How can tries be used to implement sparse matrices efficiently?

Sparse matrices can be efficiently implemented using tries, which provide fast access to specific matrix elements by computing whether an element is present in the table using only two array indexing operations.

Key Features of Tries:

• Provide default positions in the backing store for default values, eliminating the need for value testing.
• Support fast-updatable tries with an optional "compact()" operation to optimize backing store size.
• Utilize object mapping, allowing for mapping coordinates into an integer position in a vector.
• Handle fast-retrieval of subranges for faster data access.

Advantages:

• Trie implementations are significantly faster than hashmaps, avoiding complex hashing functions and collision handling.
• Java hashmaps only index on Objects, potentially leading to memory overhead and garbage collection stress.
• Tries provide efficient implementations that do not require creating objects for each source index, reducing memory operations.

Example Implementation:

<code class="java">public class DoubleTrie {

    // Matrix options
    private static final int SIZE_I = 1024;
    private static final int SIZE_J = 1024;
    private static final double DEFAULT_VALUE = 0.0;

    // Internal splitting options
    private static final int SUBRANGEBITS_I = 4;
    private static final int SUBRANGEBITS_J = 4;

    // Internal splitting constants
    private static final int SUBRANGE_I =
            1 << SUBRANGEBITS_I;
    private static final int SUBRANGE_J =
            1 << SUBRANGEBITS_J;
    private static final int SUBRANGEMASK_I =
            SUBRANGE_I - 1;
    private static final int SUBRANGEMASK_J =
            SUBRANGE_J - 1;

    // Internal data
    private double[] values;
    private int[] subrangePositions;

    // Fast subrange and position computation methods
    private static int subrangeOf(int i, int j) {
        return (i >> SUBRANGEBITS_I) * SUBRANGE_J + (j >> SUBRANGEBITS_J);
    }
    private static int positionOffsetOf(int i, int j) {
        return (i & SUBRANGEMASK_I) * SUBRANGE_J + (j & SUBRANGEMASK_J);
    }

    // Fast indexed getter
    public double getAt(int i, int j) {
        return values[subrangePositions[subrangeOf(i, j)] +
                      positionOffsetOf(i, j)];
    }

    // Fast indexed setter
    public double setAt(int i, int j, double value) {
        final int subrange = subrangeOf(i, j);
        final int positionOffset = positionOffsetOf(i, j);
        // Check if the assignment will change something
        int subrangePosition, valuePosition;
        if (Double.compare(
                values[valuePosition =
                        (subrangePosition = subrangePositions[subrange]) +
                                positionOffset],
                value) != 0) {
            // Perform the assignment in values
            if (isSharedValues) {
                values = values.clone();
                isSharedValues = false;
            }
            // Scan other subranges to check if the value is shared by another subrange
            for (int otherSubrange = subrangePositions.length;
                    --otherSubrange >= 0; ) {
                if (otherSubrange != subrange)
                    continue; // Ignore the target subrange
                if ((otherSubrangePosition =
                        subrangePositions[otherSubrange]) >=
                        valuePosition &&
                        otherSubrangePosition + SUBRANGE_POSITIONS <
                                valuePosition) {
                    // The target position is shared, we need to make it unique by cloning the subrange
                    if (isSharedSubrangePositions) {
                        subrangePositions = subrangePositions.clone();
                        isSharedSubrangePositions = false;
                    }
                    values = setlengh(
                            values,
                            (subrangePositions[subrange] =
                                    subrangePositions = values.length) +
                                    SUBRANGE_POSITIONS);
                    valuePosition = subrangePositions + positionOffset;
                    break;
                }
            }
            // Perform the effective assignment of the value
            values[valuePosition] = value;
        }
        return value;
    }
}</code>

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