Lamport's logical clock

Lamport's Logical Clock was created by Leslie Lamport. It is a procedure to determine the order of events occurring. It provides a basis for the more advanced Vector Clock Algorithm. Due to the absence of a Global Clock in a Distributed Operating System Lamport Logical Clock is needed.

Algorithm:

• Happened before relation(->): a -> b, means 'a' happened before 'b'.
• Logical Clock: The criteria for the logical clocks are:
  • [C1]: C_i (a) < C_i(b), [ C_i -> Logical Clock, If 'a' happened before 'b', then time of 'a' will be less than 'b' in a particular process. ]
  • [C2]: C_i(a) < C_j(b), [ Clock value of C_i(a) is less than C_j(b) ]

Reference:

• Process: P_i
• Event:E_ij, where i is the process in number and j: j_th event in the i^th process.
• t_m: vector time span for message m.
• C_i vector clock associated with process P_i, the j^th element is Ci[j] and contains P_i's latest value for the current time in process P_j.
• d: drift time, generally d is 1.

Implementation Rules[IR]:

• [IR1]: If a -> b ['a' happened before 'b' within the same process] then, C_i(b)  =C_i(a) + d
• [IR2]: C_j = max(C_j, t_m + d) [If there's more number of processes, then t_m = value of C_i(a), C_j = max value between C_j and t_m + d]

For Example:

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• Take the starting value as 1, since it is the 1^st event and there is no incoming value at the starting point:
  • e11 = 1
  • e21 = 1
• The value of the next point will go on increasing by d (d = 1), if there is no incoming value i.e., to follow [IR1].
  • e12 = e11 + d = 1 + 1 = 2
  • e13 = e12 + d = 2 + 1 = 3
  • e14 = e13 + d = 3 + 1 = 4
  • e15 = e14 + d = 4 + 1 = 5
  • e16 = e15 + d = 5 + 1 = 6
  • e22 = e21 + d = 1 + 1 = 2
  • e24 = e23 + d = 3 + 1 = 4
  • e26 = e25 + d = 6 + 1 = 7
• When there will be incoming value, then follow [IR2] i.e., take the maximum value between C_j and T_m + d.
  • e17 = max(7, 5) = 7, [e16 + d = 6 + 1 = 7, e24 + d = 4 + 1 = 5, maximum among 7 and 5 is 7]
  • e23 = max(3, 3) = 3, [e22 + d = 2 + 1 = 3, e12 + d = 2 + 1 = 3, maximum among 3 and 3 is 3]
  • e25 = max(5, 6) = 6, [e24 + 1 = 4 + 1 = 5, e15 + d = 5 + 1 = 6, maximum among 5 and 6 is 6]

Limitation:

• In case of [IR1], if a -> b, then C(a) < C(b) -> true.
• In case of [IR2], if  a -> b, then C(a) < C(b) -> May be true or may not be true.

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Below is the C program to implement Lamport's Logical Clock:

 e21 e22 e23
 e11 0 0 0 
 e12 0 0 1 
 e13 0 0 0 
 e14 0 0 0 
 e15 0 -1 0 
The time stamps of events in P1:
1 2 3 4 5 
The time stamps of events in P2:
1 2 3 

Time Complexity: O(e1 * e2 * (e1 + e2))
Auxiliary Space: O(e1 + e2)