Distributed Algorithms 22903 Leader election in rings Lecturer

Distributed Algorithms (22903) Leader election in rings Lecturer: Danny Hendler This presentation is based on the book “Distributed Computing” by Hagit attiya & Jennifer Welch

Message-passing systems p 0 1 1 p 1 2 1 p 3 3 2 2 1 p 2 • Processes are represented by graph nodes • Bi-directional (unless otherwise stated) labeled communication links (reliable, FIFO) • Each process pi (having r links) has state consisting of: Ø Program state Ø inbufi[m], 1 ≤ m ≤ r Ø outbufi[m], 1 ≤ m ≤ r (this is inaccessible to pi) 2

Message-passing systems (cont'd) p 0 1 1 p 1 2 1 p 3 3 2 2 1 p 2 • A Configuration is a vector of process states (q 0, …, qn-1) • In the initial configuration Ø inbuf buffers are empty Ø outbuf buffers may contain messages 3

Synchronous message-passing systems p 0 1 1 p 1 2 1 p 3 3 2 2 1 p 2 • Computation proceeds in rounds • In each round: Ø All messages in the outbuf buffers are delivered Ø Each process performs a computational step and sends at most a single message on every link. 4

Asynchronous message-passing systems p 0 1 1 p 1 2 1 p 3 3 2 2 1 p 2 • No fixed upper bound on the delay of a message or the time between computation steps • An execution is a sequence of events: Ø comp(i) – a computational step by process i Ø del(i, j, m) – the delivery of message m from pi to pj 5

Leader Election in Rings 2 p 0 1 2 1 p 2 2 p 1 1 • Each process has to decide whether it is a leader or not • Termination states are partitioned to elected and not-elected. • In every execution, exactly one process enters an elected state, all others enter a non-elected state. • We assume rings are oriented. 6

Leader election in anonymous rings 2 p 1 2 1 p p 2 1 • A ring algorithm is anonymous if processes don’t have an ID they can use in the algorithm (all perform the same code) An anonymous leader election algorithm …Is impossible!!. . . What about the asynchronous model? 7

A simple algorithm for (non-anonymous) asynchronous rings 1. Each process sends probe(ID) to its left neighbor (and then waits) 2. Upon receiving a message probe(i) (from the right) 3. 4. 5. 6. 7. 8. 9. if (i > ID) forward message to the left if (i=ID) send terminate(ID) and terminate as leader Upon receiving terminate(i) if (i > ID) forward message to the left and terminate as non-leader Message complexity is… Θ(n 2) 8

A more efficient asynchronous leader election algorithm • Processes may join the algorithm either by spontaneous wakeup at the same time or by receiving a message from an awake process • Only spontaneously awakened processes actively participate – the process with maximum ID wins • Algorithm proceeds in phases • In phase j, a process tries to become the temporary leader of its 2 j-neighborhood – Has to have maximum ID in this neighborhoud • Only temporary leaders of phase-j continue to phase j+1 9

A more efficient asynchronous leader election algorithm • Processes may join the algorithm either by spontaneous wakeup at the same time or by receiving a message from an awake process • Only spontaneously awakened processes actively participate – the process with maximum ID wins • Algorithm proceeds in phases • In phase j, a process tries to become the temporary leader of its 2 j-neighborhood – Has to have maximum ID in this neighborhoud • Only temporary leaders of phase-j continue to phase j+1 10

An O(n log n) messages asynchronous leader election algorithm (cont’d) 1. Initially, asleep=true, participating=false 2. Upon receiving no message (spontaneous wakeup) 3. asleep: =false, participating=true 4. send probe(ID, 0, 1) to left and right 5. Upon receiving probe(j, l, d) from left (resp. , right) 6. if (j=ID) terminate as leader ; a termination message should be sent 7. if (j>ID or participating) and (d<2 l) send probe(j, l, d+1) to right (resp. , left) 8. if (j>ID or participating) and (d = 2 l) send reply(j, l) to left (resp. right) 9. Upon receiving reply(j, l) from left (resp. , right) 10. if (j ≠ ID) send reply(j, l) to right (resp. , left) 11. else 12. 13. if already received reply(j, l) from right (resp. left) send probe(ID, l+1, 1) to left and right 11

The message complexity of the asynchronous leader election algorithm is O(n log n) 12

An (n log n) lower bound on the number of messages for asynchronous rings A ring algorithm is uniform if it has no knowledge of the size of the ring. It is non-uniform if it can use its knowledge about the size of the ring. We’ll see a proof for uniform algorithms where the process with the maximum ID must be elected and all other processes must learn its ID (write it to some local variable). 13

Pasting R 1 and R 2 into R (illustration 1. for Lemma 6) p 1 R 1 p 2 e 2 R 2 e 1 q 2 q 1 p 1 R 1 ep p 2 R 2 eq q 1 q 2 14

Induction step of lower bound (illustration 2. for Lemma 6) P p 1 ep R 1 p 2 R 2 eq q 2 q 1 Q 15

An O(n) messages Non-uniform Algorithm for Synchronous Rings 1. Initially phase: =0 and round : =0 2. if (phase = ID) 3. send Leader(ID) 4. terminate as leader 5. else 6. round: =round+1 7. if (round = n) 8. phase: =phase+1 9. round=0 10. Upon receiving Leader(j) 11. 12. 13. if (round < n-1) • Synchronized start • Only clockwise links are used Time complexity? n(min. ID+1) send Leader(j) terminate as non-leader 16

An O(n) messages Uniform Algorithm • Does not rely on synchronized start: a preliminary wake-up phase is added (first-phase messages) • Processes that do note wake-up spontaneously act as relays • Messages from process with ID i travel at speed 1/2 i (second-phase messages) 17

An O(n) messages uniform Algorithm (cont’d) Initially waiting is empty and status=asleep 1. Let R be the set of messages received in this computation round 2. S : = empty set ; Set of IDs to send 3. if (status=asleep) then 4. if (R is empty) ; Woke-up spontaneously 5. status: =participating 6. min : = ID ; 7. add (ID, 1) to S ; First-phase message 8. Else status=relay, min : = ∞ ; Act as a relay 9. 10. 11. 12. 13. 14. 15. 16. 17. for each <m , h > in R do if (m < min) become not elected min: =m if (status=relay) and (h=1) add < m, 1 > to S ; m stays first-phase else add <m, 2> to waiting tagged with current round number elseif (m = ID) become elected ; if m>min message is swallowed 18. 19. 20. for each <m, 2 > in waiting do if <m, 2 > was received 2 m-1 rounds ago remove <m, 2 > from waiting and add to S 21. Send S to left 18

A proof that the algorithm has linear complexity 19