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50005 CSE Lab 2

Computational Complexity of Banker's Algorithm

The main complexity of the banker's algorithm in this exercise lies within the checksafe() function, which is called during a request of resources through requestResources() by some process.

The first section initializes the temporary arrays that the algorithm will use to check the new state of system, which are temp_avail[], temp_need[][], temp_allocation[][], work[], and finally finish[] which is used to verify that each individual process is will not cause a deadlock in this new state. Given $n$ customers and $m$ resources, the resulting complexity of this series of iterative for-loops is is: $$ \mathcal{O}(2nm+2n+m) $$

assuming the assignment operation is $\mathcal{O}(1)$ each for $N$ elements, which is $\mathcal{O}(N)$ overall.

In the final section that carries out the actual safety check, a while loop is used to iterate a single for-loop of $n$ elements over a maximum possible of $n$ times (if all processes are considered safe in the next state i.e checkSafe() returns true). This for-loop also contains a comparison operation which compares two different arrays of $m$ elements each, which is implemented with a nested for-loop. If the condition passes, then another array assignment operation is carried out. Thus, the *overall computational complexity of this implementation of the banker's algorithm is: $$ \mathcal{O}(2nm+2n+m+2n^2m) = \Theta(n^2m) $$

*Formula ignores some simple $\mathcal{O}(1)$ assignment and comparison operations.