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. $\mathrm{\forall }i\mathrm{\forall }j[R_i\bigcap R_j\ne \mathrm{\varnothing }]$
2. $\mathrm{\forall }i[P_i\in R_i]$
3. $\mathrm{\forall }i[|R_i|=K]$
4. Site $P_i$ is contained in exactly $K$ request sets

Therefore:

• $|R_i|\ge \sqrt{N-1}$

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.