title: Maximum Subarray Sum Using Prefix Sum
signature: function maximum_subarray_sum_using_prefix_sum(arr, n)
test_code: 
  
  -- SOLUTION CODE
  -- ============================================
  PLACEHOLDER_CODE_BODY
  
  -- TESTING CODE 
  -- ============================================
  function isCorrect(actual, expected)
      if type(expected) ~= type(actual) then
          return false
      end
      
      if type(expected) == 'table' then
          for k,v_expected in pairs(expected) do
              
              if actual[k] == nil then
                  return false
              end
              local v_actual = actual[k]
              if not isCorrect(v_actual, v_expected) then
                  return false
              end
              actual[k]=nil
          end
      
          if next(actual) then 
              return false
          end
          return true
      else
          return actual == expected
      end
  end
  
  function validateSolution(actual, expected)
      return isCorrect(actual,expected)
  end
  
  function driver(arr, n, expected)
      local exec_result;
      local status, error_msg = pcall(function () 
          exec_result=PLACEHOLDER_FN_NAME(arr, n) 
      end)
      if status then
          if validateSolution(exec_result,expected) then
              return "PASSED"
          end
          return "FAILED"
      else
          return string.format("ERROR=%s",string.gsub(error_msg,".*:[%d]+: ",""))
      end
  end
      
  function main()
      result = driver({8, 9, 11, 17, 18, 19, 23, 24, 27, 30, 31, 31, 35, 44, 46, 47, 49, 51, 55, 58, 59, 61, 65, 67, 71, 71, 71, 71, 78, 78, 82, 91, 98}, 20, 633)
      print(string.format("TEST-0...%s",result))
  
      result = driver({-82, -28, -66, -52, -36, 36, -88, 52, -62, 46, 42, 26, -60, 18, -52, 38, 94, -68, 44, -94, 14, 36, -70}, 15, 114)
      print(string.format("TEST-1...%s",result))
  
      result = driver({0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, 19, 8)
      print(string.format("TEST-2...%s",result))
  
      result = driver({28, 36, 42, 42, 5, 52, 74, 86, 55, 82, 59, 81, 4, 90, 24, 34, 20, 99, 86, 25, 52, 48, 62, 5, 67, 83, 60, 72, 80, 73, 38, 55, 8, 70, 95}, 19, 999)
      print(string.format("TEST-3...%s",result))
  
      result = driver({-92, -52, -24, 36, 56}, 3, -24)
      print(string.format("TEST-4...%s",result))
  
      result = driver({0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0}, 13, 7)
      print(string.format("TEST-5...%s",result))
  
      result = driver({1, 1, 4, 4, 7, 7, 17, 18, 20, 26, 26, 32, 37, 38, 42, 44, 44, 46, 50, 53, 57, 58, 58, 60, 61, 61, 64, 74, 75, 77, 83, 83, 84, 84, 85, 87, 88, 90, 95, 96, 97, 98, 99, 99}, 25, 811)
      print(string.format("TEST-6...%s",result))
  
      result = driver({-86, 2, 26, 54, -16, 16, 48, 24, 50, -10, -32, -62, 48, -12, -66}, 13, 204)
      print(string.format("TEST-7...%s",result))
  
      result = driver({0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, 14, 0)
      print(string.format("TEST-8...%s",result))
  
      result = driver({58, 14, 79, 11, 31, 28, 61, 86, 25, 27, 75, 78, 32, 55, 86, 48, 15, 51, 6, 78, 23, 82, 16, 62, 35, 51, 91, 16, 79, 38, 97, 30, 23, 58, 95, 57, 82, 35, 57, 43, 22, 41, 58, 69, 25, 65, 13, 79}, 39, 1971)
      print(string.format("TEST-9...%s",result))
  
  end
  
  main()
source_py: def maximum_subarray_sum_using_prefix_sum(arr, n):
      min_prefix_sum = 0
      res = (- math.inf)
      prefix_sum = []
      prefix_sum.append(arr[0])
      for i in range(1, n):
          prefix_sum.append((prefix_sum[(i - 1)] + arr[i]))
      for i in range(n):
          res = max(res, (prefix_sum[i] - min_prefix_sum))
          min_prefix_sum = min(min_prefix_sum, prefix_sum[i])
      return res
source_cpp: int maximum_subarray_sum_using_prefix_sum(vector<int> arr, int n) {
    int min_prefix_sum = 0;
    int res = numeric_limits < int > :: min ( );
    int prefix_sum [ n ];
    prefix_sum [ 0 ] = arr [ 0 ];
    for ( int i = 1;
    i < n;
    i ++ ) prefix_sum [ i ] = prefix_sum [ i - 1 ] + arr [ i ];
    for ( int i = 0;
    i < n;
    i ++ ) {
      res = max ( res, prefix_sum [ i ] - min_prefix_sum );
      min_prefix_sum = min ( min_prefix_sum, prefix_sum [ i ] );
    }
    return res;
  }