# Dynamic programming problem and shortest path problem

I was wondering

1. if any dynamic programming problem can always be converted to a source-sink shortest path problem in a network with source and sink nodes given? And vice versa?
2. Is any dynamic programming problem essentially a linear integer programming problem?
3. if in a network, the shortest path length between every vertices defines a metric on the set of vertices of the network?

Thanks and regards!

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For the first part of (1): The answer is no. The simplest example off the top of my head is the longest substring of ones in a 0,1 string. The typical DP solution would be to use a 1D array and store the length of the longest substring up that includes the $i$-th character in the $i$-th coordinate.