The Universe of Symmetry Breaking Tasks

Abstract

Processes in a concurrent system need to coordinate using a shared memory or a message-passing subsystem in order to solve agreement tasks such as, for example, consensus or set agreement. However, often coordination is needed to "break the symmetry" of processes that are initially in the same state, for example, to get exclusive access to a shared resource, to get distinct names or to elect a leader.This paper introduces and studies the family of generalized symmetry breaking (GSB) tasks, that includes election, renaming and many other symmetry breaking tasks. Differently from agreement tasks, a GSB task is "inputless", in the sense that processes do not propose values; the task specifies only the symmetry breaking requirement, independently of the system's initial state (where processes differ only on their identifiers). Among many various characterizing the family of GSB tasks, it is shown that (non adaptive) perfect renaming is universal for all GSB tasks.Résumé : Dans un système réparti, les processus ont besoin de coordination en utilisant un sous-système de mémoire partagée ou de passage de message pour pouvoir résoudre des problèmes tels que le consensus ou l'accord ensembliste. Dans certains cas, la coordination est nécéssaire pour "casser la symétrie" entre des processus qui ont le mêmeétat initial.Ce rapport introduit la famille des tâches "generalized symmetry breaking" (GSB) qui inclut l'élection, le renommage et de nombreuses autres tâches qui cassent la symétrie.

Optimal Collision/Conflict-Free Distance-2 Coloring in Wireless Synchronous Broadcast/Receive Tree Networks

“…(18) when TERM(dest, id, sender cl) is received do % the tree rooted at process id is properly colored % (19) if (dest = idi) then discard the message (do not execute lines 20-21) end if; (20) to colori ← to colori \ {id}; d1colorsi ← d1colors i ∪ {sender cl}; (21) if (to colori = ∅) then statei ← 1 else statei ← 3 end if.…”

“…(09) when (CLOCK mod nb cl parenti) = colori) ∧ (statei ∈ {1, 3}) do (10) if (statei = 1) (11) then pairs f or childreni ← empty set of pairs; (12) palette ← sequence 0, 1, ... without the colors colori and sender cli; (13) for each k ∈ to colori do (14) cl ← first color in palette; (15) suppress cl from palette and add k, cl to pairs f or childreni (16) end for; (17) broadcast COLOR(pairs f or childreni, idi, colori, nb cli); statei ← 2 (18) else broadcast TERM(parenti, idi); statei ← 4 % pi and its neighbors are colored % (19) end if. (20) when TERM(dest, id) is received do (21) if (dest = idi) then discard the message (do not execute lines 22-25) end if; (22) to colori ← to colori \ {id}; Description of the algorithm To simplify the presentation, we assume that the initial value of CLOCK is −1. As in the sequential version, a single process p r receives the external message START(), that defines it as the root of the tree.…”

“…The distance-k coloring problem consists in assigning colors to the vertices of the graph such that no two vertices at distance at most k have the same color. Let us remember that minimum distance-1 vertex coloring is an NP-complete problem [16].Coloring is (with leader election [1] and renaming [2, 10]) one of the most important symmetry breaking problems encountered in distributed computing [20]. Solving such problems requires a pre-existing initial asymmetry from which the problem can be solved.…”

“…Coloring is (with leader election [1] and renaming [2, 10]) one of the most important symmetry breaking problems encountered in distributed computing [20]. Solving such problems requires a pre-existing initial asymmetry from which the problem can be solved.…”

“…when (CLOCK increases) (11) if (statei = 1) then colori ← the first color in 0, 1... which is not in d1colorsi ∪ d2colors i; (12) broadcast CORRECT(senderi, idi, colori, d1colorsi); statei ← 0 (13) end if; (14) if (statei = 2) then color f or child ← the first color in 0, 1.. which is not in d1colors i ∪ colori; (15) next ← any id k ∈ to colori; (16) broadcast COLOR(next, idi, colori, color f or child, d1colors i); statei ← 0 (17) end if; (18) if (statei = 3) then broadcast TERM(parenti, colori, idi); statei ← 0 end if; (19) if (statei = 4) then broadcast CORRECTED COLOR (senderi, parenti, idi, colori); statei ← 0 end if; (20) if (statei = 5) then broadcast RESUME COLORING (senderi, idi); statei ← 0 end if; (21) if (statei = 6) then broadcast CORRECTED COLOR (−1, −1, idi, corrected cli) ; statei ← 0 end if. when TERM(dest, id, color) is received do (22) if (dest = idi) then discard the message (do not execute lines 23-28) end if; (23) to colori ← to colori \ {id}; (24) if (to colori = ∅) % the neighbors of pi are properly colored % (25) then if (parenti = idi) then the root claims the graph is colored else statei ← 3 end if (26) else d1colorsi ← d1colors i ∪ {color}; next ← any id k ∈ to colori; (27) color to child ← the first color in 0, 1.. which is not in d1colorsi ∪ colori; statei ← 2 (28) end if.…”