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Input: Set of machines M, Each machine has an availability array D_m[t], 
set of activities A, for each activity j its starting time s_j,
its duration p_j and its consumption in D during processing. 

Output: True or False if the usage for  each machine at each instant is less than the availability 

for each machine m in M

  usage := array[0..L] of zeros
  
  for each activity j in A
    usage[s_j] += c_{j,k}
    usage[s_j + p_j] -= c_{j,k} 
  
  for each time t in [1, L]
    if usage == 0
       usage[t] += usage[t-1]
  
  for each time t in [0, L-1]
    if usage[t] > D_m[t]
      return false

return true

The complexity is $O(|M| * (L + |A|))$. But $L$ is not an input length, it's a numerical value. I guess it's pseudo-polynnomial (Am I right ?)

How to prove that there is no better alternative, In other words that its also $\Omega(|M| * (L + |A|))$

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I presume that $L$ is the length of $D_m$ for all machines. Hence your input is of size $O(|M|\cdot (L+|A|))$ because it contains $|A|$ activities for all machines and the $D_m$ array of length $L$ for all machines. Your runtime is $O(|M|\cdot (L+|A|))$. Thus your algorithm runs in linear time with respect to your input (meaning your algo is not pseudo-polynomial it is in fact polynonmial and even linear).

If a value in $D_m$ could be below $0$ we would have to check every value in $D_m$ for every machine at least once and for every activity that $D_m$ is large enough making the minimum runtime $\Omega (|M|L + |M||A|) = \Omega (|M|(L+|A|))$ and thus your algo optimal.

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