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

Receiving site:

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

Critical section:

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

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

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

Therefore:

• $|R_{i}|\geq {\sqrt {N-1}}$

Performance

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

See also

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

References

• 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.