# Robert G. Gallager's research while affiliated with Massachusetts Institute of Technology and other places

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## Publications (103)

Schalkwijk and Kailath (1966) developed a class of block codes for Gaussian channels with ideal feedback for which the probability of decoding error decreases as a second-order exponent in block length for rates below capacity. This well-known but surprising result is explained and simply derived here in terms of a result by Elias (1956) concerning...

Variable-length block-coding schemes are investigated for discrete memoryless channels with ideal feedback under cost constraints. Upper and lower bounds are found for the minimum achievable probability of decoding error Pe,min as a function of constraints R, P, and tau on the transmission rate, average cost, and average block length, respectively....

Variable-length block-coding schemes are investigated for discrete memoryless channels (DMC) with perfect feedback under cost constraints. Upper and lower bounds are found for the minimum achievable probability of decoding error Pepsi,min as a function of transmission rate R, cost constraint P, and expected block length taumacr. For given P and R,...

We develop on-line routing and wavelength assignment (RWA) algorithms for WDM bidirectional ring and torus networks with N nodes. The algorithms dynamically support all k-allowable traffic matrices, where k denotes an arbitrary integer vector [k<sub>1</sub>, k<sub>2</sub>,... k<sub>N</sub>], and node i, 1 ≤ i ≤ N, can transmit at most k<sub>i</sub>...

We develop an on-line wavelength assignment (WA) algorithm for a wavelength-routed WDM tree network. The algorithm dynamically supports all $bf k$ -port traffic matrices among $N$ end nodes, where $bf k$ denotes an integer vector $[k_1 ldots, k_N]$ and end node $i, , 1leq ileq N$ , can transmit at most $k_i$ wavelengths and receive at most $k_i$ wa...

We study the provision of deterministic rate guaran- tees over single crossbar switches. Birkhoff decomposition yields a general approach for this problem, but the required complexity can be very high and the quality of service can be unsatisfactory for practical traffic sources. We develop a method called rate quantization which converts the set o...

We study the provision of deterministic rate guarantees over single crossbar switches. Birkhoff decomposition yields a general approach for this problem, but the required complexity can be very high and the quality of service can be unsatisfactory for practical traffic sources. We develop a method called rate quantization which works with any resou...

We develop on-line routing and wavelength assignment (RWA) algorithms for WDM bidirectional ring and torus networks with N nodes. The algorithms dynamically support all k-allowable traffic matrices, where k denotes an arbitrary integer vector [k<sub>1</sub>, k <sub>2</sub>, ..., k<sub>N</sub>], and node i, 1≤i≤N, can transmit at most k<sub>i</sub>...

To help understand distributed processing in the context of channel coding, we study distributed processing in a single input multi output (SIMO) channel with memory. Distributed processing in the context of source coding has been studied. One problem which arises in the context of source coding is the CEO problem [2], [9], [8], [4]. A parallel bet...

We study the problem of statistical multiplexing of cell streams that have correlations at multiple time-scales. Each stream is modeled by a singularly perturbed Markov-modulated process with some state transitions occurring much less frequently than others. One motivation of this model comes from variable-rate compressed video, where the fast time...

We consider a user communicating over a fading channel with
perfect channel state information. Data are assumed to arrive from some
higher layer application and are stored in a buffer until transmitted.
We study adapting the user's transmission rate and power based on the
channel state information as well as the buffer occupancy; the
objectives are...

We show that very large bandwidths on fading multipath channels
cannot be effectively utilized by spread-spectrum systems that (in a
particular sense) spread the available power uniformly over both time
and frequency. The approach is to express the input process as an
expansion in an orthonormal set of functions each localized in time and
frequency...

We show that very large bandwidths on fading mul-tipath channels cannot be effectively utilized by spread-spectrum systems that (in a particular sense) spread the available power uniformly over both time and frequency. The approach is to express the input process as an expansion in an orthonormal set of functions each localized in time and frequenc...

Claude E. Shannon (1948) invented information theory and provided
the concepts, insights, and mathematical formulations that now form the
basis for modern communication technology. In a surprisingly large
number of ways, he enabled the information age. A major part of this
influence comes from his two-part monumental 1948 paper, “A
Mathematical The...

We investigate the time-varying additive white Gaussian noise
channel with imperfect side-information. In practical systems, the
channel gain may be estimated from a probing signal and estimation
errors cannot be avoided. The goal of this paper is to determine a power
allocation that a priori incorporates statistical knowledge of the
estimation err...

It is argued that military networks will have to operate in channels which exhibit a wider range of fundamental characteristics than are present in commercial systems. It is then demonstrated that these channel characteristics can make it difficult for modifications of existing commercial systems to operate acceptably for every possible channel. Bo...

We consider a user communicating over a flat fading channel. The user wishes to reliably communicate bursty data over this channel while minimizing both the average power and the average delay incurred. We formulate a buffer control problem which illustrates the trade-off between these quantities. This model is analyzed using dynamic programming te...

We introduce the real, discrete-time Gaussian parallel relay
network. This simple network is theoretically important in the context
of network information theory. We present upper and lower bounds to
capacity and explain where they coincide

We present a new lower bound on the feedback capacity of the colored Gaussian noise channel. Under the assumption of large power, this lower bound is shown to be strictly larger than the non-feedback capacity. Insight into the role of the feedback and the capacity-achieving strategy have been obtained

We present a new algorithm which creates redundant trees on arbitrary node-redundant or link-redundant networks. These trees are such that any node is connected to the common root of the trees by at least one of the trees in case of node or link failure. Our scheme provides rapid preplanned recovery of communications with great flexibility in the t...

Any rate tuple in the capacity region of an M-user Gaussian or
discrete memoryless multiple access channel can be achieved using a time
sharing configuration requiring no more than ½M log<sub>2</sub>
M+M single user codes

High speed slotted bus networks are attractive candidates for
MANs, backbone networks for LANs, high speed access networks, and feeder
networks for high speed ATM networks. In all of these approaches, there
are tradeoffs between utilization, fairness, access delay, overhead, and
complexity. Our objective here is to understand the relationship betwe...

We present a new algorithm for automatic protection switching
(APS) which creates node (edge) redundant trees on any node
(edge)-redundant network. These trees are desirable for performing
multicasting with APS. Our algorithm is based on constructing trees with
appropriate associated directions. The algorithm gives great flexibility
in the choice o...

© 1998 Optical Society of America

The issue of bandwidth scaling has always caused some confusion
for channels subject to additive white Gaussian noise plus multipath
fading. On the one hand, there is an old result saying that the capacity
of the channel (under a power constraint but no bandwidth constraint) is
the same as the infinite bandwidth capacity of the non-fading additive...

Tunstall codes are variable-to-fixed length codes that maximize
the expected number of source letters per dictionary string for
discrete, memoryless sources. We analyze a generalization of Tunstall
coding to sources with memory and demonstrate that as the dictionary
size increases, the number of code letters per source symbol comes
arbitrarily clos...

A popular information theoretic technique for multiaccess communication on a white Gaussian noise channel is to decode the users one by one. After each user is decoded, its encoded waveform is subtracted from the received signal, thus cancelling the interference from that user for the task of decoding subsequent users. This technique is not directl...

Many existing protocols indicate that full utilization and
fairness might be incompatible in high-speed high-latency MANs or LANs.
The purpose of this paper is to understand the relationship between full
utilization and fairness. A new protocol called FUFA (fully utilized and
fair) is designed to demonstrate some of the basic properties. We define...

We describe some of the results of the Advanced Research Projects
Agency (ARPA) sponsored Consortium on Wideband All-Optical Networks in
developing architectures, technology components, and applications for
the realization of scaleable, wideband, and transparent optical
wavelength-division multiplexing (WDM) networks. Our architecture
addresses all...

Caption title. Includes bibliographical references (p. 31-32). Supported by ARPA. MDA972-92-J-1038 Angela L. Chiu, Robert G. Gallager.

Markov chains with a countably infinite state space exhibit some types of behavior not possible for chains with a finite state space. Figure 5.1 helps explain how these new types of behavior arise. If p > 1/2, then transitions to the right occur with higher frequency than transitions to the left. Thus, reasoning heuristically, we expect Xn to be la...

The counting processes {N(t),t≥0} of Chapters 2 and 3 have the property that N(t)changesat discrete instants of time, butis definedfor all real t ≥ 0. Such stochastic processes are generally called continuous time processes. The Markov chains to be discussed in this and the next chapter are stochastic processesdefinedonly at integer values of time,...

A Poisson process is a simple and widely used stochastic process for modeling the times at which arrivals enter a system. We usually look at arrivals after some starting time, say t=0. Figure 2.1 illustrates some of the different ways to characterize random arrivals over the positive time axis. The sequence of times at which arrivals occur is denot...

The capacity of wireless systems depends critically on the
variations of the channels. The effect of the channel variations is
twofold: it creates fading and it precludes exact measurement of the
channel. We study the second effect. We establish bounds for the loss in
mutual information due to not knowing the channel at the receiver for
the single...

A stochastic process (or random process) is a probabilistic experiment or model that evolves in time. That is, each sample point (i.e., possible outcome) of the experiment is a function of time called a sample function. The sample space is the set of possible sample functions, and the events are subsets of sample functions. Finally, there is a rule...

Recall that a counting process, {N(t);t≥0} is a stochastic process in which N(t) models the number of arrivals to a system in the interval (0,t]. The corresponding arrival epochs are denoted S1,S2,…, and the inter-arrival intervals are denoted X1,X2,…. A renewal process is a counting process for which the inter-arrival intervals are positive indepe...

AMarkov processwith a countable state space is a special case of a semi-Markov process in which, first, the interval between successive transitions has an exponential distribution, and second, that interval is independent of the next state. Thus, we can take the set of possible states as {0, 1, 2,…} and the process as {X(t),t≥0} where for each real...

Stochastic processes are found in probabilistic systems that evolve with time. Discrete stochastic processes change by only integer time steps (for some time scale), or are characterized by discrete occurrences at arbitrary times. Discrete Stochastic Processes helps the reader develop the understanding and intuition necessary to apply stochastic pr...

Let {Xi; i ≥ 1} be a sequence of IID random variables, and let Sn=X1+X2+...+Xn. The integer time stochastic process {Sn;n≥1} is called arandom walk.For any given n, S, is just a sum of IID random variables, but here, we are more interested in the behavior of the random walkprocess, {Sn;n≥1}, and thus in such questions as finding the first n for whi...

The purpose of this paper is to understand the relationship between utilization, fairness and access delay in high speed slotted bus networks. We illustrate this relationship by means of a protocol called FUFA (fully utilized and fair). We define full utilization, and fairness precisely, and show that both are achieved together in the FUFA protocol...

The purpose of this paper is to understand the relationship between utilization, fairness and access delay in high speed slotted bus networks. We illustrate this relationship by means of a protocol called FUFA (fully utilized and fair). We define full utilization, and fairness precisely, and show that both are achieved together in the FUFA protocol...

We examine the effect of a randomly time-varying channel on mutual
information between the receiver and sender when the channel is m<sup>th
</sup> order Markov. Such channels often occur in mobile communications
and can affect the achievable rate. If the channel is perfectly known,
then the mutual information between a receiver and an arbitrary num...

We study the problem of statistical multiplexing of cell streams
that have correlations at multiple time-scales. Each stream is modeled
by a singularly perturbed Markov-modulated process with some state
transitions occurring much less frequently than others. One motivation
of this model comes from variable-rate compressed video, where the fast
time...

We develop and analyze a multiaccess communication model over the
additive Gaussian noise channel. The framework is information-theoretic;
nonetheless it also incorporates some queueing-theoretic aspects of the
problem

The capacity of a multiple-access, time-varying but known
multipath channel is considered in the presence of white Gaussian noise,
a bandwidth constraint, and power constraints. It is assumed that the
product of the time spread and the Doppler spread is small. The general
case is specialized to particular cases such as flat fading and specular
path...

We introduce and analyze a multi-access communication model over the additive Gaussian noise channel. The model combines information theoretic and queueing-theoretic aspects of the problem

We analyze a generalization of Tunstall (1968) coding to sources
with memory and demonstrate that this code becomes asymptotically
optimal as the dictionary size increases

Worst-case bounds on delay and backlog are derived for leaky
bucket constrained sessions in arbitrary topology networks of
generalized processor sharing (GPS) servers. The inherent flexibility of
the service discipline is exploited to analyze broad classes of
networks. When only a subset of the sessions are leaky bucket
constrained, we give succinc...

Presents a logical approach to designing an effective and efficient error detection scheme for ATM. The authors specifically look at providing error protection for the class C service of ATM, which is a connection oriented, variable bit rate service, with no required timing between source and destination. The resulting scheme is similar to the sche...

The authors analyze the expected delay for infinite precision arithmetic codes, and suggest a practical implementation that closely approximates the idealized infinite precision model

Considers the problem of optimal variable-rate lossy compression of Markov sources when the encoded bit stream is buffered before entering a fixed-rate channel. The authors discuss how to minimize the average distortion under this constraint

Includes bibliographical references (p. 12-13). Supported by the Army Research Office. ARO DAAL03-92-G-0115 Robert G. Gallager.

A decoder that has the option of not decoding and thus declaring
an erasure is called an errors and erasures decoder. We obtain improved
simultaneous exponential upper bounds to error and erasure
probabilities. We also improve the lower bound to the zero
undetected-error capacity for discrete memoryless channels

The rapid evolution in optical fiber communication has made terabit link speed a likely possibility for the future. The urge to integrate voice, data, and video within a single network, combined with increasingly image oriented data traffic, also suggest that terabit networks will someday be useful. In these integrated networks of the future, diffe...

Worst-case bounds on delay and backlog are derived for leaky bucket constrained sessions in arbitrary topology networks of generalized processor sharing servers. When only a subset of the sessions are leaky bucket constrained succinct per-session bounds that are independent of the behavior of the other sessions and also of the network topology are...

The problem of allocating network resources to the users of an integrated services network is investigated in the context of rate-based flow control. The network is assumed to be a virtual circuit, connection-based packet network. It is shown that the use of generalized processor sharing (GPS), when combined with leaky bucket admission control, all...

The technical core of a precompetitive consortium formed by
AT&T, DEC, and MIT to study the technology, architecture and
applications of wideband all-optical networks of local to national (or
international) extent is described. A general introduction to
all-optical networks is given, and some proposed applications are
discussed. The architecture, t...

The problem of allocating network resources to the users of an integrated services network is investigated in the context of rate based flow control. The authors propose the use of a packet service discipline at the nodes of the network that is based on a multiplex scheme called generalized processor sharing (GPS). This service discipline is combin...

The authors analyze the expected delay for infinite precision arithmetic codes and suggest a practical implementation that concentrates on the issue of delay.< >

An algorithm is presented that allows each node in a computer
network to maintain a correct view of the network topology despite link
and node failures. Reliability is achieved without transmitting any
information other than the operational status of links. Messages are
only sent in response to topological changes: periodic retransmission is
not re...

The research from 1984 to 1986 on Data Network Reliability had the objective of developing general principles governing the reliable and efficient control of data networks. The research was centered around three major areas: congestion control, multiaccess networks, and distributed asynchronous algorithms. The major topics within congestion control...

A broadcast network of N +1 nodes is considered in which
each binary digit transmitted by each node is received by every other
node via a binary symmetric channel of given transition probability. The
errors on these channels are independent over transmitters, receivers
and time. Each node has a binary state, and the problem is to construct
a distri...

This is a slightly edited version of a paper written, but not published, in the mid-eighties. Its age is symbolic of my much older friendship and respect for Dave Forney, and it is dedicated to that friendship. It’s age should not be taken as any claim for precedence of the results, but merely to excuse the lack of up-to-date references.
We conside...

Caption title. "November 1987." Includes bibliographical references. Supported in part by a contract from the National Science Foundation. NSF-ECS-8310698 Supported in part by a contract from ARO. DAAL03-86-K-0171 Robert G. Gallager.

This paper develops a new distributed BFS algorithm for an asynchronous communication network. This paper presents two new BFS algorithms with improved communication complexity. The first algorithm has complexity O((E+V1.5)Â¿logV) in communication and O(V1.5Â¿logV) in time. The second algorithm uses the technique of the first recursively and achiev...

Consider a broadcast network of N nodes in which each binary digit transmitted by each node is received by each other node via a binary symmetric channel whose crossover probability is independent over transmitters, receivers, and time. Each node has a binary state and the problem is to construct a distributed algorithm to find the parity of the se...

A new distributed shortest path algorithm for an asynchronous
communication network with unit edges, of the number of elementary
messages used in finding shortest paths form a given root to all other
nodes is 0(V1.6+E) where V is the number of nodes and E the number of
edges. For dense networks, with E > or = V to 1.6 power, this order
of complexit...

Bibliography: p. [65-69] "January 1985." "Grant NSF-ECS-8310698" "...DARPA...Contract ONR/N00014-84-K-0357" R.G. Gallager.

We propose a class of algorithms for finding an optimal quasi-static routing in a communication network. The algorithms are based on Gallager's method [1] and provide methods for iteratively updating the routing table entries of each node in a manner that guarantees convergence to a minimum delay routing. Their main feature is that they utilize sec...

A distributed algorithm is presented that constructs the minimum-weight spanning tree in a connected undirected graph with distinct edge weights. A processor exists at each node of the graph, knowing initially only the weights of the adjacent edges. The processors obey the same algorithm and exchange messages with neighbors until the tree is constr...

A distributed algorithm is presented that constructs the minimum-weight spanning tree in a connected undirected graph with distinct edge weights. A processor exists at each node of the graph, knowing initially only the weights of the adjacent edges. The processors obey the same algorithm and exchange messages with neighbors until the tree is constr...

The control of data communication networks (and any other large distributed systems) must be at least partly distributed because of the need to make observations and exert control at the various nodes of the network. When one also considers the desireability of letting networks grow (or shrink due to failures), it is reasonable to consider control...

The multiaccess problem as characterized by a finite user population and
a time-slotted channel with limited feedback is examined. Results
pertaining to the relationship among commonly used performance measures
are derived, and insight as to the nature of and difficulty in finding
an optimal protocol is given. In addition, three restricted but
reas...

In honor of the twenty-fifth anniversary of Huffman coding, four new results about Huffman codes are presented. The first result shows that a binary prefix condition code is a Huffman code iff the intermediate and terminal nodes in the code tree can be listed by nonincreasing probability so that each node in the list is adjacent to its sibling. The...

Bibliography: leaf 25. "September 1978." Suppported in part by Codex Corporation. Supported in part by the Advanced Research Project Agency (monitored by ONR) under Contract no. N00014-75-C-1183 A. Segall, P.M. Merlin and R.G. Gallager.

Two familiar techniques for encoding message lengths are considered. One technique breaks messages into packets, with each but the last message packet having the same length. The message length is encoded by specifying the last packet and its length. The other technique uses a special bit sequence called a flag to terminate the message and slightly...

Given the traffic flow from each source to each destination in a network and given the aggregate traffic in each link, we want to find if there is any looping of traffic. A careful definition of looping shows that the question is equivalent to whether some of the aggregate link flows can be reduced without increasing any of the others. It is then s...

An algorithm is defined for establishing routing tables in the individual nodes of a data network. The routing table at a node i specifies, for each other node j , what fraction of the traffic destined for node j should leave node i on each of the links emanating from node i . The algorithm is applied independently at each node and successively upd...

Caption title. Includes bibliographical references. Supported in part by ARPA under Grant ONR/N00014-75-C-1183 National Science Foundation Grant NSF/ENG76-24447 Robert G. Gallager.

We consider basic limitations on the amount of protocol information that must be transmitted in a data communication network to keep track of source and receiver addresses and of the starting and stopping of messages. Assuming Poisson message arrivals between each communicating source-receiver pair, we find a lower bound on the required protocol in...

Let P(i)= (1 - theta)theta^i be a probability assignment on the set of nonnegative integers where theta is an arbitrary real number, 0 < theta < 1 . We show that an optimal binary source code for this probability assignment is constructed as follows. Let l be the integer satisfying theta^l + theta^{l+1} leq 1 < theta^l + theta^{l-1} and represent e...

The Huffman source coding algorithm is a well-known algorithm for encoding the letters of a finite source alphabet into a uniquely decipherable code of minimum expected codeword length. Since the algorithm operates by successively ″merging″ the least probable letters in the alphabet, it cannot be directly applied to infinite source alphabets. In th...

A simple algorithm is developed for mapping the outputs of a source into the set of code sequences generated by a tree code. The algorithm is analyzed for a special case in which the source produces discrete independent equiprobable letters, and the distortion measure satisfies a symmetry condition. Letting R be the code rate and D^{ast} be the min...

New lower bounds are presented for the minimum error probability that can be achieved through the use of block coding on noisy discrete memoryless channels. Like previous upper bounds, these lower bounds decrease exponentially with the block length N. The coefficient of N in the exponent is a convex function of the rate. From a certain rate of tran...

Upper bounds are derived on the probability of error that can be achieved by using block codes on general time-discrete memoryless channels. Both amplitude-discrete and amplitude-continuous channels are treated, both with and without input constraints. The major advantages of the present approach are the simplicity of the derivations and the relati...

A low-density parity-check code is a code specified by a parity-check matrix with the following properties: each column contains a small fixed number j geq 3 of l's and each row contains a small fixed number k > j of l's. The typical minimum distance of these codes increases linearly with block length for a fixed rate and fixed j . When used with m...

Caption title. Includes bibliographical references (p. 18). Supported by the NSF, DARPA and ATT Bell Laboratories. Jane M. Simmons and Robert G. Gallager.

Caption title. Includes bibliographical references (p. 11). David Tse, Robert Gallager and John Tsitsiklis.

We present a new algorithm which creates redundant trees on arbitrary node-redundant or link-redundant networks. These trees are such that any node is connected to the common root of the trees by at least one of the trees in case of node or link failure. Our scheme provides rapid preplanned recovery of communications with great flexibility in the t...

Cover title. Includes bibliographical references (p. 14). Research supported by the Army Research Office. DAAL03-86-K-0171 I. Emre Telatar, Robert G. Gallager.

Caption title. Includes bibliographical references (p. 47-48). Research supported by a Vinton Hayes Fellowship. Abhay K. Parekh and Robert G. Gallager.

## Citations

... Assuming the underlying graph is ergodic, eachÃ (k) possesses the same right-eigenvector p 1 = 1 with corresponding eigenvalue λÃ (k) 1 = 1. All other eigenvalues are strictly less in absolute value as given by the Perron-Frobenius theorem [38] and stochastic processes [39]. Given a minimum edge weight bound ϵ > 0, we build on the insight from Wu et al. [23] that any product Π ∞ k=0Ã (k) also converges to a matrix with constant rows. ...

... This model contains the following parameters. Let S=S1, S2………SN where S=state, S1 is the first state, S2 is the second state and so on [34]. where Pij characterized to move likelihood from Si to Sj. Geometrically, Pij is just significant if Si, Sj are neighborhood states. ...

... A distribution π = [π 1 , . . . , π K ] on X = [K] is said to be a stationary or invariant distribution of the Markov chain if πP = π [29]. ...

Reference: Missing Mass Estimation from Sticky Channels

... However, this is rarely the case in a real system, where channel estimation errors result in a performance degra- dation [17]–[19]. Of particular relevance in this work is the fact that the SR, once the PM is successfully decoded, cannot exactly remove its interference from the received signal, due to the mismatch between the true channel and its estimate, resulting in a residual error term which degrades the performance of interference cancellation techniques [18]. This can be circumvented by defining a worst case model where the channel is replaced by its estimate, and the residual interference from channel mismatch is modeled as additive Gaussian noise [19]. ...

... The traditional statistical point process is the basis of the neural point process, which can be roughly organized into five categories: Poisson process [49], enhanced Poisson process [50], Hawkes process [14], active point process [51] and self-tuning process [52]. The simplest and most basic model is Poisson process, whose CIF is constant. ...

... Moreover, it was noted in [1] that it is rather easy to translate a self-stabilizing spanning tree construction protocol into a reset protocol. As mentioned below, the important work of Spinelli and Gallager [48] can also be transformed into a reset protocol if we ignore the complexity. ...

... The completely fair scheduler (CFS) has been the standard scheduler for desktop tasks since its introduction to the Linux kernel v2.6.23. CFS implements the weighted fair queuing policy [49] but also provides some prioritization via the nice value which is essentially a priority field. CFS attempts to emulate the fairness of generalized processor sharing (an ideal policy in which the time is infinitely divisible) through the concept of "virtual runtime". ...

... The problem of scheduling packets from multiple deterministic traffic flows through an IQ or CIOQ switch, to achieve strict QoS guarantees, is another well-known problem which is summarized in [49], [56]- [59]. It has been shown that IQ and CIOQ switches can achieve up to 100% throughput, when carrying deterministic traffic flows along with strict QoS guarantees, however the scheduling algorithms are often NP-complete. ...

... Two arc-disjoint trees may traverse a common link in opposite directions only, and they can provide 1:1 dedicated protection, i.e., resources for the backup tree are reserved and are used to carry bit streams when any link fails [9]. Earlier works in [11], [12] have exploited the notion of arc-disjoint trees to develop algorithms to compute primary and backup trees in an undirected graph. In [13], it is shown how arc-disjoint trees can protect a multicast session from any fiber cut in a network even when the link traversed by both primary and backup tree fails. ...

... It is less simple to choose the direction of the added paths and, hence, the and directed sub-graphs. The technique we present relies in part on results presented [6], [19], [21], and [22]. We now present the algorithm followed by the proof of its correctness. ...