The provided code defines a function `initiate_global_dp(m, n)` that initializes...

September 1, 2025 at 01:20 PM

def initiate_global_dp(m,n):
    S = np.zeros((m+1, n+1))       # An (m+1)*(n+1) matrix, initiated with 0's
    # Careful! S and trace are shape (m+1)*(n+1). The sequences lengths are m and n. Remember that in the future!
    # For the trace matrix we use a three dimentional matrix of booleans
    # where 
    #   trace(x,y,0) indicates a match in x,y
    #   trace(x,y,1) indicates an insert in x,y (fix column)
    #   trace(x,y,2) indicates a delete in x,y (fix row)
    trace = np.zeros((m+1, n+1, 3), dtype=np.bool_) # An (m+1)*(n+1)*3 boolean matrix, initiated with (False,False,False)

    # HINT: try printing S and trace for debugging.
    # trace is difficult to 3D-visualize out of the box with numpy, so use the function pretty_trace(trace) for a better result!
    # you can also use the function pretty_trace_arrows(trace) for an even clearer (but less abstract) result

    # First initiate the origin of S (0,0) here:
    S[0,0] = 0

    # Now, fill in the first row and the first column of the matrices S and trace
    # The first column of S (S[i, 0]) is filled with values based on the gap penalty.
    # Initiate the first column of S and trace here:
    for i in range (1,m+1):
        S[i,0] = i * gap_penalty()
        trace[i,0,2] = (-1,0,2)
        
    # Initiate the first row of S and trace here:
    for j in range(1,n+1):
        S[0,j] = j * gap_penalty()
        trace[0,j,1] = (0,-1,1)

The provided code defines a function initiate_global_dp(m, n) that initializes two data structures essential for dynamic programming, particularly for problems like global sequence alignment in bioinformatics or related fields where operations like "match," "insert," and "delete" are key.

Here is what the code does step-by-step:

Input

m: Length of the first sequence.
n: Length of the second sequence.

Initializations

Matrix S - Dynamic Programming Table:
- S is a 2D matrix (size (m+1) x (n+1)), initialized with zeros (np.zeros).
- This S is often referred to as the scoring matrix, where S[i, j] will eventually hold the optimal alignment score up to the i-th character of the first sequence and the j-th character of the second sequence.
- The size (m+1, n+1) is used because most dynamic programming solutions include an extra row and column to represent alignment against gaps (empty sequences).
Matrix Trace - Backtracking Information:
- trace is a 3D boolean matrix of dimensions (m+1) x (n+1) x 3.
- Each cell (i, j) in trace has three boolean flags for tracking alignment paths:
  - trace[i, j, 0]: Indicates a "match/mismatch" move.
  - trace[i, j, 1]: Indicates an "insert" operation (insertion in the first sequence; fixed column).
  - trace[i, j, 2]: Indicates a "delete" operation (deletion in the first sequence; fixed row).
This matrix will aid in reconstructing the optimal alignment paths.

Setting the Base Cases

Origin Initialization ((0, 0)):
- The value at S[0, 0] is explicitly set to 0, representing no penalty for aligning two empty sequences.
First Column Initialization (Aligning the First Sequence solely with Gaps):
- For every row index i (from 1 to m), S[i, 0] is updated to be i * gap_penalty(), where gap_penalty() is presumably a function that returns the gap-penalty value (not defined in the given code snippet).
- These values represent the cumulative cost of aligning the first i characters of the first sequence with gaps in the second sequence.
- The corresponding trace[i, 0, 2] is set, indicating that these moves are "delete" operations (moving vertically in the DP table).
First Row Initialization (Aligning the Second Sequence solely with Gaps):
- For every column index j (from 1 to n), S[0, j] is updated to j * gap_penalty(), representing the cost of aligning the first j characters of the second sequence with gaps in the first sequence.
- The corresponding trace[0, j, 1] is set, indicating that these moves are "insert" operations (moving horizontally in the DP table).

Gaps in Code and Bugs:

Typo in trace Assignment:
- Example: trace[i, 0, 2] = (-1, 0, 2) is not valid syntax because trace is a boolean matrix. The value assigned should instead be True (e.g., trace[i, 0, 2] = True).
Undefined gap_penalty() Function:
- The function gap_penalty() is invoked but not defined. It presumably returns a scalar penalty value for introducing a gap (could be positive or negative, depending on the scoring scheme).
Debugging Notes:
- Hints provided for debugging suggest that S can be easily printed, while visualizing the 3D trace matrix may require helper functions like pretty_trace(trace) and pretty_trace_arrows(trace). These functions are not provided but are meant to visually clarify the alignment paths.

Potential Applications

This code is most likely setting up a framework for global sequence alignment (like Needleman-Wunsch algorithm) or a similar problem involving two sequences and dynamic programming. Here’s how:

S would hold the computed alignment scores.
trace would store the path information to backtrack and reconstruct the optimal alignment at the end.

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