Why is the knapsack problem pseudo-polynomial?

I know that Knapsack is NP-complete while it can be solved by DP. They say that the DP solution is pseudo-polynomial, since it is exponential in the “length of input” (i.e. the numbers of bits required to encode the input). Unfortunately I did not get it. Can anybody explain that pseudo-polynomial thing to me slowly ?


The running time is O(NW) for an unbounded knapsack problem with N items and knapsack of size W. W is not polynomial in the length of the input though, which is what makes it pseudo-polynomial.

Consider W = 1,000,000,000,000. It only takes 40 bits to represent this number, so input size = 40, but the computational runtime uses the factor 1,000,000,000,000 which is O(240).

So the runtime is more accurately said to be O(N.2bits in W), which is exponential.

Also see:

Source : Link , Question Author : Michael , Answer Author : Community

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