Leader election. message passing asynchronous. motivation who starts? Leader election, maximum finding, spanning tree.

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1 eader election message passing asynchronous? 9 (eader election) motivation who starts? eader election, maximum finding, spanning tree

2 (eader election) Unidirectional ring Bidirectional rings Complete networks General networks (eader election) Unidirectional ring phases, unique execution (eader election) Bidirectional ring sense of direction

3 (eader election) Bidirectional ring no sense of direction Sense of Direction: For each process p in a bidirectional ring, its left and right neighbors are termed and respectively. If for every, then there is a sense of direction (otherwise no sense of direction) eann s algorithm! " # $ "! # "

4 (eann s algorithm) messages: time: % (eann s algorithm) Theorem: eann s algorithm terminates, and exactly one processor is in! " Chang and oberts algorithm! " # $ "! # "

5 (Chang and oberts algorithm) or: "! # $ " # " (Chang and oberts algorithm) messages: = 0 time: (Chang and oberts algorithm) Theorem: Chang and oberts s algorithm terminates, and exactly one processor is in (worst) "

6 (Chang and oberts algorithm) Theorem: The average message complexity of Chang and oberts s algorithm is assume all rings equally probably (for the proof assume ids are,,, n) (Chang and oberts algorithm) i i k n i P( i, k) = n n k k K P(i,k) probability that id i makes exactly k steps (Chang and oberts algorithm) n i n n n + k p( i, k) = n + = i= k = k = k + n( ) 0.9n log n + O() n

7 (Chang and oberts algorithm) or: Consider all n! rings Each id makes step n! times Identity of P i : makes nd step iff it is largest n! among Pi, Pi+, which happens times Identity of Pi: makes rd step iff it is largest among n! Pi, Pi+,Pi+, which happens times, etc Bidirectional rings messages:? time:? Hirschberg and Sinclair s algorithm Phases,, active k processors start phase k active = n activek activek no. of phases log n messages n log n time =

8 Franklin s algorithm messages:? time:? (Franklin s algorithm) messages:? (Franklin s algorithm) no. of phases log n messages n log n time = &""& #!#"#"'

9 Peterson s st Algorithms P DK This algorithm is a modification of Franklin s algorithm for unidirectional ring. The basic idea is, during a phase, each active process receives the temporary identifier of its nearest active neighbor and that neighbor s nearest active neighbor s temporary identifier, then applies Franklin s strategy. (Peterson s st Algorithms) Each node maintains four variables: { # # $ temporary identity first id received second id received := ; := ; while do begin [start phase] send( ); receive( ); if = then := ; if > then send( ); else send( ); receive( ); if = then := ; if max(, ) then := else := ; end; (now = ) 9

10 (now = ) while do begin receive( ); if = then := ; send( ); end tid ntid nntid tid:=id; [start phase] : send(tid); receive(ntid); % tid ntid nntid if tid > ntid then send(tid); else send(ntid); receive(nntid); %& 0

11 tid ntid nntid if ntid max(tid, nntid) then tid:=ntid else state := relay; % tid ntid nntid [start phase] : send(tid); receive(ntid); ' tid ntid nntid & if tid > ntid then send(tid); else send(ntid); receive(nntid); '

12 tid ntid nntid if ntid max(tid, nntid) then tid:=ntid else state := relay; ' tid ntid nntid ( [start phase] : send(tid); receive(ntid); ' &(" &) * +' ""#',""#('

13 ) processor holding * ) %##' + &+, - # &!&,.!# ( /% 0 ( %"!! ) ) # ) #) /% #'#) +% * % # * Theorem: -.( " /0 " " " %& & ( &"&# ## #

14 Peterson s nd Algorithm improvement of Peterson s st algorithm Instead of comparing its id with both neighbors in the same time, a process first compares itself with its left neighbor, then its right neighbor. (Peterson s nd Algorithms) Each node maintains four variables: { # # $ temporary identity id received := ; := ; while do begin [compare to left, odd phase] send( ); receive( ); if = then := ; if < then := ; end; begin [compare to right, even phase] send( ); receive( ); if = then := ; if > then := else := ; end; (now = )

15 (now = ) while do begin receive( ); if = then := ; send( ); end tid ntid &! % tid ntid &! $ ; ' %&

16 tid ntid &! "-. " / %& ( &"&" " " *(* ( +% ( ( (

17 & /% -# ""#""!#" " " " ( "!" #"( p #"( #" p #" p -!! " #"( #" q p? #" q p $/

18 9#"#"(# " q #"( #" p " "!* & /% ( ( + : ( k + + O() ( : + p+ + O() #" ( & (

19 eferences E. Chang and. oberts, An improved algorithm for decentralized extrema-finding in circular configurations of processes, Communications of the ACM},,, 99, pp. -. eferences D. Dolev, M. Klawe and M. odeh, An O(n log n) unidirectional distributed algorithm for extrema finding in a circle, Journal of Algorithms,, 9, pp. -0. eferences W.. Franklin, On an improved algorithm for decentralized extrema finding in circular configurations of processors, Communication of the ACM,, 9, pp. -. 9

20 eferences D. S. Hirschberg and J. B. Sinclair, Decentralized extrema-finding in circular configuration of processors, Communications of the ACM,, 90, pp. -. eferences G. eann, Distributed systems - towards a formal approach, Information Processing etters, 9, pp. -0. eferences G.. Peterson An O(nlogn) unidirectional algorithm for the circular extrema problem. ACM Trans. Program. ang. Syst., (Oct. 9), -. 0

21 eferences N. Santoro, Sense of direction, topological awareness and communication complexity, SIGACT News,,, Summer 9, pp. 0-.