code: def circular_kantorovich(x, y, p=1):
      x_sum = 0.0
      y_sum = 0.0
      for i in range(x.shape[0]):
          x_sum += x[i]
          y_sum += y[i]
  
      x_cdf = x / x_sum
      y_cdf = y / y_sum
  
      for i in range(1, x_cdf.shape[0]):
          x_cdf[i] += x_cdf[i - 1]
          y_cdf[i] += y_cdf[i - 1]
  
      mu = np.median((x_cdf - y_cdf) ** p)
  
      # Now we just want minkowski distance on the CDFs shifted by mu
      result = 0.0
      if p > 2:
          for i in range(x_cdf.shape[0]):
              result += np.abs(x_cdf[i] - y_cdf[i] - mu) ** p
  
          return result ** (1.0 / p)
  
      elif p == 2:
          for i in range(x_cdf.shape[0]):
              val = x_cdf[i] - y_cdf[i] - mu
              result += val * val
  
          return np.sqrt(result)
  
      elif p == 1:
          for i in range(x_cdf.shape[0]):
              result += np.abs(x_cdf[i] - y_cdf[i] - mu)
  
          return result
  
      else:
          raise ValueError("Invalid p supplied to Kantorvich distance")
  
imports: import numpy
  import numba
  from pynndescent.optimal_transport import allocate_graph_structures, initialize_graph_structures, initialize_supply, initialize_cost, network_simplex_core, total_cost, ProblemStatus, sinkhorn_transport_plan
label_desc: Use image node_id 42 for calling a global function with example usage: circular_kantorovich(x, y, p) and returns: unknown, np, result