One cave, one fire. Only ONE Ugg cook at a time.

Distributed mutual exclusion (DME) — many caves, many Ugg, all want the one magic fire-pit (the critical section, CS). Only ONE Ugg in fire-pit at once. In a single cave, easy: grab bone-lock, cook, drop bone-lock. But these caves far apart. No shared bone-lock, no big chief to boss everyone. Only shout-messages between caves. So how make sure only one Ugg cook? Ugg teach you. Five ways. Two tribes.

What a good fire-rule must promise

Every fire-rule keep three promises:

• Safety — never two Ugg in fire-pit same moment. Most important.
• Liveness — every Ugg who ask, EVENTUALLY cook. No Ugg wait forever.
• Fairness — Ugg cook in the order they ask, by timestamp (shiny-stone clock number). First ask, first cook.

Ugg measure a fire-rule four ways:

• Message complexity — how many shout per one cook.
• Synchronisation delay (SD) — how long fire-pit sit empty between one Ugg leave and next Ugg enter.
• Response time — from ask until done cook.
• Throughput $=1/(\text{SD}+E)$, where $E$ = how long one Ugg cook. Small SD → more dinner per moon.

Tribe One: ask-permission (non-token)

Lamport — old grandfather rule. Need FIFO shout-paths (shouts arrive in the ORDER sent). Every Ugg keep a request queue — line of stones sorted by timestamp.

• Want cook: shout $\text{REQUEST}(t{s}_i,i)$ to all; drop own stone in own queue.
• Other Ugg hear it: queue the stone, shout back a timestamped REPLY.
• Enter fire-pit only when BOTH true: L1 — got a shout with bigger timestamp from EVERY other cave (any shout counts, not only a REPLY); L2 — own stone sit at the TOP of own queue.
• Done: throw own stone away, shout RELEASE. All Ugg remove that stone.

Cost: $3(N-1)$ shouts per cook (REQUEST + REPLY + RELEASE to each other cave). SD $=T$ (one shout-time).

Ricart-Agrawala — smart grandchild. Kill the RELEASE shout. No FIFO needed!

• Shout REQUEST.
• Hear a REQUEST: shout REPLY if you idle OR your own timestamp is bigger. Else defer — hold the REPLY in fist, send later.
• Enter after REPLY from all $N-1$ caves.
• Done: now send every held-back REPLY.

Cost: $2(N-1)$ shouts. SD $=T$. Beats grandfather. Roucairol-Carvalho trick: once you got REPLY from cave-j, no need ask cave-j again to re-cook — unless you sent cave-j a REPLY in between. Shouts sink to $0$ up to $2(N-1)$.

Maekawa — do NOT ask everyone. Ask only a small hunting-party, the quorum $R_i$. Party rules: you sit in your own party ($i\in R_i$); every two parties SHARE at least one Ugg ($R_i\cap R_j\ne \varnothing$ — shared Ugg is the judge); all parties same size $K$; each Ugg sit in $D$ parties. Best size $K=D=\sqrt{N}$, because the sharing rule forces $K^2\ge N$.

V1: $3\sqrt{N}$ shouts, SD $=2T$. BUT V1 can deadlock — three parties lock each other in a ring, all stuck. V2 fix adds three shouts to break the ring: FAILED ("sorry, already promised a better Ugg"), INQUIRE ("you actually cooking, or can give lock back?"), YIELD ("fine, I let go my lock"). V2: up to $5\sqrt{N}$ shouts, SD still $2T$.

Tribe Two: pass-the-magic-bone (token)

Suzuki-Kasami — one magic token bone. Hold bone = cook. Want cook? Shout to all; bone-holder toss you the bone IF your ask is FRESH. Bone carries a FIFO queue $Q$ of waiting Ugg plus $LN[\mathrm{1..}n]$ (last cook-number done for each Ugg). Each Ugg keep $RN[\mathrm{1..}n]$ (biggest ask-number ever heard from each Ugg).

• Ask: $R{N}_i[i]$++, shout $\text{REQUEST}(i,R{N}_i[i])$.
• Hear $\text{REQUEST}(i,sn)$: set $R{N}_j[i]=\max (R{N}_j[i],sn)$. If you hold bone, idle, AND $R{N}_j[i]=LN[i]+1$ (fresh — the very next ask), toss bone to Ugg-i.
• Done: $LN[i]=R{N}_i[i]$. Any Ugg-j with $R{N}_i[j]=LN[j]+1$, add to $Q$. Pop front of $Q$, toss bone there.

Cost: $0$ (already hold bone) or $N$. SD $=0$ or $T$. The $RN[i]=LN[i]+1$ test throws away OLD stale asks from slow shouts.

Raymond — token on a tree, no shouting to all. Every Ugg has a Holder pointer aiming toward the bone (the root), plus a FIFO queue $Q_i$.

• Ask: push self on $Q_i$; if not the holder and $Q_i$ was empty, shout REQUEST to Holder (your parent).
• Middle Ugg hears REQUEST: queue it; if no ask sent up yet, pass REQUEST up to Holder.
• Root hears REQUEST: send bone toward asker, flip its Holder pointer.
• Get bone: pop $Q$ front; if self, cook; else pass bone on and flip Holder. If $Q$ still not empty, shout REQUEST up again.

Cost: $O(\log N)$ shouts (balanced tree). SD $=(\log N)/2\cdot T$. Danger: everything funnels through the root — root bottleneck.

The big stone-tablet (MEMORIZE)

| Rule | Tribe | Shouts | SD | Watch out | |---|---|---|---|---| | Lamport | non-token | $3(N-1)$ | $T$ | need FIFO | | Ricart-Agrawala | non-token | $2(N-1)$ | $T$ | no FIFO | | Maekawa V1 | quorum | $3\sqrt{N}$ | $2T$ | deadlocks | | Maekawa V2 | quorum | up to $5\sqrt{N}$ | $2T$ | deadlock-free | | Suzuki-Kasami | token | $0$ or $N$ | $0$ or $T$ | bone can get lost | | Raymond | token tree | $O(\log N)$ | $(\log N)/2\cdot T$ | root bottleneck |

Why grandfather needs FIFO but grandchild does not

Lamport leans on REQUEST-then-RELEASE order. RELEASE removes a stone that REQUEST first put in the queue. If RELEASE could run PAST an earlier REQUEST (no FIFO), an Ugg tries to remove a stone not queued yet. Broken! Ricart-Agrawala has NO release shout at all — the held-back REPLYs keep the timestamp order by themselves. No FIFO needed.

Ugg warn: no do this!

• No mix up shout-counts. Lamport $3(N-1)$, R-A $2(N-1)$, Maekawa V1 $3\sqrt{N}$, V2 up to $5\sqrt{N}$, S-K $0$ or $N$, Raymond $O(\log N)$.
• No say "R-A needs FIFO." Wrong — R-A's whole beauty is NO FIFO.
• No forget Lamport needs L1 AND L2 both. Either one alone not enough.
• No say "Maekawa V1 is deadlock-free." Wrong — V1 deadlocks, V2 fixes it.
• S-K bone test is $RN[i]=LN[i]+1$, EXACT equal — not bigger, not smaller.

Ugg remember

• Two tribes: ask-permission (Lamport, R-A, Maekawa) vs pass-the-bone (Suzuki-Kasami, Raymond). Token = fewer shouts, but the bone can be lost or its holder can die.
• Three promises (safety, liveness, fairness) + throughput $=1/(\text{SD}+E)$. Smaller SD, more dinners.
• The table is the whole exam: shouts + SD + gotcha for all six rows. Memorize it cold.
• Lamport: L1+L2, FIFO, $3(N-1)$. R-A: no FIFO, $2(N-1)$, deferred REPLY.
• Maekawa $K=\sqrt{N}$, V1 deadlocks, V2 = FAILED/INQUIRE/YIELD. S-K fresh test $RN[i]=LN[i]+1$. Raymond Holder pointer, $O(\log N)$, root bottleneck.