; t) requires that link (i; j)

be silent.

3We use the notation (i; j) ; (s; t) to denote the case when there is a path from link (i; j) to

link (s; t) in the directed link graph. We have the following lemma regarding to the property

of the directed link graph.

Lemma 1 (Proof in the Appendix) Let x

¤

ij

and x

¤

st

respectively denote the lexicographic

max-min fair rates for the two links (i; j) and (s; t). If there is a path from (i; j) to (s; t) in a

directed link graph, i.e. (i; j) ; (s; t), then we have x

¤

ij · x

¤

st

.

For a directed graph G = (V; E), the set of predecessors of u 2 V is de¯ned as Pu = fv 2

V : v ; ug

S

fug. Also, for any vertex set U µ V , we de¯ne GU = (U; EU ) as a subgraph of G

for U, where EU = ((u; v) : (u; v) 2 E; u 2 U; v 2 U).

3.1.2 Component Graph

In the directed graph GL = (VL; EL), a strongly connected component is a maximal set of

vertices C µ V such that for every pair of vertices u and v in C, we have both v ; u and

u ; v, that is, vertices u and v are reachable from each other. The following corollaries easily

follow from Lemma 1.

Corollary 1 The lexicographic max-min fair rates of all links belonging to the same strongly

connected component of GL = (VL; EL) is the same.

Corollary 2 Let C1 and C2 be two strongly connected components in the directed link graph

GL = (VL; EL), and x

¤

1

and x

¤

2

be the lexicographic max-min fair rates for C1 and C2, respec-

tively. If u 2 C1 and v 2 C2 such that u ; v, then we have x

¤

1 · x

¤

2

.

For a directed link graph GL = (VL; EL), we can decompose it into its strongly con-

nected components, and construct the component graph GL = (VL; EL), which we de¯ne as

follows. Suppose GL has strongly connected components C1, C2, ..., Ck. The vertex set VL

is fv1; v2; :::; vkg, and it contains a vertex vi for each strongly connected component Ci of GL.

There is a directed edge (vi

; vj ) 2 EL if GL contains a directed edge (x; y) for some x 2 Ci

and some y 2 Cj . Viewed another way, we obtain GL from GL by contracting all edges whose

incident vertices belong to the same strongly connected component of GL. From Lemma 22.13

of [6], it follows that the component graph is a directed acyclic graph.

For any v 2 VL, we denote C(v) as the set of links in C, where C is the corresponding

strongly connected component in the directed link graph. For a set of vertices U µ VL, we

de¯ne C(U) =

S

vi2U C(vi).

3.1.3 Illustrative Example

We use a small wireless ad hoc network to illustrate the concept of a directed link graph.

The network we consider is composed of 8 nodes and 9 links, and is shown in Fig. 1. In this

graph, the set of undirected edges (computed based on the symmetric hearing matrix), E, is

given by E = f(A; B);(B; C);(C; D);(D; E);(D; F);(F; G);(F; H);(G; H)g.

48

A

B

C

D

E

F

G

H

0

1

2

3

4

5

6

7

Figure 1: An example wireless ad hoc network.

In the directed link graph, there are 9 vertices, representing the 9 links. It can be seen

from Fig. 1 that a successful transmission on link 0 requires that node C and its neighboring

nodes (node D) keep silent. Therefore there are edges (7; 0), (1; 0) and (6; 0) in the directed

link graph. Also, when link 0 is scheduled from node B, all other links from node B should

not be scheduled, i.e. there is edge (8; 0) in the directed link graph. Similarly we ¯nd all other

edges in the directed link graph, and the result is shown in Fig. 2.

7

0

8

1

6

2

3

4

5

Figure 2: The directed link graph for the wireless ad hoc network considered.

It is obvious in Fig. 2 that there are two strongly connected components in this directed

link graph, both of which are highlighted by dashed square box. The ¯rst strongly connected

component, denoted as C1, contains link 0, link 1, link 6, link 7, and link 8, and the second

strongly connected component, denoted as C2, contains link 2, link 3, link 4 and link 5. Also

it can be seen that there are edges from vertices in C1 to vertices in C2, and therefore at the

lexicographic max-min fairness, x

¤

1 · x

¤

2

, where x

¤

1

and x

¤

2

are the lexicographic max-min fair

rates for links in C1 and C2, respectively.

3.2 Max-Min Fair Rate Allocation Problem

The objective of max-min rate allocation is to maximize the minimum rate over all links.

Note that whereas the lexicographic max-min fairness optimizes the entire sorted vector of link

rates in a lexicographic manner, max-min fairness only maximizes the minimum component

in the rate vector. Therefore, max-min fairness is a weaker notion of fairness compared to

lexicographic max-min fairness. We will show, however, that we can solve the lexicographic

5max-min fair rate allocation problem by solving a sequence of max-min fair rate allocation

problems.

In our context, the max-min fair rate allocation problem can be formulated as follows [7]:

max x;

s.t. x · xij (p); 8(i; j)2L;

p 2 Pf ;

(2)

where x is the max-min rate, and xij (p) is given in (1). It is worth noting that (2) is equivalent

to the following convex program:

max f(z);

s.t. hij (z) · 0; 8(i; j)2L;

(3)

where z = (y; p) and y = log(x), i.e. the logarithmic value of the max-min rate. f(z) = y, and

hij (z) = y ¡ log(xij (p)). Note that hij (z) is the transformed function of capacity constraint on

link (i; j) 2 L and is convex. Also note that p 2 Pf is removed as the logarithmic function

automatically ensures the feasibility of p.

3.3 Bottleneck Link

Next we de¯ne the notion of a bottleneck link. Loosely speaking, a bottleneck link is a link

that has the minimum rate in (2) and hence decides the max-min rate.

De¯ne gij (x; p) = x ¡ xij (p), and denote an optimal solution to (2) as (x

¤

; p

¤

). It is easy

to argue that x

¤

is unique while p

¤

could be non-unique. If gij (x

¤

; p

¤

) = 0 for any optimal

solution (x

¤

; p

¤

), i.e. the constraint for link (i; j) is active at all optimal solutions, then link

(i; j) is called a bottleneck link.

An alternative de¯nition of a bottleneck link is as follows. Consider perturbing (3) with a

perturbation on link (i; j),

max f(z);

s.t. hij (z) · ¡²;

huv(z) · 0; 8(u; v) 2 L n f(i; j)g:

(4)

The optimal value of (4) is a function on ², and we denote it as U

¤

ij

(²). We de¯ne link (i; j) as a

bottleneck link if U

¤

ij

(0) > U

¤

ij

(²) for any positive ². It can be easily argued that this de¯nition

of a bottleneck link is consistent with the previous one.

The result below follows directly from Lemma 1.

Corollary 3 If link (i; j) is a bottleneck link, and if links (i; j) and (s; t) belong to the same

strongly connected component, then link (s; t) is also a bottleneck link.

Furthermore, the following property also holds:

Lemma 2 (See the proof in the appendix) If link l is a bottleneck link, and l 2 C(v)

where v is a vertex in the component graph, then all links that belong to C(Pv) are bottleneck

links, where Pv is the set of predecessors for v in the component graph.

From Corollary 3 and Lemma 2, we can identify one or more strongly connected components

(in the directed link graph) that consist only of bottleneck links.

64 Algorithm Based on Identifying a Subset of the Bottleneck

Links

4.1 Identifying Bottleneck Links Using Lagrange Multipliers

Direct identi¯cation of bottleneck links, using the de¯nition or the alternative de¯nition,

has extremely high computational cost and maybe practically infeasible. In this section we

discuss how we can identify at least one bottleneck link, using Lagrange multipliers, in an

e±cient manner.

We consider (3), the transformed convex program for the max-min fair rate problem. It is

clear that the Slater Constraint Quali¯cation holds for equation (3). Thus the global optimality

of a feasible solution of (3) is characterized by the Karush-Kuhn-Tucker (KKT) conditions:

0 = rf(z) +

X

(i;j)2L

¸ij rhij (z); 0 · ¸ ? h(z) · 0; (5)

where h(z) is the jLj-vector with components hij (z) and ¸ is the jLj-vector with components ¸ij ,

the Lagrange multipliers for hij . The ? notation means orthogonality or the complementary

slackness condition.

The relationship between Lagrange multipliers and bottleneck links is stated in the following

lemma. Its proof uses the cross complementarity property of the solutions of the KKT system,

which results from the convexity of such solutions. Namely, if (z

i

; ¸

i

) for i = 1; 2 are two KKT

pairs, then we must have ¸

1

ij hij (z

2

) = 0 for all (i; j) 2 L.

Lemma 3 For any KKT pair (z

¤

;¸

¤

), if ¸

¤

ij > 0, then link (i; j) is a bottleneck link.

The lemma above is a special case of Lemma 4. Note that, at least one Lagrange multiplier

is non-zero for a KKT pair, as rf must be non-zero at optimality. Therefore we can always

identify at least one bottleneck link using Lagrange multipliers. Following Corollary 3 and

Lemma 2, we can further identify a set of bottleneck links.

However, it is worth noting that we cannot identify all bottleneck links using an arbitrary

Lagrange multiplier, since it is possible that a bottleneck link has zero Lagrange multipliers.

In fact, identifying all bottleneck links is in general a di±cult problem. We will discuss this in

more detail in Section 5.

4.2 Solution Approach

To attain lexicographic max-min fairness, we adopt the following procedure:

1. For a given wireless Aloha network G = (N; E), compute the directed link graph GL =

(VL; EL), and construct the component graph GL = (VL; EL) for the directed link graph;

2. Set k = 0, Gk = GL, Vk = VL, and Ek = EL;

73. Solve the transformed convex program of (2) for all links that belong to C(Vk),

max y

s.t. y · log(xij (p)); 8(i; j) 2 C(Vk);

p 2 Pf :

(6)

Denote the max-min fair rate as x

¤

k = e

y

¤

, where y

¤

is the optimal solution for (6);

4. Find at least one bottleneck link using the Lagrange multipliers, and ¯nd the correspond-

ing vertex v in the component graph;

5. Set Uk = Pv. The lexicographic max-min rate for the links in C(Uk) is x

¤

k

;

6. Fix the link attempt probabilities for all the links that belong to C(Uk), and set Vk+1 =

Vk n Uk;

7. If Vk+1 is nonempty, we construct the subgraph of Gk for Vk+1 and denote it as Gk+1,

i.e. Gk+1 = GVk+1 = (Vk+1; EVk+1

), increment k by 1, and go to step 3;

8. Terminate if Vk+1 is empty.

Intuitively, the above procedure repeatedly solves the problem (6), which maximizes the

next minimum rate in the network. Note that, at the optimum of (6), we can identify at least

one bottleneck using Lagrange multipliers. By following Corollary 3 and Lemma 2, we can

furthermore identify one or more strongly connected components (in the directed link graph)

that consist only of bottleneck links. We ¯x the attempt probabilities for those bottleneck

links identi¯ed, and go to the next step, i.e. solving (6) again for the rest of the links in the

network. Note that the rate on a link, once identi¯ed as a bottleneck link, remain unchanged in

the later steps, and the procedure is repeated until all links have been identi¯ed as bottleneck

links (at di®erent step k) and their attempt probabilities have been ¯xed.

The following theorem states that the above procedure converges to a lexicographic max-

min fair rate allocation.

Theorem 1 (Proof in the appendix) Denote x

¤

and p

¤

as the vector of link rates and link

attempt probabilities attained by the procedure 1) to 8) given above. Then x

¤

is the lexicographic

max-min fair rate allocation, and p

¤

is the link attempt probabilities that makes x

¤

feasible,

i.e. x

¤

ij · xij (p

¤

) for any link (i; j).

In addition we can show that the optimal solution of x

¤

and p

¤

are unique, and x

¤

ij = xij (p

¤

)

for any link (i; j).

It is worth noting that the procedure above can be implemented in a distributed manner.

The construction of a strongly connected component is realized if each vertex in the directed

link graph ¯nds the set of vertices that it has a path to, and the set of vertices who have a path

to it, and hence can be achieved through a distributed path-search algorithm. Also, (6) can

be solved in a distributed manner to obtain both primal variables and Lagrange multipliers

(refer [7] for details). Also note that the procedure only identify a subset of bottleneck links

at each step. In the worst case it might identify only one strongly connected component at a

step, and solve (6) repeatedly more than necessary. This motivates a solution approach based

on identifying all bottleneck links discussed in Section 5.

85 Algorithm Based on Identifying All Bottleneck Links

5.1 Bottleneck Links, Maximally Complementary Solutions, and the Inte-

rior Point Methods

The concept of a bottleneck link is closely tied to that of a maximally complementary

solution [8, 9] of a monotone nonlinear complementarity problem (NCP) [5] derived from

the KKT conditions of a convex program satisfying a suitable CQ. In order to de¯ne the

corresponding concept of maximal complementarity for the KKT system (5), we introduce

three basic sets, ®(z; ¸) = f(i; j) : ¸ij > 0 = hij (z) g, ¯(z; ¸) = f(i; j) : ¸ij = 0 = hij (z) g,

and °(z; ¸) = f(i; j) : ¸ij = 0 > hij (z) g, associated with every KKT pair (z; ¸) satisfying

(5).

A KKT pair (bz;¸b) is said to be maximally complementary if the index set ®(bz;¸b)[°(bz; ¸b) is

maximal among all KKT pairs; i.e., if (z; ¸) is another KKT pair such that ®(bz;¸b) [ °(bz; ¸b) µ

®(z; ¸) [ °(z; ¸), then equality holds in the above inclusion. It is not di±cult to show (see [9,

page 627]) that if (bz; ¸b) is a maximally complementary solution, then ®(bz;¸b) and °(bz; ¸b) are

respectively maximal as two separate sets among all the KKT pairs.

An important fact is that the respective index sets ®(bz;¸b), ¯(bz; ¸b), and °(bz;¸b) are the

same among all maximally complementary KKT pairs. This fact is easy to prove using the

convexity of the solutions to the KKT system. Therefore we can label the common sets as ®b,

¯b, and °b, respectively.

By de¯nition, link (i; j) 2 L is a bottleneck link if its capacity constraint hij is satis¯ed as

an equality by all optimal solutions of (3). Let LB ½ L denote the set of all bottleneck links,

and LNB ½ L denote the set of all non-bottleneck links. Obviously L = LB

S

LNB. The main

connection between a bottleneck link and the maximally complementary solution is described

in the result below.

Lemma 4 (Proof in the appendix) Link (i; j) is a bottleneck link, i.e. hij (z) is satis¯ed as

an equality by all optimal solutions, if and only if (i; j) 2 ®b

S

¯b, i.e. LB = ®b

S

¯b and LNB = °b.

This relation of maximally complementarity and bottleneck links can be illustrated by

considering the case below. Suppose we have three links in an Aloha network, namely link

1, link 2, and link 3. The network is setup in a way such that x1(p) = p1(1 ¡ p2), x2(p) =

p2(1 ¡ p1), and x3(p) = p3(1 ¡ p2). The max-min fair rate problem is therefore formulated as

follows:

max x;

s.t. x · p1(1 ¡ p2);

x · p2(1 ¡ p1);

x · p3(1 ¡ p2):

(7)

Obviously the optimization solution is x

¤ = 0:25 when p1 = 0:5, p2 = 0:5, and 0:5 · p3 · 1.

Note that only link 1 and link 2 are bottleneck links, i.e. B = f1; 2g. Also note that ®b = f1; 2g,

¯b = Á, and °b = f3g in this case. Therefore we have LB = ®b

S

¯b and LNB = °b.

The signi¯cance of Lemma 4 is that the index sets ®b and ¯b can be obtained by an interior-

point algorithm (see [9], [5, Chapter 11]) applied to the KKT formulation (5). Nevertheless,

9enjoying both polynomial complexity and local quadratic convergence, such an interior-point

algorithm cannot easily be implemented in a distributed manner.

5.2 The Barrier Method

5.2.1 Identifying All the Bottleneck Links Using the Barrier Method

In this section, we discuss how to identify all the bottleneck links using the barrier method.

For the convex program of the max-min fair rate optimization, (3), the barrier problem is

formulated below.

min µ(¹); s.t. ¹ ¸ 0; (8)

where µ(¹) = inff¡f(z) + ¹B(z) : hij (z) < 0; 8(i; j) 2 L; p 2 Pf g. Here B is the barrier

function that is nonnegative and continuous over the region fz : hij (p) < 0; p 2 Pf g, and

approaches 1 as the boundary of the region fz : hij (p) < 0; p 2 Pf g is approached from the

interior.

More speci¯cally, the barrier function B is de¯ned by

B(z) =

X

(i;j)2L Á [hij (z)] ;

where Á is a function of one variable that is continuous over fs : s < 0g and satis¯es:

Á(s) ¸ 0 if s < 0 and lim

s!0¡

Á(s) = 1

One typical Á(s) is Á(s) = ¡1=s, and another is Á(s) = log(¡s). We refer to the function

¡f(z) + ¹B(z) as the auxiliary function.

Intuitively, when solving the max-min fair rate optimization problem (3) using the barrier

method, a very large penalty (the barrier function) is added in the objective to convert the

originally constrained optimization problem into an unconstrained optimization problem (8).

Lemma ?? (see the statement of the lemma in the appendix) ensure that for any positive

¹, there exists z¹ so that µ(¹) = ¡f(z¹) + ¹B(z¹), and that the limit of any convergent

subsequence of fz¹g is an optimal solution to the primal problem (3) when ¹ approaches to

zero, and therefore ensure the validity of the barrier method.

It is worth noting that, when searching for the optimal solution of µ(¹) at any given positive

¹, the solution tries to stay away from the boundaries as a solution close to the boundary always

incurs a very large penalty on the objective. In this manner, the constraints are active only for

those bottleneck links. For non-bottleneck links that can be inactive at some optimal solutions,

the constraint will not be active. Therefore, the optimal solution given by the barrier method

naturally divides all the links into the set of bottleneck links and the set of non-bottleneck

links. We make this argument rigorous in the following theorem.

Theorem 2 (Proof in the appendix) Suppose Á(s) satis¯es that the auxiliary function f(z)+

¹B(z) is strictly convex on z. If the limit point of a convergent subsequence of fz¹g is denoted

by z

¤

, then hij (z

¤

) = 0 for all (i; j) 2 LB and hij (z

¤

) < 0 for all (i; j) 2 LNB, where LB and

LNB denote the set of bottleneck links and the set of non-bottleneck links respectively.

105.2.2 Distributed Implementation

To provide lexicographic max-min fairness in a distributed manner, we need to solve the

max-min fair rate problem (8) in a distributed manner.

In [7] it has been shown that (3) is equivalent to the convex problem below,

min

P

(i;j)2L

y

2

ij

;

s.t. yij ¡ log (xij (p)) · 0; 8(i; j)2L;

yij · yst

; 8(i; j) 2 L;(s; t) 2 L(i; j);

p 2 Pf :

(9)

where yij is the logarithmic value of the max-min fair rate on link (i; j). L(i; j) is de¯ned as the

neighboring links of link (i; j) in the directed link graph, i.e. L(e) = fe^ : (e; e^) 2 EL or (^e; e) 2

EL; e^ 2 VL)g for e = (i; j) 2 VL.

Intuitively, (9) introduces yij for each link (i; j) and forces them to be equal (in the second

constraint) so that the originally centralized problem can be solved in a distributed manner.

It applies logarithmic transformation to each capacity constraint to make it a convex set. The

objective function is rewritten as min

P

y

2

ij

since yij · 0 for any link (i; j) and we want to

make yij maximized (hence its square should be minimized).

Based on the convex program (9), we construct a barrier problem for ¹ > 0 as below,

min

P

(i;j)2L

y

2

ij + ¹

P

(i;j)2L

1

log (xij (p)) ¡ yij

;

s.t. yij · yst

; 8(i; j) 2 L;(s; t) 2 L(i; j);

p 2 Pf :

(10)

Note that (10) actually transfers the capacity constraints to the objective by using the barrier

function Á(s) = ¡1=s. From Lemma 6 (refer to the appendix), (10) converges to the optimal

solution of (9) when ¹ ! 0.

According to the scalar composition [2], for Á : R ! R and g : Rn ! R, Á ± g is convex

if Á is convex and nondecreasing, and g is convex. For the barrier problem considered in (10),

Á(s) = ¡1=s and gij (y; p) = yij ¡ log (xij (p)) for link (i; j), where y = (yij : (i; j) 2 L).

Therefore the barrier function B(y; p) de¯ned as

B(y; p) =

X

(i;j)2L

1

log (xij (p)) ¡ yij

is a convex function on (y; p). Therefore (10) is a convex program, and can be solved in a

distributed manner. We now present a distributed algorithm to solve (10) iteratively.

Let p

(n)

ij

and y

(n)

ij

denote the attempt probability on link (i; j) and the logarithmic value

of the link rate at the nth iteration respectively. De¯ne the \link relation indicator" for link

(i; j) and its neighboring link (s; t), º

(n)

ij;st

, as

º

(n)

ij;st =

½

0 when uij · ust

;

1 when uij > ust

:

(11)

11Let · be a positive constant, and °n be the step size at the nth iteration. The logarithmic

value of rate on link (i; j), yij , is updated as

y

(n+1)

ij = y

(n)

ij ¡ °

0

@2y

(n)

ij + ¹

@B

@yij

+ ·

X

(s;t)2L(i;j)

³

º

(n)

ij;st ¡ º

(n)

st;ij

´

1

A; (12)

and the attempt probability on link (i; j), pij , is updated as

p

(n+1)

ij = p

(n)

ij ¡ °¹

@B

@pij

: (13)

Following the procedure based on the subgradient method [3], we can show that the max-

min rates (the exponential of yij for link (i; j)) and link attempt probabilities converge to a

neighborhood around the optimum value, and the size of the neighborhood becomes arbitrarily

small with decreasing stepsize.

5.3 Solution Approach

If we let C(Uk) be all the bottleneck links identi¯ed when we repeatedly solve (3) for the

kth time, not only the procedure given in Section 4.2 applies here, but also Theorem 1 on

the convergence holds true. However, there is some considerable di®erence between these

two procedures that is worth noting here. Since the procedure given above can identify all

bottleneck links every time when (3) is solved, (3) will be solved for the least number of times

and hence greatly lower the computation cost. Also, in the barrier method, identi¯cation of

the set of bottleneck links does not require any information on the structure of the component

graph, and this can greatly reduce the complexity in the distributed implementation. This

simplicity in computation/implementation comes at a cost: note that in practice, the barrier

method would only yield approximate solutions, since its solution converges to the optimum

only when ¹ approaches zero. However, the solution provided by the barrier method will

become closer to the optimum when ¹ is decreased, and can be arbitrarily close to the optimum

by making ¹ su±ciently small.

6 Conclusion

In this paper, we address the problem of providing lexicographic max-min fair rate allo-

cations at the link layer in a wireless ad hoc network with random access. We propose two

e±cient approaches which attain the globally optimal solutions and are amenable to distributed

implementation.

Our algorithms are based on solving the problem of maximizing the minimum rate repeat-

edly, and identifying bottleneck links at each iteration. In this respect, our approaches share

an intuitive similarity with the well-known bottleneck-based algorithm for computing the lex-

icographic max-min fair rates in a wired network [1, Chapter 6]. However, the lexicographic

max-min fair rate allocation problem in our context is signi¯cantly more complex than that for

12wired networks. In particular, whereas the problem constraints in our case are non-linear, non-

convex and non-separable, the corresponding link capacity constraints in a wired network are

linear. Naturally, the notion of a bottleneck link, and the proof of optimality of our approaches

are considerably more involved than their counterparts for wired networks. Finally, note that

the bottleneck-based algorithm in [1, Chapter 6] considers multi-hop end-to-end sessions in a

wired network, as is therefore designed to attain fair rates at the level of the transport layer

(the corresponding problem at the link layer, where we we need to consider single-hop connec-

tions, is trivial to solve). In contrast, the approaches in our paper are applicable at the level

of the link layer, where only single-hop connections need to be considered. The question of

attaining lexicographic max-min fair rates for end-to-end multi-hop wireless sessions remains

open for future investigation.

Appendix

A Proof of Lemma 1

Proof: First we show that, if there is an edge from (i; j) to (s; t) in a directed link graph,

we have x

¤

ij · x

¤

st

.

By the de¯nition of an edge in the directed link graph, it must be one of the following two

cases to have an edge from (i; j) to (s; t),

1. link (i; j) and (s; t) have the same source nodes;

2. i is either the receiver t or a neighboring node of receiver t for link (s; t).

We then show that in either of the above cases, x

¤

ij · x

¤

st

.

Assume that x

¤

ij > x

¤

st

, and at the lexicographic max-min fairness the corresponding at-

tempt probabilities are p

¤

uv

for (u; v) 2 E.

In case 1), i and s are the same node. Obviously we can ¯nd ± > 0, and de¯ne p

0

ij = p

¤

ij ¡±,

p

0

st = p

¤

st + ±, and p

0

uv = p

¤

uv

such that

x

0

ij = p

0

ij

(1 ¡ P

0

j

)

Y

k2Kjnfig

(1 ¡ P

0

k

) = p

0

ij

(1 ¡ P

¤

j

)

Y

k2Kjnfig

(1 ¡ P

¤

k

)

x

0

st = p

0

st

(1 ¡ P

0

t

)

Y

k2Ktnfsg

(1 ¡ P

0

k

) = p

0

st

(1 ¡ P

¤

t

)

Y

k2Ktnfsg

(1 ¡ P

¤

k

)

It is worth noting that rates of all other links except (i; j) and (s; t) remain unchanged. Since

rate of (s; t) is increased while rate of all the links whose rate is smaller than x

¤

st

remains

unchanged, this contradicts the fact that x

¤

st

is the lexicographic max-min fair rate.

In case 2), i is either the receiver of link (s; t) or a neighboring node of node t. We can ¯nd

± > 0, and de¯ne p

0

ij = p

¤

ij ¡ ± and p

0

uv = p

¤

uv otherwise, such that x

¤

st < x

0

st < x

0

ij < x

¤

ij

. Note

that when changing from p

¤

to p

0

, rates of all links except (i; j) are non-decreased. Since rate

of (s; t) is increased while rate of all the links whose rate is smaller than x

¤

st

is not decreased,

this contradicts the fact that x

¤

st

is the lexicographic max-min fair rate.

13We can then conclude that in either case 1) or 2), there is contradiction. Therefore if x

¤

is

the vector of lexicographic max-min fair rates, then x

¤

ij · x

¤

st

.

If (i; j) ; (s; t) in the directed link graph, we can denote the links along the path as (u1; v1),

(u2; v2), ..., (un¡1; vn¡1), (un; vn). Since there is an edge from (i; j) to (u1; v1), x

¤

ij · x

¤

u1v1

.

Similarly, we have x

¤

u1v1 · x

¤

u2v2

, ..., x

¤

un¡1vn¡1 · x

¤

unvn

, and x

¤

unvn · x

¤

st

. Therefore x

¤

ij · x

¤

st

.

This completes the proof.

B Proof of Lemma 2

Proof: Denote the max-min fair rate for (2) as x

¤

. For any link r in the directed link

graph, denote its lexicographic max-min rate as x

¤

r

. Obviously x

¤

r ¸ x

¤

, as x

¤

won't be the

max-min rate for (2) otherwise. Since r 2 C(Pv), r ; l in the directed link graph. According

to Lemma 1, x

¤

r · x

¤

l

, where x

¤

r and x

¤

l

are the lexicographic max-min rates for link r and

link l respectively. Therefore the lexicographic max-min fair rates for link r and link l must

be equal. Since l is a bottleneck link, link r is also a bottleneck link.

C Proof Outline of Theorem 1

Proof: First, we show that for the bottleneck links in C(Uk), their rates will remain ¯xed

in any steps l > k. By the de¯nition of a directed link graph, if the rate of link (i; j) depends

on the attempt probability of link (s; t), then (s; t) ; (i; j) in the directed link graph, i.e., the

rate of a link (i; j) depends on the attempt probability of link (s; t) only if (s; t) is a predecessor

of link (i; j). From Lemma 2, Pv ½

S

l=0;1;:::;k Ul

for any v 2 Uk. Since all links that belong to

C(

S

l=0;1;:::;k Ul) have ¯xed link attempt probabilities for iteration l > k, the rates for the links

that belong to C(Uk) will remain ¯xed.

We now show that attempt probabilities of the links in C(Uk), solved from (6), are unique.

Lemma 5 Consider the max-min fair rate problem

max y;

s.t. y · log(xij (p)); 8(i; j) 2 C(Vk);

p 2 Pf :

At the optimal solution, attempt probabilities of the links in C(Uk) are unique.

Proof: If l1 2 C(Vk n Uk) and l2 2 C(Uk), we have x

¤

l1

¸ x

¤

l2

since l2 ; l1 in the

directed link graph, where x

¤

l1

and x

¤

l2

denote the lexicographic max-min fair rate for link l1

and l2 respectively. Therefore we can remove the constraints that correspond to the links in

C(Vk n Uk) and obtain the following equivalent problem.

max y;

s.t. y · log(xij (p)); 8(i; j) 2 C(Uk);

p 2 Pf :

(14)

As all the links in C(Uk) are bottleneck links, all the constraints in (14) are active at the

optimum. The max-min rate x and the attempt probabilities for the links in C(Uk) will be

decided in (14).

14Denote p

Uk = (pij : (i; j) 2 C(Uk)). Note that if (i; j) 2 C(Uk), then xij (p) only depends

on p

Ul

, where l = 1; :::; l ¡ 1. Note that p

Ul

is ¯xed for l < k, we can write xij (p

Uk

).

Denote the optimal value of (14), as y

¤

. Assume that both p

Uk

1

and p

Uk

2

are both optimal

solutions. Since all constraints of (14) are active at the optimum, we have y

¤ = log(xij (p

Uk

1

))

and y

¤ = log(xij (p

Uk

2

)).

Denote p

Uk

¸ = ¸p

Uk

1 +(1¡¸)p

Uk

2

, for 0 < ¸ < 1. Since log(xij (p

Uk

) is strictly concave on p

Uk

,

for any (i; j) 2 C(Uk) we have log(xij (p

Uk

¸

) = log(xij (¸p

Uk

1 +(1¡¸)p

Uk

2

) > ¸y

¤+(1¡¸)y

¤ = y

¤

.

This contradicts with the fact that y

¤

is the optimal value for (14). Therefore (14) has a unique

solution.

From Lemma 2, we conclude that all links in C(Uk) are bottleneck links, and their max-min

fair rate is x

¤

k

. We have also shown that the max-min rate for those bottleneck links will remain

¯xed in the later steps.

From Corollary 2 and Lemma 2 we can easily see that x

¤

0 · x

¤

1 · x

¤

2 · ::: · x

¤

n

for step 1

to step n.

We now show that, if the algorithm terminates at the nth step, x

¤

0 · x

¤

1 · x

¤

2 · ::: · x

¤

n

is the lexicographic max-min fair rate. From the above discussion, it can be seen that our

procedure ¯rst maximizes the minimum rate in the network, and then maximizes the rate

that's second to the minimum, and so on. Therefore the procedure guarantees that the rate of

any link cannot be increased without decreasing the rate of a link that has smaller rate. Also,

we have shown that for each step, the attempt probabilities solved from (14) is unique, and

hence there is no possibilities that the rate of a link is increased while the rates of the links,

who have smaller rates, remain unchanged. Therefore our procedure gives the lexicographic

max-min fair rate.

D Proof of Lemma 4

Proof: If (i; j) is a bottleneck link, then it is clear that (i; j) 62 °b by the cross complemen-

tarity property. Conversely, suppose that (i; j) is not a bottleneck link. Then there exists an op-

timal solution ez of (3) such that hij (ez) < 0. Let ¸e be any KKT multiplier corresponding to ez and

let (bz; ¸b) be a maximally complementary KKT pair. The pair (z

1=2

; ¸

1=2

) ´

1

2

(bz; ¸b) +

1

2

(ez; ¸e)

is also a KKT pair, by the convexity of the solution set of the KKT system. Clearly, we have

°b = °(bz; ¸b) µ °(z

1=2

; ¸

1=2

). The maximality of °b implies that (i; j) 2 °(z

1=2

;¸

1=2

) = °b.

E Proof of Theorem 2

Proof: The following lemma ensures the validity of using barrier functions for solving a

constrained problem by converting them into a single unconstrained problem or into a sequence

of unconstrained problems.

Lemma 6 The following statements hold for the barrier method applied to our problem:

1. For each ¹ > 0 there exists an z¹ 2 Z with g(z¹) < 0 such that

µ(¹) = f(z¹) + ¹B(z¹)

= infff(z) + ¹B(z) : g(z) < 0; z 2 Zg;

152. infff(z) : g(z) · 0; z 2 Zg · inffµ(¹) : ¹ > 0g

3. For ¹ > 0, f(z) and µ(¹) are nondecreasing functions of ¹, and B(z¹) is a non-increasing

function of ¹,

4. minff(z) : g(z) · 0; z 2 Zg = lim¹!0¡ µ(¹) = inf¹>0 µ(¹),

5. The limit of any convergent subsequence of fz¹g, at least one of which must exist, is an

optimal solution to the primal problem, and furthermore ¹B(z¹) ! 0 as ¹ ! 0

+

.

Proofs of Lemma 6 follows from standard results for the barrier method [4].

Let A¹(y; p) denote the auxiliary function for the barrier problem (8) at the given ¹ > 0,

i.e.

A¹(y; p) = ¡y + ¹

X

(i;j)2L

Á(hij (y; p)); (15)

and let (y

¤

; p

¤

) denote the limit point of a convergent of subsequence of fz¹g = f(y¹; p¹)g.

Since Á is assumed to make A¹(y; p) a strictly convex function, (y

¤

; p

¤

) is an optimal solution

to (3) from Lemma 6.

We write p = (p

B

; p

NB

), where p

B

is the vector of attempt probabilities for bottleneck

links, and p

NB

is the vector of attempt probabilities for non-bottleneck links.

Since the limit point (y

¤

; p

¤

) is an optimal solution to (3), by de¯nition of a bottleneck

link we have

hij (y

¤

; p

¤

) = 0; 8(i; j) 2 LB: (16)

According to the de¯nition of a non-bottleneck link, there exists an p~ such that for any

(i; j) 2 LNB, we have

hij (y

¤

; p~) < 0:

Therefore there exists an ³ > 0 such that

hij (y

¤

; p~) < ¡³; 8(i; j) 2 LNB: (17)

We now show that for any convergent subsequence f(y¹; p¹)g and for any ² > 0, there exists

±

1

²

such that for any 0 < ¹ < ±

1

²

, there exists (y¹; p~¹) that satis¯es for any (i; j) 2 LNB we

have

hij (y¹; p~¹) < ¡³ + ²; (18)

where jLNBj denotes the size of the set LNB, i.e. the number of non-bottleneck links.

Noting that p

B¤

is unique from Lemma 5, we conclude that the convergent subsequence

f(y¹; p¹)g satis¯es that fp

B

¹ g converges to p~

B = p

B¤

.

We construct p~¹ = (p

B

¹

; p~

NB

), and therefore for any ² > 0 there exists ±

1

² > 0, such that

for any 0 < ¹ < ±

1

² and any (i; j) 2 LNB we have

jy¹ ¡ y

¤

j < 0:5²; jlog(xij (p~)) ¡ log(xij (p~¹))j < 0:5²:

16Therefore we have

jhij (y¹; p~¹)j = jy¹ ¡ log(xij (p~¹))j

= jy¹ ¡ y

¤ + y

¤ ¡ log(xij (p~)) + log(xij (p~)) ¡ log(xij (p~¹))j

¸ jy

¤ ¡ log(xij (p~))j ¡ jy¹ ¡ y

¤

j ¡ jlog(xij (p~))¡log(xij (p~¹))j

> ³ ¡ ²:

Since hij (y¹; p~¹) < 0, it follows that

hij (y¹; p~¹) < ¡³ + ²; 8(i; j) 2 LNB:

We then show that for a convergent subsequence f(y¹; p¹)g, the limit point (y0; p0) must

satisfy that hij (y0; p0) < 0. We prove this result by contradiction.

Suppose that for link l 2 LNB, hl(y0; p0) = 0. Therefore for any ² > 0, we can ¯nd ±

2

²

such

that for any 0 < ¹ < ±

2

²

, it holds ¡² < hl(y¹; p¹) < 0. From the previous discussion, we see

that when 0 < ¹ < ±

1

²

, we can ¯nd (y¹; p~¹) such that hij (y¹; p~¹) < ¡³ + ² for all (i; j) 2 LNB.

Since Á(s) approaches in¯nity when s approaches 0 where s < 0, there exists ²1 > 0 such

that for any 0 < ² < ²1, we have Á(¡²) > jLNBjÁ(¡0:5³). We then de¯ne ²2 = 0:5³. Denote

²0 = minf²1; ²2g, and denote ±²0 = minf±

1

²0

; ±

2

²0

g.

Noting that for a bottleneck link (i; j), hij only depends on p

B

, and that p~¹ = (p

B

¹

; p~

NB

),

we have hij (p~¹) = hij (p¹) for any link (i; j) 2 LB. Therefore, for any 0 < ¹ < ±²0 we then have

A¹(y¹; p¹) ¡ A¹(y¹; p~¹)

=

"

¡y¹ + ¹

P

(i;j)2L

Á(hij (y¹; p¹))

#

¡

"

¡y¹ + ¹

P

(i;j)2L

Á(hij (y¹; p~¹))

#

= ¹

P

(i;j)2LB

[Á(hij (y¹; p¹)) ¡ Á(hij (y¹; p~¹))]

+¹

P

(i;j)2LNB

[Á(hij (y¹; p¹)) ¡ Á(hij (y¹; p~¹))]

= ¹

P

(i;j)2LNB

[Á(hij (y¹; p¹)) ¡ Á(hij (y¹; p~¹))]

¸ ¹

"

Á(hl(y¹; p¹)) ¡

P

(i;j)2LNB

Á(hij (y¹; p~¹))

#

¸ ¹(Á(¡²0) ¡ jLNBÁ(¡0:5³)j) > 0:

This contradicts with the fact that (y¹; p¹) is the optimal solution to A¹(y; p). Therefore the

assumption that hl(y0; p0) = 0 for link l 2 LNB is incorrect, and we conclude that hl(y0; p0) < 0

for any link l 2 LNB.

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18