Maekawa's algorithm

Maekawa's algorithm is an algorithm for mutual exclusion on a distributed system. The basis of this algorithm is a quorum-like approach where any one site needs only to seek permissions from a subset of other sites.

Algorithm

Terminology

• A site is any computing device which runs the Maekawa's algorithm
• For any one request of entering the critical section:
  • The requesting site is the site which is requesting to enter the critical section.
  • The receiving site is every other site which is receiving the request from the requesting site.
• ts refers to the local time stamp of the system according to its logical clock

Algorithm

Requesting site:

• A requesting site ${\displaystyle P_{i}}$ sends a message ${\displaystyle {\text{request}}(ts,i)}$ to all sites in its quorum set ${\displaystyle R_{i}}$.

Receiving site:

• Upon reception of a ${\displaystyle {\text{request}}(ts,i)}$ message, the receiving site ${\displaystyle P_{j}}$ will:
  • If site ${\displaystyle P_{j}}$ does not have an outstanding ${\displaystyle {\text{grant}}}$ message (that is, a ${\displaystyle {\text{grant}}}$ message that has not been released), then site ${\displaystyle P_{j}}$ sends a ${\displaystyle {\text{grant}}(j)}$ message to site ${\displaystyle P_{i}}$.
  • If site ${\displaystyle P_{j}}$ has an outstanding ${\displaystyle {\text{grant}}}$ message with a process with higher priority than the request, then site ${\displaystyle P_{j}}$ sends a ${\displaystyle {\text{failed}}(j)}$ message to site ${\displaystyle P_{i}}$ and site ${\displaystyle P_{j}}$ queues the request from site ${\displaystyle P_{i}}$.
  • If site ${\displaystyle P_{j}}$ has an outstanding ${\displaystyle {\text{grant}}}$ message with a process with lower priority than the request, then site ${\displaystyle P_{j}}$ sends an ${\displaystyle {\text{inquire}}(j)}$ message to the process which has currently been granted access to the critical section by site ${\displaystyle P_{j}}$. (That is, the site with the outstanding ${\displaystyle {\text{grant}}}$ message.)
• Upon reception of a ${\displaystyle {\text{inquire}}(j)}$ message, the site ${\displaystyle P_{k}}$ will:
  • Send a ${\displaystyle {\text{yield}}(k)}$ message to site ${\displaystyle P_{j}}$ if and only if site ${\displaystyle P_{k}}$ has received a ${\displaystyle {\text{failed}}}$ message from some other site or if ${\displaystyle P_{k}}$ has sent a yield to some other site but have not received a new ${\displaystyle {\text{grant}}}$.
• Upon reception of a ${\displaystyle {\text{yield}}(k)}$ message, site ${\displaystyle P_{j}}$ will:
  • Send a ${\displaystyle {\text{grant}}(j)}$ message to the request on the top of its own request queue. Note that the requests at the top are the highest priority.
  • Place ${\displaystyle P_{k}}$ into its request queue.
• Upon reception of a ${\displaystyle {\text{release}}(i)}$ message, site ${\displaystyle P_{j}}$ will:
  • Delete ${\displaystyle P_{i}}$ from its request queue.
  • Send a ${\displaystyle {\text{grant}}(j)}$ message to the request on the top of its request queue.

Critical section:

• Site ${\displaystyle P_{i}}$ enters the critical section on receiving a ${\displaystyle {\text{grant}}}$ message from all sites in ${\displaystyle R_{i}}$.
• Upon exiting the critical section, ${\displaystyle P_{i}}$ sends a ${\displaystyle {\text{release}}(i)}$ message to all sites in ${\displaystyle R_{i}}$.

Quorum set (${\displaystyle R_{x}}$):
A quorum set must abide by the following properties:

1. ${\displaystyle \forall i\,\forall j\,[R_{i}\bigcap R_{j}\neq \emptyset ]}$
2. ${\displaystyle \forall i\,[P_{i}\in R_{i}]}$
3. ${\displaystyle \forall i\,[|R_{i}|=K]}$
4. Site ${\displaystyle P_{i}}$ is contained in exactly ${\displaystyle K}$ request sets

Therefore:

• ${\displaystyle |R_{i}|\geq {\sqrt {N-1}}}$

Performance

• Number of network messages; ${\displaystyle 3{\sqrt {N}}}$ to ${\displaystyle 6{\sqrt {N}}}$
• Synchronization delay: 2 message propagation delays
• The algorithm can deadlock without protections in place. ^{[1] [2]}

See also

• Lamport's bakery algorithm
• Lamport's Distributed Mutual Exclusion Algorithm
• Ricart–Agrawala algorithm
• Raymond's algorithm

References

1. "Maekawa's Mutual Exclusion Algorithm: Voting approach".
2. "Distributed Mutual Exclusion" (PDF).

• M. Maekawa, "A √N algorithm for mutual exclusion in decentralized systems”, ACM

Transactions in Computer Systems, vol. 3., no. 2., pp. 145–159, 1985.

• Mamoru Maekawa, Arthur E. Oldehoeft, Rodney R. Oldehoeft (1987). Operating Systems: Advanced Concept. Benjamin/Cummings Publishing Company, Inc.
• B. Sanders (1987). The Information Structure of Distributed Mutual Exclusion Algorithms. ACM Transactions on Computer Systems, Vol. 3, No. 2, pp. 145–59.