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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Introduction
In this notebook, we implement a simpler version of the BUC (Bottom up cubing) algorithm. 
The original BUC algorithm is a highly optimized and practically efficient algorithm to materialize the entire data cube given a fact table. We instead experiment with a simpler version, called BUC_rec (BUC recursive), based on our teaching slides. The implementation here emphasize more on the conceptual simplicity rather than efficiency. 
Once you understand BUC_rec, you may try to understand BUC if you want to learn more by yourself. 
The recursive formula used in BUC_rec is
Step1:  Now we can implement the buc_rec() algorithm and test it.
Step2:  With the following pivot table, we can easily see the output is correct (i.e., all the (non-empty) aggregates are computed). 
But did you notice anything else interesting?
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Python Code:
import sys
import pandas as pd
import numpy as np
ALL = -1
# DEBUG = True
DEBUG = False
##============================================================
# Data file format: 
# * tab-delimited input file
# * 1st line: dimension names and the last dimension is assumed to be the measure
# * rest of the lines: data values. 
def read_data(filename):
    df = pd.read_csv(filename, sep='\t')
    dims = df.shape[1] - 1 # the last dim is the measure
    return (df, dims)
def dump_input2(input):
    if DEBUG: 
        print("\n.. BUC_rec invoked on:")
        print(input)
        print("......................\n")
        
# helper functions
def project_data(input, d):
    # Return only the d-th column of INPUT
    return input.iloc[:, d]
def select_data(input, d, val):
    # SELECT * FROM INPUT WHERE INPUT.d = VAL
    col_name = input.columns[d]
    return input[input[col_name] == val]
def remove_first_dim(input):
    # Remove the first dim of the input
    return input.iloc[:, 1:]
def slice_data_dim0(input, v):
    # syntactic sugar to get R_{ALL} in a less verbose way
    df_temp = select_data(input, 0, v)
    return remove_first_dim(df_temp)
def output(val):
    print('=>\t{}'.format(val))
data, d = read_data('./asset/a_.txt')
print(d)
data
project_data(data, 0)
select_data(data, 1, 2)
slice_data_dim0(data, 2)
Explanation: Introduction
In this notebook, we implement a simpler version of the BUC (Bottom up cubing) algorithm. 
The original BUC algorithm is a highly optimized and practically efficient algorithm to materialize the entire data cube given a fact table. We instead experiment with a simpler version, called BUC_rec (BUC recursive), based on our teaching slides. The implementation here emphasize more on the conceptual simplicity rather than efficiency. 
Once you understand BUC_rec, you may try to understand BUC if you want to learn more by yourself. 
The recursive formula used in BUC_rec is: 
$$
Cube(R, {A,B,C,\ldots,M}) = \bigcup_{i \in {1, 2, \ldots, m, \text{ALL}}} Cube(R_i, {B,C,\ldots,M}) 
$$
where $R_i$ is $\pi_{B,C,\ldots,M}(\sigma_{A = a_i}(R))$ and $R_{\text{ALL}}$ is $\pi_{B,C,\ldots,M}(R)$.
End of explanation
# Note that input is a DataFrame
def buc_rec(input):
    # Note that input is a DataFrame
    dump_input2(input)
    dims = input.shape[1]
    
    if dims == 1:
        # only the measure dim
        input_sum = sum( project_data(input, 0) )
        output(input_sum)
    else:
        # the general case
        dim0_vals = set(project_data(input, 0).values)
        for dim0_v in dim0_vals:
            sub_data = slice_data_dim0(input, dim0_v)
            buc_rec(sub_data)
        ## for R_{ALL}
        sub_data = remove_first_dim(input)
        buc_rec(sub_data)
buc_rec(data)
Explanation: Now we can implement the buc_rec() algorithm and test it.
End of explanation
data.pivot_table(index = ['A'], columns = ['B'], aggfunc = np.sum, margins = True)
Explanation: With the following pivot table, we can easily see the output is correct (i.e., all the (non-empty) aggregates are computed). 
But did you notice anything else interesting?
End of explanation
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