What is a pseudo-polynomial algorithm?
A pseudo-polynomial algorithm is an algorithm whose worst-case time complexity is polynomial in the numeric value of input (not number of inputs).
For example, consider the problem of counting frequencies of all elements in an array of positive numbers. A pseudo-polynomial time solution for this is to first find the maximum value, then iterate from 1 to maximum value and for each value, find its frequency in array. This solution requires time according to maximum value in input array, therefore pseudo-polynomial. On the other hand, an algorithm whose time complexity is polynomial in the number of elements in array (not value) is considered as polynomial time algorithm.
Pseudo-polynomial and NP-Completeness
Some NP-Complete problems have pseudo-polynomial time solutions. For example, Dynamic Programming Solutions of 0-1 Knapsack, Subset-Sum and Partition problems are Pseudo-Polynomial. NP complete problems that can be solved using a pseudo-polynomial time algorithms are called weakly NP-complete.
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