CS 603 Dining Philosophers Problem

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CS 603Dining Philosophers Problem

• February 15, 2002

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Project 2 Starts Today

• The winner
• NTP Client
• Basic Program that accepts NTP server as
  argument, gets and returns time from that server
• Three points for well document and tested
  solution
• Extras (worth one additional point)
• Fault Tolerant averaging solution Accepts up to
  four servers and gives average after throwing
  away bad servers
• Class library Initialize sets offset from local
  clock, get time returns local offset without
  sending message

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Dining Philosophers Problem(Dijkstra 71)
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Dining Philosophers Solutions

• Simple waiting state
• Enter waiting state when neighbors eating
• When neighbors done, get forks
• Neighbors cant enter waiting state if neighbor
  waiting
• Problem
• Doesnt prevent starvation
• Requires checking both neighbors at once
• Race condition

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Fully Distributed Solution(Lehman and Rabin 81)

• Problem with previous solutions
• Not truly distributed Requires some sort of
  central coordination or global state
• Non-Symmetric Different philosophers run
  different algorithms
• Additional properties
• Deadlock free Eventually someone new gets to
  eat
• Lockout free Eventually every hungry
  philosopher gets to eat
• Adversary One philosopher may try to starve
  another
• Cant just hold the fork indefinitely
• Communication only between adjacent philosophers
• No global state
• Cant communicate with both at same time

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No Deterministic Solution

• Proof Assume solution for philosophers 1..n
• Philosophers dont know their number!
• Philosophers activated in order from 1..n
• Each takes one step
• Claim If symmetric at beginning of round, will
  be symmetric at end of round
• If anyone eating, all would be!

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Probabilistic Solution

• Assume Random coin toss
• Guaranteed with probability 1 to break symmetry
• Idea Try to get one first
• Then get other
• If cant get other, put first down and try again
• But dont go for the same fork first every time

• Think
• trying true or die
• While trying
• s random(left,right)
• Wait for fork s then take it
• If fork s available take itelse drop fork s
• Eat
• drop srandom(left,right)
• drop fork s

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Lemmas Assume Plato sitting to left of Aristotle

• If Plato picks up fork infinite number of times,
  Aristotle finite number, then
• P(Plato eats infinite number of times)1
• If deadlocked, every philosopher picks up fork
  infinite number of times with probability 1
• If after t steps, both trying to eat, tried to
  get same fork. Then with probability ? ½,
• One picks up fork only finite number of times in
  future, or
• One gets to eat in next two draws
• If at time t the last set of random draws is A,
  then with probability 1 there is a later
  configuration B ? A where two neighbors try to
  get the same fork first

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Theorem P(Deadlock) 0

• Assume P(Deadlock) gt 0
• By Lemma 2, if deadlocked everyone performs
  infinite draws
• By Lemma 4, with probability 1 there will be
  infinite sequence of configuration of last draws
  A0, A1, satisfying condition of Lemma 3
• By Lemma 3, ?n some philosopher eats between An
  and An2 with probability 1
• Therefore if deadlocked, non-deadlocked condition
  will occur with probability 1

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Problem Not Lockout-freeCourteous Philosophers

• Possible for all but one to starve
• Solution If eating and neighbor trying to eat,
  once done wait until neighbor has eaten before
  trying again
• Requires more shared variables
• signal to neighbor On/Off
• Share last with neighbor Left, Neutral, Right
• Initialized to Neutral
• Only need mutual exclusion with one neighbor at a
  time

• Think
• trying true left_signal right_signal on
• s random(left,right)
• Wait until s down and(s-neighbor-signal off
  ors-last neutral or s-last s)then lift fork
  s
• If s down thenlift s trying false
• else drop s
• Eat
• left-signal right-signal off
• left-last right right-last left
• Drop forks

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Lesson Non-Determinism Gives Additional Power

• In fully distributed system, random variable
  solves problems that cant otherwise be solved
• Used in practice
• Ethernet Random backoff if collision
• Makes proving correctness harder
• Consider such solutions when building distributed
  systems!