Self Stabilization 1 Goals of the lecture: Self-stabili zation � F ault-tolerance � De�nition of self-stabilizing � Algo rithm with K -state Machines � Pro of � Algo rithm with 3 -state Machines � Pro of References: Dijkstra 74, Dijkstra 86 � Vija c y K. Ga rg Distributed Systems F all 94
Self Stabilization 2 F ault-tolerance � systems which recover from faults � self-stabilization: highly fault-tolerant � a fault can change any data � system view ed as consisting of legal and illegal states � self-stabilzation: should reach a legal state in �nite moves � Vija c y K. Ga rg Distributed Systems F all 94
Self Stabilization 3 T erminolo gy � Underlying top ology: connection graph � neighb o rs � p rivilege: b o olean function of � o wn state, � states of its neighb o rs � legal state: application dep endent � Vija c y K. Ga rg Distributed Systems F all 94
Self Stabilization 4 Requirements on legal state � In each legal state, one o r mo re p rivileges � any move from a legal state leads to a legal state � each p rivilege p resent in at least one legal state � fo r any pair of legal states, there exist a sequence of trans- ferring moves De�nition of self-stabilzati on: Rega rdless of initial state, and p rivilege selected each time, the system is gua ranteed to reach a legal state after a �nite numb er of moves. � Vija c y K. Ga rg Distributed Systems F all 94
Self Stabilization 5 Example: Mutual Exclusion legal state: exactly one p rivilege � N+1 machines numb ered 0..N � L,S,R: states of left, self, right � b ottom machine: machine 0 � fo rmat: if p rivilege then co rresp onding move � c � Vija y K. Ga rg Distributed Systems F all 94
Self Stabilization 6 Algo rithm I: K -state machine ( K > N ) Bottom: if (L=S) then S := S+1 mo d K � F o r other machines: if ( L 6 = S ) then S := L � c � Vija y K. Ga rg Distributed Systems F all 94
Self Stabilization 7 Example 4 4 4 t t t 1 1 2 2 t 2 t 2 t t t t ) ) t t t t t t 3 3 4 B B B 4 4 4 t t t 3 4 4 + 4 4 4 t t t 4 4 4 2 t 4 t 4 t t t t ( ( t t t t t t 4 4 4 B B B 4 4 4 t t t 4 4 5 c � Vija y K. Ga rg Distributed Systems F all 94
Self Stabilization 8 Pro of Lemma 0: If the system is in a legal state, then it will sta y legal. Lemma 1: A sequence of moves in which Bottom do es not move is �nite. Lemma 2: Given any con�guration, either (1) no other machine has the same state as the b ottom, o r (2) there exists a value which is di�erent from all machines. Lemma 3: With in a �nite numb er of moves, pa rt one of Lemma 2 will b e true. Theo rem 1: Within �nite numb er of moves, the system will reach a legal state. c � Vija y K. Ga rg Distributed Systems F all 94
Self Stabilization 9 Algo rithm I I: 3 -state machine Ring of at least 3 machines Bottom: B, No rmal: N, T op: T con�guration view ed as a string of 0,1,2 Bottom: if ( B + 1 = R ) then B := B + 2 ; No rmal: if ( L = S + 1) o r ( R = S + 1) then S := S + 1 ; T op: if ( L = B ) and ( T 6 = B + 1) then T := B + 1 c � Vija y K. Ga rg Distributed Systems F all 94
Self Stabilization 10 Viewing the string with a rro ws y = # of left-p ointing + 2# of right-p ointing Bottom : (0) B R to B ! R � y = +1 No rmal Machine: (1) L ! S R to L S ! R � y = 0 (2) L S R to L S R � y = 0 (3) L ! S R to L S R � y = � 3 (4) L ! S ! R to L S R � y = � 3 (5) L S R to L ! S R � y = 0 T op Machine (p rivilege also dep ends on B): (6) L ! T to L T � y = +1 (7) L T to L T � y = +1 c � Vija y K. Ga rg Distributed Systems F all 94
Self Stabilization 11 Example 1 ! 0 ! 2 0 z z z z T B 1 1 2 0 z z z z T B 1 2 2 0 z z z z T B 0 ! 2 2 0 z z z z T B 0 0 ! 2 0 z z z z T B 0 0 0 0 z z z z T B 0 0 0 1 z z z z T B � Vija c y K. Ga rg Distributed Systems F all 94
Self Stabilization 12 Pro of Claim: Single a rro w implies it sta ys that w a y . Claim: string free from a rro w creates one in a single move. No w sho w that if multiple a rro w then y will b e decreased in �nite moves. c � Vija y K. Ga rg Distributed Systems F all 94
Self Stabilization 13 Pro of contd Lemma 0: Bet w een t w o successive moves of T op at least one move of Bottom tak es place. Lemma 1: A sequence of moves in which Bottom do es not move is �nite. Pro of: su�cient to consider no rmal machines. (3),(4),(5) decrease numb er of a rro ws. (1) and (2) moves �nite due to top ology . Theo rem: Within �nite moves, there is one a rro w in the string. Pro of:b et w een successive moves of b ottom, falsi�cation of \left- most a rro w exists and p oints to the right" happ en in (3), (4), o r (6). if (6) then done. If (3) o r (4), y decreases b y 3. y can increase b y at most 2 p er move of Bottom, thus y is decreased b y 1. c � Vija y K. Ga rg Distributed Systems F all 94
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