Narrowing power vs efficiency in synchronous set agreement: Relationship, algorithms and lower bound
The k -set agreement problem is a generalization of the uniform consensus problem: each process proposes a value, and each non-faulty process has to decide a value such that a decided value is a proposed value, and at most k different values are decided. It has been shown that any algorithm that sol...
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| Veröffentlicht in: | Theoretical computer science Jg. 411; H. 1; S. 58 - 69 |
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| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
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Elsevier B.V
2010
Elsevier |
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| ISSN: | 0304-3975, 1879-2294 |
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| Abstract | The
k
-set agreement problem is a generalization of the uniform consensus problem: each process proposes a value, and each non-faulty process has to decide a value such that a decided value is a proposed value, and at most
k
different values are decided. It has been shown that any algorithm that solves the
k
-set agreement problem in synchronous systems that can suffer up to
t
crash failures requires
⌊
t
k
⌋
+
1
rounds in the worst case. It has also been shown that it is possible to design early deciding algorithms where no process decides and halts after
min
(
⌊
f
k
⌋
+
2
,
⌊
t
k
⌋
+
1
)
rounds, where
f
is the number of actual crashes in a run (
0
≤
f
≤
t
).
This paper explores a new direction to solve the
k
-set agreement problem in a synchronous system. It considers that the system is enriched with base objects (denoted has
[
m
,
ℓ
]
_SA objects) that allow solving the
ℓ
-set agreement problem in a set of
m
processes (
m
<
n
). The paper makes several contributions. It first proposes a synchronous
k
-set agreement algorithm that benefits from such underlying base objects. This algorithm requires
O
(
t
ℓ
m
k
)
rounds, more precisely,
⌊
t
Δ
⌋
+
1
rounds, where
Δ
=
m
⌊
k
ℓ
⌋
+
(
k
mod
ℓ
)
. The paper then shows that this bound, that involves all the parameters that characterize both the problem (
k
) and its environment (
t
,
m
and
ℓ
), is a lower bound. The proof of this lower bound sheds additional light on the deep connection between synchronous efficiency and asynchronous computability. Finally, the paper extends its investigation to the early deciding case. It presents a
k
-set agreement algorithm that directs the processes to decide and stop by round
min
(
⌊
f
Δ
⌋
+
2
,
⌊
t
Δ
⌋
+
1
)
. These bounds generalize the bounds previously established for solving the
k
-set agreement problem in pure synchronous systems. |
|---|---|
| AbstractList | The
k
-set agreement problem is a generalization of the uniform consensus problem: each process proposes a value, and each non-faulty process has to decide a value such that a decided value is a proposed value, and at most
k
different values are decided. It has been shown that any algorithm that solves the
k
-set agreement problem in synchronous systems that can suffer up to
t
crash failures requires
⌊
t
k
⌋
+
1
rounds in the worst case. It has also been shown that it is possible to design early deciding algorithms where no process decides and halts after
min
(
⌊
f
k
⌋
+
2
,
⌊
t
k
⌋
+
1
)
rounds, where
f
is the number of actual crashes in a run (
0
≤
f
≤
t
).
This paper explores a new direction to solve the
k
-set agreement problem in a synchronous system. It considers that the system is enriched with base objects (denoted has
[
m
,
ℓ
]
_SA objects) that allow solving the
ℓ
-set agreement problem in a set of
m
processes (
m
<
n
). The paper makes several contributions. It first proposes a synchronous
k
-set agreement algorithm that benefits from such underlying base objects. This algorithm requires
O
(
t
ℓ
m
k
)
rounds, more precisely,
⌊
t
Δ
⌋
+
1
rounds, where
Δ
=
m
⌊
k
ℓ
⌋
+
(
k
mod
ℓ
)
. The paper then shows that this bound, that involves all the parameters that characterize both the problem (
k
) and its environment (
t
,
m
and
ℓ
), is a lower bound. The proof of this lower bound sheds additional light on the deep connection between synchronous efficiency and asynchronous computability. Finally, the paper extends its investigation to the early deciding case. It presents a
k
-set agreement algorithm that directs the processes to decide and stop by round
min
(
⌊
f
Δ
⌋
+
2
,
⌊
t
Δ
⌋
+
1
)
. These bounds generalize the bounds previously established for solving the
k
-set agreement problem in pure synchronous systems. The k-set agreement problem is a generalization of the uniform consensus problem: each process proposes a value, and each non-faulty process has to decide a value such that a decided value is a proposed value, and at most k different values are decided. It has been shown that any algorithm that solves the k-set agreement problem in synchronous systems that can suffer up to t crash failures requires ?tk?+1 rounds in the worst case. It has also been shown that it is possible to design early deciding algorithms where no process decides and halts after min(?fk?+2,?tk?+1) rounds, where f is the number of actual crashes in a run (0?f?t). This paper explores a new direction to solve the k-set agreement problem in a synchronous system. It considers that the system is enriched with base objects (denoted has [m,?]_SA objects) that allow solving the ?-set agreement problem in a set of m processes (m < n). The paper makes several contributions. It first proposes a synchronous k-set agreement algorithm that benefits from such underlying base objects. This algorithm requires O(t?mk) rounds, more precisely, ?t?+1 rounds, where m?k??+(k mod ?). The paper then shows that this bound, that involves all the parameters that characterize both the problem (k) and its environment (t, m and ?), is a lower bound. The proof of this lower bound sheds additional light on the deep connection between synchronous efficiency and asynchronous computability. Finally, the paper extends its investigation to the early deciding case. It presents a k-set agreement algorithm that directs the processes to decide and stop by round min(?f?+2,?t?+1). These bounds generalize the bounds previously established for solving the k-set agreement problem in pure synchronous systems. The k-set agreement problem is a generalization of the uniform consensus problem: each process proposes a value, and each non-faulty process has to decide a value such that a decided value is a proposed value, and at most k different values are decided. It has been shown that any algorithm that solves the k-set agreement problem in synchronous systems that can suffer up to t crash failures requires t/k+1 rounds in the worst case. It has also been shown that it is possible to design early deciding algorithms where no process decides and halts after min(f/k+2, t/k+1) rounds, where f is the number of actual crashes in a run (f |
| Author | Travers, Corentin Mostéfaoui, Achour Raynal, Michel |
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| Keywords | Lower bound t -resilience Synchronous system Efficiency Round-based algorithm Set agreement Consensus Asynchronous Computer theory Solving Algorithm Failures Computability t-resilience Proof Problem solving Environment Failure Power t-Resilience |
| Language | English |
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| References | Fischer, Lynch (b11) 1982; 14 Herlihy, Shavit (b19) 1999; 46 Keidar, Rajsbaum (b20) 2003; 85 Attiya, Welch (b3) 2004 Raynal, Travers (b29) 2006 Saks, Zaharoglou (b30) 2000; 29 Dolev, Reischuk, Strong (b10) 1990; 37 Fischer, Lynch, Paterson (b12) 1985; 32 Herlihy, Rajsbaum (b18) 2000; 10 Awerbuch (b4) 1985; 32 Wang, Teo, Cao (b32) 2005; 96 Herlihy, Penso (b17) 2005; 18 Mostéfaoui, Raynal (b26) 2000 Santoro (b31) 2007 Mostéfaoui, Raynal (b27) 2001 Charron-Bost, Schiper (b7) 2004; 51 Aguilera, Toueg (b1) 1999; 71 Lynch (b22) 1996 Bazzi, Neiger (b5) 2001; 48 Gafni, Guerraoui, Pochon (b14) 2005 Chaudhuri, Herlihy, Lynch, Tuttle (b9) 2000; 47 Aguilera, Le Lann, Toueg (b2) 2002; vol. 2508 Mostéfaoui, Rajsbaum, Raynal, Travers (b24) 2008; 38 Mostéfaoui, Rajsbaum, Raynal (b25) 2006; 18 Borowsky, Gafni (b6) 1993 Mostéfaoui, Rajsbaum, Raynal (b23) 2003; 50 Herlihy (b16) 1991; 13 Gafni (b13) 1998 Raynal (b28) 2002 Guerraoui, Herlihy, Pochon (b15) 2009; 410 Chaudhuri (b8) 1993; 105 L. Lamport, M. Fischer, Byzantine generals and transaction commit protocols, Unpublished manuscript, 16 pages, April 1982 Aguilera (10.1016/j.tcs.2009.09.002_b1) 1999; 71 Charron-Bost (10.1016/j.tcs.2009.09.002_b7) 2004; 51 Awerbuch (10.1016/j.tcs.2009.09.002_b4) 1985; 32 Borowsky (10.1016/j.tcs.2009.09.002_b6) 1993 Herlihy (10.1016/j.tcs.2009.09.002_b16) 1991; 13 Attiya (10.1016/j.tcs.2009.09.002_b3) 2004 Mostéfaoui (10.1016/j.tcs.2009.09.002_b25) 2006; 18 Fischer (10.1016/j.tcs.2009.09.002_b11) 1982; 14 Herlihy (10.1016/j.tcs.2009.09.002_b17) 2005; 18 Mostéfaoui (10.1016/j.tcs.2009.09.002_b26) 2000 Herlihy (10.1016/j.tcs.2009.09.002_b19) 1999; 46 Mostéfaoui (10.1016/j.tcs.2009.09.002_b23) 2003; 50 Bazzi (10.1016/j.tcs.2009.09.002_b5) 2001; 48 Mostéfaoui (10.1016/j.tcs.2009.09.002_b24) 2008; 38 Gafni (10.1016/j.tcs.2009.09.002_b13) 1998 Guerraoui (10.1016/j.tcs.2009.09.002_b15) 2009; 410 Santoro (10.1016/j.tcs.2009.09.002_b31) 2007 Chaudhuri (10.1016/j.tcs.2009.09.002_b9) 2000; 47 Raynal (10.1016/j.tcs.2009.09.002_b28) 2002 Fischer (10.1016/j.tcs.2009.09.002_b12) 1985; 32 Keidar (10.1016/j.tcs.2009.09.002_b20) 2003; 85 Gafni (10.1016/j.tcs.2009.09.002_b14) 2005 Wang (10.1016/j.tcs.2009.09.002_b32) 2005; 96 Mostéfaoui (10.1016/j.tcs.2009.09.002_b27) 2001 Herlihy (10.1016/j.tcs.2009.09.002_b18) 2000; 10 Lynch (10.1016/j.tcs.2009.09.002_b22) 1996 Saks (10.1016/j.tcs.2009.09.002_b30) 2000; 29 Dolev (10.1016/j.tcs.2009.09.002_b10) 1990; 37 Chaudhuri (10.1016/j.tcs.2009.09.002_b8) 1993; 105 Raynal (10.1016/j.tcs.2009.09.002_b29) 2006 10.1016/j.tcs.2009.09.002_b21 Aguilera (10.1016/j.tcs.2009.09.002_b2) 2002; vol. 2508 |
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-set agreement problem is a generalization of the uniform consensus problem: each process proposes a value, and each non-faulty process has to decide a... The k-set agreement problem is a generalization of the uniform consensus problem: each process proposes a value, and each non-faulty process has to decide a... |
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| SubjectTerms | [formula omitted]-resilience Algorithmics. Computability. Computer arithmetics Applied sciences Artificial intelligence Computer Science Computer science; control theory; systems Consensus Distributed, Parallel, and Cluster Computing Efficiency Exact sciences and technology Logic and foundations Lower bound Mathematical logic, foundations, set theory Mathematics Miscellaneous Problem solving, game playing Recursion theory Round-based algorithm Sciences and techniques of general use Set agreement Synchronous system Theoretical computing |
| Title | Narrowing power vs efficiency in synchronous set agreement: Relationship, algorithms and lower bound |
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