Question

A ring consists of n nodes. Each node holds a value known only to itself. Design an efficient single-initiator protocol to find the minimum value in a ring. Prove its correctness and analyze its costs.

Answer 1

Note: Each node has 2 neighbors in the ring.

Without lose of generality, lets say we have n nodes from 1....n.

Lets san The ring looks like this 1-2-3.....-n-1

Not that n is linked back to 1 which forms ring, if that edge was not there then there won't be a ring formation by this connection.

Algorithm:

Lets say node 1 is initiator.

• Node 1 sends its value to its neighbor say 2.
• Node 2 sends min(value got from 1, its own value) to its neighbor who have not sent it the result. 2 has neighbors 1 and 3. 1 sent 2 the value so now 2 will send value to only 3.
• By doing step 2 on rest of nodes now we look at what happens at n.
• n gets value from n-1. It sends min(its own value, value got from n-1) to 1.
• So now 1 have got the minimum value

Cost-

Each node sends message once so,

we are sending n messages to find minimum value in ring.

Correctness-

Proof that the value find is indeed minimum.

Assume minimum value was m at node i.

So when node i sends value to next node it does

x = min(m,value it got)

since m is minimum so x evaluates to m.

So node i sends m to next node.

Now this m will reach 1 without reducing further as m was the minimum value of a node in ring.