09/04/2019 Distributed Algorithm Soyez Efficace, Rembobinez Be efficient, Rewind Stéphane Devismes (Verimag, UGA) Stephane.Devismes@univ-grenoble-alpes.fr Initial Configuration Colette Johnen (LaBRI, Univ. Bordeaux) johnen@labri.fr Goal of the algorithm Distributed Algorithm 1 2 Unison : clocks synchronization Unison : clocks synchronization v in the neighborhood of u, we have c(v) {c(u), c(u)+1%K} Each process u has a clock : c(u) c(u) := (c(u)+1) %K • the difference of clock values between neighbors is at most 1 (safety) • Each process increments its clock infinitely often 0 0 0 0 0 (liveness) 0 0 Distributed algorithm using a bounded clock – 0 0 0 0 0 a bound on network size is known (K) v in the neighborhood of u, we have c(v) {c(u), c(u)+1%K} c(u) := (c(u)+1) %K The initial configuration is a safe configuration The initial configuration is a safe configuration K > size of networks K > size of networks 3 4 Unison : clocks synchronization Unison : clocks synchronization v in the neighborhood of u, we have c(v) {c(u), c(u)+1%K} v in the neighborhood of u, we have c(v) {c(u), c(u)+1%K} c(u) := (c(u)+1) %K c(u) := (c(u)+1) %K 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 The initial configuration is a safe configuration The initial configuration is a safe configuration K > size of networks K > size of networks 5 6 1
09/04/2019 Unison : clocks synchronization Unison : clocks synchronization v in the neighborhood of u, we have c(v) {c(u), c(u)+1%K} v in the neighborhood of u, we have c(v) {c(u), c(u)+1%K} c(u) := (c(u)+1) %K c(u) := (c(u)+1) %K 1 2 1 1 2 2 2 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 The initial configuration is a safe configuration The initial configuration is a safe configuration K > size of networks K > size of networks 7 8 Unison : clocks synchronization Unison : clocks synchronization v in the neighborhood of u, we have c(v) {c(u), c(u)+1%K} v in the neighborhood of u, we have c(v) {c(u), c(u)+1%K} c(u) := (c(u)+1) %K c(u) := (c(u)+1) %K 2 2 3 2 2 2 3 2 3 3 1 2 1 2 1 0 0 0 1 1 1 0 1 1 The initial configuration is a safe configuration The initial configuration is a safe configuration K > size of networks K > size of networks 9 10 Unison : clocks synchronization Unison : clocks synchronization v in the neighborhood of u, we have c(v) {c(u), c(u)+1%K} v in the neighborhood of u, we have c(v) {c(u), c(u)+1%K} c(u) := (c(u)+1) %K c(u) := (c(u)+1) %K 4 3 3 4 4 3 3 3 3 4 2 3 2 3 2 1 0 1 2 2 1 0 1 2 The initial configuration is a safe configuration The initial configuration is a safe configuration K > size of networks K > size of networks 11 12 2
09/04/2019 Unison : clocks synchronization Unison : clocks synchronization v in the neighborhood of u, we have c(v) {c(u), c(u)+1%K} v in the neighborhood of u, we have c(v) {c(u), c(u)+1%K} c(u) := (c(u)+1) %K c(u) := (c(u)+1) %K 4 5 4 4 5 5 5 4 4 4 3 3 3 3 0 1 2 0 1 2 2 1 2 1 The initial configuration is a safe configuration The initial configuration is a safe configuration K > size of networks K > size of networks 13 14 Self-stabilizing algorithm Unison : clocks synchronization v in the neighborhood of u, we have c(v) {c(u), c(u)+1%K} Chaos c(u) := (c(u)+1) %K Self-stabilizing algorithm 5 4 4 5 6 3 3 2 1 0 1 2 Goal of the algorithm The initial configuration is a safe configuration K > size of networks 15 16 To rewind (To reset) To reuse Unison : clocks synchronization v in the neighborhood of u, we have c(v) {c(u), c(u)+1%K} Chaos c(u) := (c(u)+1) %K to reach an correct configuration 8 7 6 10 9 Correct Configuration 5 11 0 1 2 3 4 Goal of the algorithm DEADLOCK : the configuration is not a safe configuration Distributed Algorithm Unison algorithm is not self-stabilizing K > 12 17 18 3
09/04/2019 To rewind (To reset) To reuse Self-stabilizing reseting algorithm Chaos to reach a correct configuration Arora, Gouda: Distributed reset. IEEE Trans. Computer. 1994 • identified network • mono initiator Goal of the algorithm Distributed • weakly fair scheduler Algorithm SDR algorithm - Self-stabilizing Distributed Reset algorithm • Awerbuch, Patt-Shamir, Varghese: self-stabilization by local • anonymous network, checking and global reset (extended abstract). FOCS ’91 • no network knowledge • multi initiators • Awerbuch, Patt-Shamir, Varghese, Dolev : self-stabilization • unfair scheduler by local checking and global reset (extended abstract). • linear number of process moves WDAG’94 • Unbounded timestamps 19 20 Chaos Correct behavior Self stabilizing reseting Total mess I : distributed algorithm solving a task (static or dynamic) from a correct configuration Chaos Correct Configuration Chaos SDR o I reseting 21 COA DESCARTES/ESTATE - 2019 22 Correct configuration of I Local reseting in I I : distributed algorithm solving a task (static or dynamic) I : distributed algorithm solving a task (static or dynamic) from a correct configuration from a correct configuration Correct configuration of I Local reseting in I P_ICorrect(u) : predicate on u and u’s neighbor variables reset(u) : macro reseting the value of u’ variables P_ICorrect(u) is closed along I P_reset(u) : predicate on u’s variable : it is true iff u’s variable are reseted u we have (P_ICorrect(u) == true) the configuration is correct ( ≠ legitimate) If u and all u’s neighbors verify P_reset then P_ICorrect(u) == true Pr(u) is closed along I iff Pr(u) stays verified along any execution of I 23 24 4
09/04/2019 Unison – resetable ? SDR – Overall presentation v in the neighborhood of u, we have c(v) {c(u), c(u)+1%K} • A process u starts the reseting because c(u) := (c(u)+1) %K P_ICorrect(u) != true • The reseting is propagated (a DAG rooted at u is P_ICorrect(u) is the safety predicate built) v in the neighborhood of u, we have c(v) {c(u)-1%K, c(u), c(u)+1%K} • The DAG is frozen from the leaves to the initiator of reset(u) : c(u) := 0 the reseting (u) P_reset(u) : c(u) == 0 If u and all u’s neighbors verify c==0 then • Processes go back to the Initial algorithm/task from P_ICorrect(u)==true the initiator (u) to the leaves of the DAG K > size of networks 25 26 SDR – variables on u SDR – variables on u • To store the DAG structure : • To store the DAG structure : – d(u) : distance to an initiator of the reseting – d(u) : distance to an initiator of the reseting 27 0 18 27 28 SDR – variables on u SDR – variables on u • To store the DAG structure : • To store the DAG structure : – d(u) : distance to an initiator of the reseting – d(u) : distance to an initiator of the reseting • The status of the reseting : st(u) : RB, RF, or C • The status of the reseting : st(u) : RB, RF, or C – a reseting is in progress – a reseting is in progress • RB : propagation of the reseting • RB : propagation phase • RF : propagation of the ending of broadcast • RF : freezing phase phase – no reseting is in progress : C – no reseting is in progress : C 29 30 5
09/04/2019 SDR – Self-stabilizing Distributed Reset SDR – Self-stabilizing Distributed Reset R_R : a process u starts the reseting because R_B : the reseting is propagated (a DAG is built) P_ICorrect(u) != true or … R_B : If (st (u)==C) and a’ u neighbor v verified ( st(v)==RB) R_R: If (st(u)==C) and P_ICorrect(u) reset(u); st(u) := RB; d(u) := d(v)+1; reset(u); st(u) := RB; d(u) := 0; A process cannot join two times the same DAG structure A process performs at most one time R_R rule a process can be the root of a single DAG At most n DAG structure is built during any execution A process performs at most n-1 times the rule R_B ( n being the network size) ( n being the network size) 31 32 SDR – Self-stabilizing Distributed Reset SDR – Self-stabilizing Distributed Reset R_C : Processes goes back to the Initial algorithm/task R_F: the DAG is frozen from the leaves to the initiator of the reseting R_C: If (st(u)==RF), u is a root of the DAG and R_F: If (st(u)==RB) and the reseting is over in the u-DAG the reseting propagation is over in the DAG rooted at u st(u) := C; reset(u); st(u) := RF; A process takes the status RB at most n times A process takes the status RF at most n times A process performs at most n+1 times the rule R_F A process performs at most n times the rule R_C ( n being the network size) ( n being the network size) 33 34 Unison : clocks synchronization Unison : clocks synchronization v in the neighborhood of u, we have c(v) {c(u), c(u)+1%K} v in the neighborhood of u, we have c(v) {c(u), c(u)+1%K} c(u) := (c(u)+1) %K c(u) := (c(u)+1) %K 8 7 6 8 7 6 10 9 10 9 5 5 11 0 0 0 1 2 3 4 0 1 2 3 4 0 st= RB st= RF st= RB st= RF st= c st= c K > 12 K > 12 d d d d An Unsafe Configuration Reseting Propagation phase 35 36 6
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