The Idea. Leader Election. Outline. Why Rings? Network. We study leader election in rings. Specification of Leader Election YAIR. Historical reasons

Transcription

1 The Idea Leader Electio Network We study leader electio i rigs Why Rigs? Outlie Specificatio of Leader Electio Historical reasos origial motivatio: regeerate lost toke i toke rig etworks Illustrates techiques ad priciples Good for lower bouds ad impossibility results YAIR Leader Electio i Asychroous Rigs A O( 2 ) algorithm A O( log()) algorithm The Revege of the Lower Boud Leader electio is sychroous rigs Breakig the Ω( log()) barrier

2 The Problem Lots of variatios... Processes ca be i oe of two fial states elected o-elected I every executio, exactly oe process (the leader) is elected All other processes are o-elected Rig ca be uidirectioal or bidirectioal Processes ca be idetical or ca somehow be distiguishable from each other The umber of processes may or may ot be kow if ot, uiform algorithms Commuicatios may be sychroous or asychroous Aoymous Networks Call me Ishmael Processes have o uique IDs (idetical automata)...but ca distiguish betwee left ad right Processes have uique IDs from some large totally ordered set (e.g. ) N + Operatios used to maipulate IDs ca be urestricted or limited (e.g. oly comparisos)

3 Commuicatio: Sychroous/Asychroous A Impossibility Result I rouds Sychroous I each roud, a process delivers all pedig messages takes a executio step (possibly sedig oe or more messages)! Asychroous No upper boud o message delivery time No cetralized clock No boud o relative sped of processes Theorem There is o ouiform aoymous algorithm for leader electio i sychroous rigs A Impossibility Result A O( 2 ) Algorithm Le La ( 77), Chag & Roberts ( 79) Theorem There is o ouiform aoymous algorithm for leader electio i sychroous rigs Proof Suppose there exists a aoymous ouiform algorithm A for R s.t. R > 1 Lemma For every roud k of A i R, the states of all the processes at the ed of roud k are the same Proof By iductio o k If some process eters the leader state, they all do upo receivig o message! sed uid i to left (clockwise) upo receivig m from right case! m.uid > uid i :!! sed m to left! m.uid < uid i :!! discard m! m.uid = uid i :!! leader := i!! sed <termiate, i > to left!! termiate edcase upo receivig <termiate, i> from right! leader := i! sed <termiate, i > to left! termiate Asychroous ad Uiform Process with highest uid is elected leader - all other uids are swallowed Time complexity: O() O( 2 ) Boud is tight:!!

4 I each phase k, p i : seds uid i toke left ad right toke iteded to travel distace! ad tur back cotiues outboud oly if greater tha tokes o path processes always forward iboud toke p i leader if it receives ow toke while goig outboud leader of -eighborhood Phase 1 leader of -eighborhood i phase k, protocol elects leader of -eighborhood

5 Phase 2 Phase 3 leader of -eighborhood leader of -eighborhood 4 +1 log( 1) before 1 wier

6 log( 1) {4 k before 1 wier per wier +1 log( 1) before 1 wier 4 { per wier 1 +1 # of wiers per phase +1 log( 1) termiatio before 1 wier 4 { per wier 1 +1 # of wiers per phase +1 Message complexity log( 1) termiatio before 1 wier 4 { per wier 1 < 8(log + 2) # of wiers per phase 1 +1

7 The Revege of the Lower Boud We have see: Facts a simple O( 2 ) algorithm a more clever O( log ) algorithm Ω( log ) lower boud i asychroous etworks Ω( log ) lower bouds i sychroous etworks whe usig oly comparisos Breakig through Ω( log ) Sychroous rigs UID are positive itegers, maipulated usig arbitrary operatios No Uiform is kow to all!uidirectioal! commuicatio! O( ) messages! Uiform! is ot kow!uidirectioal! commuicatio!o( ) messages! What about time complexity? The power of sychroy No uiform algorithm Iformatio ca be coveyed by ot sedig a message! whe your phoe does t rig, it s me Rus i phases, each lastig rouds I phase i if there is a process with uid = i the process i elects itself leader sed all toke with its uid Time complexity: UID mi