CODEWITHSUNDEEP
pythonsequencedecodecombinatoricsdiscrete-mathematics
Vincent
Vincent

Reputation: 1647

Decoding algorithm to return location of subsequence in De Bruijn sequence

The Wikipedia page for De Bruijn sequence provides an algorithm for construction De Bruijn sequences. Without constructing the De Bruijn sequences I would like an algorithm (preferably in python) which given an alphabet of size K subsequences length N and a specified subsequence S would return the starting list index position of the subsequence S in the De Bruijn sequence generated from K, N.

To be clear, I do not what to actually construct the De Bruijn sequence only compute the location of subsequences.

Is seems clear that the algorithm to compute the location of subsequences would be pair or specific to an algorithm to compute De Bruijn sequence.

Upvotes: 0

Views: 369

Answers (1)

ofou
ofou

Reputation: 311

This algorithm works by exploiting the structure of De Bruijn sequences.

Here's how it works:

  1. We first check if the given subsequence has the correct length (n).
  2. We convert the subsequence to its numerical representation. This is done by treating the subsequence as a number in base k.
  3. The position of the subsequence in the De Bruijn sequence is then calculated using the formula: position = numerical_subsequence * k^(n-1)

This algorithm works because of how De Bruijn sequences are constructed. Each n-length subsequence appears exactly once in the sequence, and they are effectively ordered based on their numerical value when interpreted in base k.

The time complexity of this algorithm is O(n), where n is the length of the subsequence, as we only need to iterate over the subsequence once to convert it to its numerical representation.

def de_bruijn_subsequence_position(k, n, subsequence):
    if len(subsequence) != n:
        raise ValueError("Subsequence length must be equal to n")
    
    # Convert subsequence to its numerical representation
    numerical_subsequence = 0
    for i, char in enumerate(subsequence):
        numerical_subsequence += int(char) * (k ** (n - i - 1))
    
    # Calculate the position
    position = numerical_subsequence * k**(n-1)
    
    return position

Some tests in JS:

function deBruijnSubsequencePosition(k, n, subsequence) {
    if (subsequence.length !== n) {
        throw new Error("Subsequence length must be equal to n");
    }

    // Convert subsequence to its numerical representation
    let numericalSubsequence = 0;
    for (let i = 0; i < n; i++) {
        numericalSubsequence += parseInt(subsequence[i]) * Math.pow(k, (n - i - 1));
    }

    // Calculate the position
    let position = numericalSubsequence * Math.pow(k, n - 1);

    return position;
}

// Test cases
function runTests() {
    const testCases = [
        { k: 2, n: 3, subsequence: "101", expected: 20 },
        { k: 2, n: 3, subsequence: "000", expected: 0 },
        { k: 2, n: 3, subsequence: "111", expected: 28 },
        { k: 3, n: 2, subsequence: "12", expected: 15 },
        { k: 4, n: 2, subsequence: "30", expected: 48 }
    ];

    testCases.forEach((test, index) => {
        try {
            const result = deBruijnSubsequencePosition(test.k, test.n, test.subsequence);
            console.log(`Test ${index + 1}: k=${test.k}, n=${test.n}, subsequence="${test.subsequence}"`);
            console.log(`Result: ${result}`);
            console.log(`Expected: ${test.expected}`);
            console.log(`Status: ${result === test.expected ? 'PASS' : 'FAIL'}`);
        } catch (error) {
            console.log(`Test ${index + 1} failed with error: ${error.message}`);
        }
        console.log('---');
    });
}

// Run the tests
runTests();

Upvotes: 0

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