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Coordinate-based Dynamic Programming


Coordinate-based Dynamic Programming are associated with the coordinates on a array or matrix. You are usually given an array/sequence or a matrix/grid as the input, and the question asks you to find either a subarray, subsequence, vector or path that satisfies a set of conditions. This is one of easier type of Dynamic Programming because it is intuitive to understand.


  • Usually given an array/sequence or grid/matrix as your input
  • Need to find a subsequence/vector/path for
    • maximizing/minimizing a certain property
    • counting
    • true/false
  • State Representation:
    • State is usually represented by index (i) or (i, j)
    • is represented by dp[i] and dp[i][j], which are the most optimal solutions for a[i] or a[i][j]
  • Transition:
    • Initalization is usually just dp[0] with a[0] or a[dp.length-1] with a[a.length-1]
    • At each index i, the most optimal solutions of the surrounding neighbors should be found already
    • Update dp[i] or dp[i,j] with an aggregate function with the values of its neighbors