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We study the problem of computing {\\em almost shortest paths} (ASP) for all pairs in $S \\times V$ in both classical centralized and parallel (PRAM) models of computation. Consider the regime of multiplicative approximation of $1+\\epsilon$, for an arbitrarily small constant $\\epsilon > 0$ . In this regime existing centralized algorithms require $\\Omega(\\min\\{|E|s,n^\\omega\\})$ time, where $\\omega < 2.372$ is the matrix multiplication exponent. 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