## Sense of Direction in Distributed Computing (1998)

Venue: | In 12th International Symposium on Distributed Computing (DISC |

Citations: | 27 - 10 self |

### BibTeX

@INPROCEEDINGS{Flocchini98senseof,

author = {Paola Flocchini and Bernard Mans and Nicola Santoro},

title = {Sense of Direction in Distributed Computing},

booktitle = {In 12th International Symposium on Distributed Computing (DISC},

year = {1998},

pages = {1--15},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

Sense of Direction is a property of labeled graphs which has been shown to have a definite impact on computability and complexity in systems of communicating entities, and whose applicability ranges from the analysis of graph classes to distributed object systems. The full consequences of this property are still not known; in fact, the ongoing investigations continue to bring new (often surprising) results, to establish unsuspected links with other research and/or application areas, and to pose more questions than they answer. The aim of this paper is to provide a view of the current status of research, describing some of the relevant results, and providing pointers to future research directions. 1 Introduction In its more general formulation, a distributed system is a collection of computational entities communicating by exchanging finite amounts of information, which we shall call messages. The exact nature of the entities (i.e., processors, processes, network nodes, agents,...

### Citations

1533 |
Distributed Algorithms
- LYNCH
- 1996
(Show Context)
Citation Context ...h entity has local memory and processing capabilities. Different models exist, depending on how the communication is achieved. We shall restrict ourselves to systems based on the point-to-point model =-=[46]-=-. In such systems, each entity can communicate directly and individually with a subset of the other entities, called its neighbors, and can distinguish between them. For such systems, the communicatio... |

85 | Computing on an Anonymous Ring
- Attiya, Snir, et al.
- 1988
(Show Context)
Citation Context ...t of research has been devoted to the study of computability in anonymous systems; i.e., the study of what problems can be solved when there are no distinct identifiers associated to the nodes (e.g., =-=[2, 9, 5, 42, 43, 51, 68]-=-). Clearly, which problems can be solved depends on many factors including the structural properties of the system as well as the amount and type of structural knowledge available to the system entiti... |

64 | T.: Computing on anonymous networks: Part I – characterizing the solvable cases
- Yamashita, Kameda
- 1996
(Show Context)
Citation Context ...t of research has been devoted to the study of computability in anonymous systems; i.e., the study of what problems can be solved when there are no distinct identifiers associated to the nodes (e.g., =-=[2, 9, 5, 42, 43, 51, 68]-=-). Clearly, which problems can be solved depends on many factors including the structural properties of the system as well as the amount and type of structural knowledge available to the system entiti... |

42 |
D.B.: Election in complete networks with a sense of direction
- Matsushita
- 1988
(Show Context)
Citation Context ... went mostly unnoticed; those who did notice were puzzled. The solution to the puzzle was provided by the result that, with chordal SD, the election process can be carried out with only O(n) messages =-=[45]-=-; this was the first clear indication of the impact of SD. This result 8 Depth First Traversal Election Fault-Tolerant Election local orientation with topological awareness O(n 2 ) \Theta(n log n) [41... |

35 | Symmetry breaking in anonymous networks: characterizations
- Codenotti, Gemmell, et al.
- 1996
(Show Context)
Citation Context ...t of research has been devoted to the study of computability in anonymous systems; i.e., the study of what problems can be solved when there are no distinct identifiers associated to the nodes (e.g., =-=[2, 9, 5, 42, 43, 51, 68]-=-). Clearly, which problems can be solved depends on many factors including the structural properties of the system as well as the amount and type of structural knowledge available to the system entiti... |

35 | A modular technique for the design of efficient distributed leader finding algorithms
- Korach, Kutten, et al.
- 1986
(Show Context)
Citation Context ...impler problems can be used as building blocks for solving the complex ones; for instance, a solution to SPT can be constructed starting from any network traversal protocol (e.g., a solution to DFT ) =-=[40]-=-. Furthermore, large-scale problems and applications are likely to be (and are) modularly decomposed in terms of these basic problems; hence, the intrinsic cost of solving such basic problems has a di... |

33 |
Sense of Direction, Topological Awareness and Communication Complexity
- Santoro
- 1984
(Show Context)
Citation Context ...complexity equivalences and relationships among some of these problems. For example, in absence of any type-T knowledge (and assuming distinct identities), the problems LE , SPT and MF are equivalent =-=[59]-=-; the communication complexity of B is not greater than that of LE [26]; and, clearly, any solution to DFT is also a solution to B. 6 consistency Broadcast Election Depth First Traversal Spanning Tree... |

32 | Optimal Election in Labeled Hypercubes
- Flocchini, Mans
- 1993
(Show Context)
Citation Context ...bitrary S; jSj = k O(nk + n log n) ? Table 4: Chordal Ring. Broadcast Election local orientation O(n) [15, 22] O(n log log n) [19] neighboring SD O(n) [3] O(n log log n) dimensional SD O(n) \Theta(n) =-=[24, 57, 64, 66]-=- chordal SD O(n) O(1) [24] arbitrary SD O(n) O(n) [18] Table 5: Hypercube. sense of direction. However, the same bound has been obtained assuming only local orientation and topological awareness [15, ... |

32 | Sense of direction: definitions, properties, and classes - Flocchini, Mans, et al. - 1998 |

26 | Optimal distributed algorithms in unlabel tori and chordal rings
- Mans
- 1997
(Show Context)
Citation Context ...o identified. Subsequent investigations have reduced the size of the structure necessary to achieve linear election algorithms from O(log n) to O(log log n) [39], to O(log log log n) [54], to O(1) in =-=[47, 65]-=-. 3.3.3 Hypercube Network The results for broadcasting in the hypercube show a general insensitivity of this topology to sense of direction. In fact, broadcasting can be performed with O(n) messages i... |

24 | Complexity of deciding sense of direction
- Boldi, Vigna
- 1999
(Show Context)
Citation Context ...SD that comprises contracted, chordal, and cartographic SDs has been de ned in [48]. It has been shown that WSD does not imply SD; in fact there are labeled graphs which haveWSD but do not have anySD =-=[4]-=-. 1.3 Sense of Direction and Local Names The de nition of both coding and decoding (and, thus, of sense of direction) can be restated in terms of \translation" capability of local names. Each nodexref... |

23 |
Towards optimal distributed election on chordal rings
- KALAMBOUKIS, MANTZARIS
- 1991
(Show Context)
Citation Context ...und with the same number of chords were also identified. Subsequent investigations have reduced the size of the structure necessary to achieve linear election algorithms from O(log n) to O(log log n) =-=[39]-=-, to O(log log log n) [54], to O(1) in [47, 65]. 3.3.3 Hypercube Network The results for broadcasting in the hypercube show a general insensitivity of this topology to sense of direction. In fact, bro... |

23 | On the Impact of Sense of Direction in Arbitrary Networks
- Mans, Santoro
- 1994
(Show Context)
Citation Context ...ow. Arbitrary Networks Broadcast Election Depth First Traversal Spanning Tree Min Finding local orientation (e) (e +n logn) [24, 43] neighboring SD (min(e;n 1+ (1) )) [3, 24]O(e+nlogn) chordal SD (n) =-=[35]-=-O(e +n logn) any SD 2n, 2 [17] 3n logn +O(n) [17, 36,35] Here and in the followingndenotes the number of nodes of the network ande the number of edges.sNote that Depth First Traversal can be performed... |

22 | On the impact of sense of direction on message complexity
- Flocchini, Mans, et al.
- 1997
(Show Context)
Citation Context ... For example, in absence of any type-T knowledge (and assuming distinct identities), the problems LE , SPT and MF are equivalent [59]; the communication complexity of B is not greater than that of LE =-=[26]-=-; and, clearly, any solution to DFT is also a solution to B. 6 consistency Broadcast Election Depth First Traversal Spanning Tree Minimum Finding local orientation \Omega\Gamma e) [26] \Omega\Gamma e ... |

21 |
A Distributed Algorithm for Minimum Spanning Tree
- Gallager, Humblet, et al.
- 1983
(Show Context)
Citation Context ...led graphs and in absence of additional type-T knowledge, the three problems LE , SPT , and MF are equivalent and require\Omega\Gamma e+n log n) messages [58]; such a bound can be achieved (e.g., see =-=[35]-=-). An improvement was shown to be possible if each entity knows the identities of all its neighbors, that is in presence of the neighboring sense of direction; in this case, O(n log n) messages suffic... |

20 | A tradeoff between information and communication in broadcast protocols - Awerbuch, Goldreich, et al. - 1990 |

19 |
Efficient algorithms for elections in meshes and complete networks
- Peterson
- 1985
(Show Context)
Citation Context ...ch area. The knowledge to date is limited to some vertex-symmetric regular graphs of constant degree. In particular, the election problem can be solved with the exchange of \Theta(n) messages in tori =-=[55, 47]-=-, in the class of chordal rings h1; p ni n [47], as well as in butterflies and cube-connected-cycles [21]. Broadcast can be easily performed with O(n) messages in all those graphs, since they are spar... |

18 |
Graph Traversal Techniques and the Maximum Flow Problem in Distributed Computation
- Cheung
- 1983
(Show Context)
Citation Context ...knowledge, both the broadcast and the depth first traversal problems require\Omega\Gamma e) messages even if the entities have distinct identities [26]; such a bound can be easily achieved (e.g., see =-=[14]-=-). An improvement in the message complexity had been shown to exist if each entity has a distinct identity and knows the identities of all its neighbors; in this case, O(n) messages suffice to solve 7... |

18 | Leader election in the presence of link failures
- SINGH
- 1996
(Show Context)
Citation Context ...]. Subsequent research has focused on improvements of the constant factor of the message complexity and on improvements of the time complexity [49, 62], and adding fault-tolerance to the requirements =-=[48, 50, 52, 61]. In the r-=-esults for fault-tolerant election, f is the number of faulty processors, k is the number of initiators, and the faults are "fail-stop". In the case of synchronous complete networks, a good ... |

18 | Network Orientation
- Tel
- 1994
(Show Context)
Citation Context ...osts of many distributed algorithms, the problem of its construction (e.g., in preprocessing phase) is of obvious relevance. The problem of constructing SD in an unlabeled network has been studied in =-=[63, 65]-=-. It has been shown that any algorithm constructing the traditional sense of direction for cliques, hypercubes, 15 Cost Type of SD Complete Network \Theta(n 2 ) [63] chordal SD Hypercube \Theta(n log ... |

17 |
Efficient elections in chordal ring networks
- Attiya, Leeuwen, et al.
- 1989
(Show Context)
Citation Context ...ng is known only for the subset S of the edges. The first result was the observation that the chord structure S = f1; 2; : : : ; kg with k = O(log n) suffice to obtain a \Theta(n) election algorithms =-=[1]-=-; other chord structures achieving the same bound with the same number of chords were also identified. Subsequent investigations have reduced the size of the structure necessary to achieve linear elec... |

17 |
A fully distributed (minimal) spanning tree algorithm
- Lavalléé, Roucairol
- 1986
(Show Context)
Citation Context ...oadcast Election Depth First Traversal Spanning Tree Minimum Finding local orientation \Omega\Gamma e) [26] \Omega\Gamma e + n log n) [58] neighboring SD \Theta(min(e; n 1+\Theta(1) )) [3] O(n log n) =-=[44]-=- \Theta(n) [60] chordal SD \Theta(n) [25] O(n log n) [48] c-group SD \Theta(n) [65] arbitrary SD 2n \Gamma 2 [26] 3n log n +O(n) [26] Table 1: Universal Protocols The studies have concentrated both on... |

16 |
Corrigenda: Corrections to a distributed depth-first search algorithm
- Kumar, Iyengar, et al.
- 1990
(Show Context)
Citation Context ...n Depth First Traversal Spanning Tree Minimum Finding local orientation \Omega\Gamma e) [26] \Omega\Gamma e + n log n) [58] neighboring SD \Theta(min(e; n 1+\Theta(1) )) [3] O(n log n) [44] \Theta(n) =-=[60]-=- chordal SD \Theta(n) [25] O(n log n) [48] c-group SD \Theta(n) [65] arbitrary SD 2n \Gamma 2 [26] 3n log n +O(n) [26] Table 1: Universal Protocols The studies have concentrated both on the inherent l... |

15 | Computing vector functions on anonymous networks
- Boldi, Vigna
- 1997
(Show Context)
Citation Context |

15 | Minimal sense of direction and decision problems for Cayley graphs
- Boldi, Vigna
- 1997
(Show Context)
Citation Context ...tion of minimal SD in terms of Cayley graphs: a regular graph with symmetric labeling has a minimal SD iff it is a Cayley graph with a Cayley labeling [29]; this result holds also for directed graphs =-=[10]-=-. b b b a a a b a t t t t Figure 8: A minimal SD with a non-symmetric labeling. If the graph is not symmetric, this necessary and sufficient condition does not hold. Consider, for example, the labeled... |

15 | Computing on anonymous networks with sense of direction
- Flocchini, Roncato, et al.
- 2003
(Show Context)
Citation Context ... the system as well as the amount and type of structural knowledge available to the system entities (the nodes). The computational power of sense of direction in anonymous systems has been studied in =-=[28]-=- and shown to be linked to the notion of surrounding, introduced in [29]. 11 Before describing the notion of surrounding, we need the following definition of labeled graph isomorphism. Given two label... |

15 | Topological constraints for sense of direction
- Flocchini, Santoro
- 1998
(Show Context)
Citation Context ... the motivation for studying the interplay between the topology of the system and the properties that a labeling must satisfy for having sense of direction. Such an interplay has been investigated in =-=[31, 32]-=-, where the following questions have been considered and answered: ffl In which graphs, every labeling guarantees the existence of sense of direction? ffl For which graphs, the fact that the labeling ... |

15 |
Sense of Direction in Processor Networks
- Tel
- 1995
(Show Context)
Citation Context ...((x; y); (x 0 ; y 0 )) = (x + x 0 ; y + y 0 ). Other examples of labelings with SD, not necessarily universal, can be found in [27]; an elegant and compact definition of a large class can be found in =-=[65]-=-. 2.4 Translation of Local Views The notion of consistent coding, decoding, and sense of direction, can be rephrased in terms of local names. Each entity x refers to the other nodes using local names ... |

14 | Minimal sense of direction in regular networks
- Flocchini
- 1997
(Show Context)
Citation Context ...irection is that, for every string ff 2 \Sigma + of labels, there exists a walk with that sequence of labels; actually, there is one starting from every node. 16 7.1 Cycle Symmetry and Minimality. In =-=[23]-=- it has been shown that the existence of a minimal sense of direction is related to the notion of cycle symmetry of a graph: a graph is cycle symmetric if each node belongs to the same number of cycle... |

14 |
Time-message trade-offs for the weak unison problem, Nordic
- Israeli, Kranakis, et al.
- 1997
(Show Context)
Citation Context ...\Theta(n + kf) [50] arbitrary SD \Theta(n) [26] \Theta(n) [18] ? Table 2: Complete Networks. Wake-Up Time \Theta Bits local orientation \Theta(n 2 ) with topological awareness chordal SD O(n log 2 n) =-=[37]-=- Table 3: Synchronous Complete Network. implies that consistency of the labeling is the key to exploit the greater hardware capabilities of the complete network with respect to the ring; these implica... |

13 | Snap-stabilizing PIF algorithm in tree networks without sense of direction
- Bui, Datta, et al.
- 1999
(Show Context)
Citation Context ...of the positive impact is already quite strong, as this paper hopes to have shown. For example, the notion of sense of direction has just recently found applications in the area of self-stabilization =-=[12, 13]-=-. The scope of the investigations on sense of direction and on the consistency of the labelings continues to expand. For example, research has recently focused to consider cases where the communicatio... |

13 |
On the message complexity of Distributed Problems
- Santoro
- 1984
(Show Context)
Citation Context ...rly, any solution to DFT is also a solution to B. 6 consistency Broadcast Election Depth First Traversal Spanning Tree Minimum Finding local orientation \Omega\Gamma e) [26] \Omega\Gamma e + n log n) =-=[58]-=- neighboring SD \Theta(min(e; n 1+\Theta(1) )) [3] O(n log n) [44] \Theta(n) [60] chordal SD \Theta(n) [25] O(n log n) [48] c-group SD \Theta(n) [65] arbitrary SD 2n \Gamma 2 [26] 3n log n +O(n) [26] ... |

12 |
The Optimality of Distributive Constructions of Minimum Weight and Degree Restricted Spanning Trees in a Complete Network of Processors
- Korach, Moran, et al.
- 1987
(Show Context)
Citation Context ...e of direction, \Omega\Gamma n 2 ) messages are required, even if there is a leader; this follows from the lower bound on the construction of degree-restricted spanning-trees of the complete graph of =-=[41]-=-. On the other hand, with any sense of direction, this problem can be solved in O(n) messages using the universal protocol of [26], even if the network is anonymous. Historically, the election problem... |

11 |
Choosing a leader on a hypercube
- Robbins, Robbins
- 1990
(Show Context)
Citation Context ...bitrary S; jSj = k O(nk + n log n) ? Table 4: Chordal Ring. Broadcast Election local orientation O(n) [15, 22] O(n log log n) [19] neighboring SD O(n) [3] O(n log log n) dimensional SD O(n) \Theta(n) =-=[24, 57, 64, 66]-=- chordal SD O(n) O(1) [24] arbitrary SD O(n) O(n) [18] Table 5: Hypercube. sense of direction. However, the same bound has been obtained assuming only local orientation and topological awareness [15, ... |

11 |
Tight lower and upper bounds for a class of distributed algorithms for a complete network of processors
- Korach, Moran, et al.
- 1984
(Show Context)
Citation Context ...e size of messages is unbounded. 3.2 Regular Networks Complete Networks Broadcast Depth First Traversal Election local orientation (n , 1)O(n 2 ) (n 2 ) [24] local orientation (n , 1)O(n 2 ) (n logn) =-=[28]-=- with topological awareness chordal SD (n , 1) (n , 1) (n) [31, 37, 45] The results for the Leader Election problem show explicitly that, despite maximum connectivity, there are strict (reachable) low... |

10 | P.: Linear Broadcasting and N log log N Election in Unoriented Hypercubes
- Dobrev
- 1997
(Show Context)
Citation Context ... log n) O(n log n) [54] S = f1; p ng O(n) [47] O(n) Fabri&Tel95 in [65] arbitrary S; jSj = k O(nk + n log n) ? Table 4: Chordal Ring. Broadcast Election local orientation O(n) [15, 22] O(n log log n) =-=[19]-=- neighboring SD O(n) [3] O(n log log n) dimensional SD O(n) \Theta(n) [24, 57, 64, 66] chordal SD O(n) O(1) [24] arbitrary SD O(n) O(n) [18] Table 5: Hypercube. sense of direction. However, the same b... |

10 |
Effect of preprocessing on election in a complete network with a sense of direction
- Masapati, Ural
- 1991
(Show Context)
Citation Context ... 8 Depth First Traversal Election Fault-Tolerant Election local orientation with topological awareness O(n 2 ) \Theta(n log n) [41] \Omega\Gamma n log k + kf) [38] chordal SD \Theta(n) [25] \Theta(n) =-=[45, 49, 62]-=- \Theta(n + kf) [50] arbitrary SD \Theta(n) [26] \Theta(n) [18] ? Table 2: Complete Networks. Wake-Up Time \Theta Bits local orientation \Theta(n 2 ) with topological awareness chordal SD O(n log 2 n)... |

10 | About the local detection of termination of local computations in graphs
- Métivier, Muscholl, et al.
- 1997
(Show Context)
Citation Context |

10 | Efficient leader election using sense of direction
- Singh
- 1997
(Show Context)
Citation Context ... 8 Depth First Traversal Election Fault-Tolerant Election local orientation with topological awareness O(n 2 ) \Theta(n log n) [41] \Omega\Gamma n log k + kf) [38] chordal SD \Theta(n) [25] \Theta(n) =-=[45, 49, 62]-=- \Theta(n + kf) [50] arbitrary SD \Theta(n) [26] \Theta(n) [18] ? Table 2: Complete Networks. Wake-Up Time \Theta Bits local orientation \Theta(n 2 ) with topological awareness chordal SD O(n log 2 n)... |

9 | Object identifiers, keys, and surrogates - objects identifiers revisited. Theory and Practice of Object Systems
- WIERINGA, J
(Show Context)
Citation Context ...uld be stable and reliable, that is, names must remain valid, and keep denoting the same objects when the system changes its state. In order to achieve these goals, the existing naming schemes (e.g., =-=[36, 53, 67]-=-) either use an approach based on global names (e.g., each object uses as its name an intrinsic property which distinguishes it from the other objects) or they use hierarchical names based on the loca... |

8 |
Distributed traversal and broadcasting in arbitrary network with distance sense of direction
- Flocchini, Mans, et al.
- 1994
(Show Context)
Citation Context ...anning Tree Minimum Finding local orientation \Omega\Gamma e) [26] \Omega\Gamma e + n log n) [58] neighboring SD \Theta(min(e; n 1+\Theta(1) )) [3] O(n log n) [44] \Theta(n) [60] chordal SD \Theta(n) =-=[25]-=- O(n log n) [48] c-group SD \Theta(n) [65] arbitrary SD 2n \Gamma 2 [26] 3n log n +O(n) [26] Table 1: Universal Protocols The studies have concentrated both on the inherent limits (lower bounds) and o... |

8 | Optimal distributed t-resilient election in complete networks
- Itai, Kutten, et al.
- 1990
(Show Context)
Citation Context ...dication of the impact of SD. This result 8 Depth First Traversal Election Fault-Tolerant Election local orientation with topological awareness O(n 2 ) \Theta(n log n) [41] \Omega\Gamma n log k + kf) =-=[38]-=- chordal SD \Theta(n) [25] \Theta(n) [45, 49, 62] \Theta(n + kf) [50] arbitrary SD \Theta(n) [26] \Theta(n) [18] ? Table 2: Complete Networks. Wake-Up Time \Theta Bits local orientation \Theta(n 2 ) w... |

8 |
Distributed computing on anonymous hypercubes
- Kranakis, Krizanc
- 1997
(Show Context)
Citation Context |

8 |
A fault-tolerant algorithm for election in complete networks with a sense of direction
- Masuzawa, Nishikawa, et al.
- 1991
(Show Context)
Citation Context ...ction Fault-Tolerant Election local orientation with topological awareness O(n 2 ) \Theta(n log n) [41] \Omega\Gamma n log k + kf) [38] chordal SD \Theta(n) [25] \Theta(n) [45, 49, 62] \Theta(n + kf) =-=[50]-=- arbitrary SD \Theta(n) [26] \Theta(n) [18] ? Table 2: Complete Networks. Wake-Up Time \Theta Bits local orientation \Theta(n 2 ) with topological awareness chordal SD O(n log 2 n) [37] Table 3: Synch... |

8 |
Fault-tolerant distributed algorithm in complete networks with link and processor failures
- Nishikawa, Masuzawa, et al.
- 1991
(Show Context)
Citation Context ...]. Subsequent research has focused on improvements of the constant factor of the message complexity and on improvements of the time complexity [49, 62], and adding fault-tolerance to the requirements =-=[48, 50, 52, 61]. In the r-=-esults for fault-tolerant election, f is the number of faulty processors, k is the number of initiators, and the faults are "fail-stop". In the case of synchronous complete networks, a good ... |

8 |
A near-optimal multi-stage distributed algorithm for finding leaders in clustered chordal rings. Information Sciences
- Pan
- 1994
(Show Context)
Citation Context ...f chords were also identified. Subsequent investigations have reduced the size of the structure necessary to achieve linear election algorithms from O(log n) to O(log log n) [39], to O(log log log n) =-=[54]-=-, to O(1) in [47, 65]. 3.3.3 Hypercube Network The results for broadcasting in the hypercube show a general insensitivity of this topology to sense of direction. In fact, broadcasting can be performed... |

8 | Optimal fault-tolerant leader election in chordal rings
- Mans, Santoro
- 1998
(Show Context)
Citation Context ...irst Traversal Spanning Tree Min Finding local orientation (e) (e +n logn) [24, 43] neighboring SD (min(e;n 1+ (1) )) [3, 24]O(e+nlogn) chordal SD (n) [35]O(e +n logn) any SD 2n, 2 [17] 3n logn +O(n) =-=[17, 36,35]-=- Here and in the followingndenotes the number of nodes of the network ande the number of edges.sNote that Depth First Traversal can be performed inO(n) with any sense of direction even if the graph is... |

7 |
A tradeoff between information and communication in broadcast protocols
- Goldreich, Vainish
- 1988
(Show Context)
Citation Context ... consistency Broadcast Election Depth First Traversal Spanning Tree Minimum Finding local orientation \Omega\Gamma e) [26] \Omega\Gamma e + n log n) [58] neighboring SD \Theta(min(e; n 1+\Theta(1) )) =-=[3]-=- O(n log n) [44] \Theta(n) [60] chordal SD \Theta(n) [25] O(n log n) [48] c-group SD \Theta(n) [65] arbitrary SD 2n \Gamma 2 [26] 3n log n +O(n) [26] Table 1: Universal Protocols The studies have conc... |

7 | Self-stabilizing network orientation algorithms in arbitrary rooted networks
- Datta, Gurumurthy, et al.
(Show Context)
Citation Context ...of the positive impact is already quite strong, as this paper hopes to have shown. For example, the notion of sense of direction has just recently found applications in the area of self-stabilization =-=[12, 13]-=-. The scope of the investigations on sense of direction and on the consistency of the labelings continues to expand. For example, research has recently focused to consider cases where the communicatio... |

7 | Backward consistency and sense of direction in advanced distributed systems
- Flocchini, Roncato, et al.
(Show Context)
Citation Context ...larly interesting open problem is the characterization of the computability relationship between sense of direction and other forms of consistency; some results have already been recently obtained in =-=[30]-=-. 5 Impact of Sense of Direction on Systems: Object Naming Sense of direction can be used to construct an efficient naming scheme in systems of distributed objects [4]. A distributed object system con... |