How to Handle Negative Numbers in Modular Arithmetic for Correct Array Indexing?

Modular Arithmetic and Negative Indices

Standard modulo operations (%) on negative integers can produce unexpected results when used for array indexing. The remainder can be negative, leading to invalid index values. To ensure correct positive array indices, we need a modified modulo function.

A common solution is to use this formula:

<code>GetArrayIndex(i, arrayLength) = (i % arrayLength + arrayLength) % arrayLength</code>

This guarantees a positive index within the range [0, arrayLength - 1], irrespective of the input i's sign.

Custom Modulo Function

For cleaner code, a custom mod function is helpful:

<code class="language-java">public static int mod(int x, int m) {
    return (x % m + m) % m;
}</code>

This function handles negative remainders by adding m to ensure a positive result.

Optimized Modulo Function

For improved efficiency (fewer modulo operations), consider this alternative:

<code class="language-java">public static int mod(int x, int m) {
    int r = x % m;
    return r < 0 ? r + m : r;
}</code>

This version directly checks if the remainder r is negative, adding m only when necessary.

Examples

Using either custom mod function, we get the expected array index behavior:

GetArrayIndex(4, 3) == 1 GetArrayIndex(3, 3) == 0 GetArrayIndex(2, 3) == 2 GetArrayIndex(1, 3) == 1 GetArrayIndex(0, 3) == 0 GetArrayIndex(-1, 3) == 2 GetArrayIndex(-2, 3) == 1 GetArrayIndex(-3, 3) == 0 GetArrayIndex(-4, 3) == 2

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