Kruskal Minimum Spanning Tree in Java

// Kruskal Minimum Spanning Tree in Java
//
// PURPOSE
// Kruskal adds edges in ascending weight order whenever they join different components.
//
// METHOD
// Sort edge indices by weight using a stable bottom-up merge. Use union by size and path
// halving to accept edges joining different roots. Stop after n-1 accepted edges.
//
// COMPLEXITY
// O(E log E + V) time and O(V+E) auxiliary space.
//
// CONTRACT
// n>=1, parallel edge arrays have m entries and valid vertex IDs. Weights nonnegative; total
// fits signed 32-bit. Returns -1 for a disconnected graph.
// Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
// counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
// Sorted order, correct tree topology, and caller-owned data-structure invariants are
// preconditions; checking them on every query would change query complexity.
// Scalar and shape checks are included. Value-range checks are included where needed to
// prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
// workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
// uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
// partly updated after arithmetic failure. These are single-threaded, in-memory
// implementations.
//
// LANGUAGE DETAILS
// Java 17+: primitive int arrays, Arrays.binarySearch for LIS, and Math exact arithmetic.
// Invalid input throws IllegalArgumentException; arithmetic overflow throws
// ArithmeticException.
//
// EXAMPLE
// Expected return: 6.
// Run: javac AlgorithmExample.java && java AlgorithmExample
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

class AlgorithmExample {
    static void alg_require(boolean valid) { if (!valid) throw new IllegalArgumentException("invalid algorithm input"); }
    static int alg_size(int n) { alg_require(n>=0 && n<=4000000); return n; }

    static int kruskal(int[] from_vertex, int[] to_vertex, int[] weight, int n) {
        java.util.Objects.requireNonNull(from_vertex, "from_vertex");
        java.util.Objects.requireNonNull(to_vertex, "to_vertex");
        java.util.Objects.requireNonNull(weight, "weight");
        int m = from_vertex.length;
        alg_require(((n >= 0) && (n <= 1000000)));
        alg_require(((m >= 0) && (m <= 1000000)));
        alg_require((from_vertex.length == m));
        alg_require((to_vertex.length == m));
        alg_require((weight.length == m));
        alg_require((n >= 1));
        for (int alg_i = 0; alg_i < from_vertex.length; alg_i++) {
            int alg_value = from_vertex[alg_i];
            alg_require(((alg_value >= 0) && (alg_value < n)));
        }
        for (int alg_i = 0; alg_i < to_vertex.length; alg_i++) {
            int alg_value = to_vertex[alg_i];
            alg_require(((alg_value >= 0) && (alg_value < n)));
        }
        int[] order = new int[alg_size(Math.addExact(m, 1))];
        int[] buffer = new int[alg_size(Math.addExact(m, 1))];
        int[] parent = new int[alg_size(n)];
        int[] size = new int[alg_size(n)];
        int i = 0;
        int j = 0;
        int k = 0;
        int width = 1;
        int left = 0;
        int mid = 0;
        int right = 0;
        int edge = 0;
        int u = 0;
        int v = 0;
        int temp = 0;
        int accepted = 0;
        int total = 0;
        while ((i < m)) {
            order[i] = i;
            i = Math.addExact(i, 1);
        }
        while ((width < m)) {
            left = 0;
            while ((left < m)) {
                mid = Math.min(Math.addExact(left, width), m);
                right = Math.min(Math.addExact(left, Math.multiplyExact(2, width)), m);
                i = left;
                j = mid;
                k = left;
                while ((k < right)) {
                    if (((i < mid) && ((j >= right) || (weight[order[i]] <= weight[order[j]])))) {
                        buffer[k] = order[i];
                        i = Math.addExact(i, 1);
                    } else {
                        buffer[k] = order[j];
                        j = Math.addExact(j, 1);
                    }
                    k = Math.addExact(k, 1);
                }
                left = Math.addExact(left, Math.multiplyExact(2, width));
            }
            i = 0;
            while ((i < m)) {
                order[i] = buffer[i];
                i = Math.addExact(i, 1);
            }
            width = Math.multiplyExact(width, 2);
        }
        i = 0;
        while ((i < n)) {
            parent[i] = i;
            size[i] = 1;
            i = Math.addExact(i, 1);
        }
        while (((edge < m) && (accepted < Math.subtractExact(n, 1)))) {
            u = from_vertex[order[edge]];
            v = to_vertex[order[edge]];
            while ((parent[u] != u)) {
                parent[u] = parent[parent[u]];
                u = parent[u];
            }
            while ((parent[v] != v)) {
                parent[v] = parent[parent[v]];
                v = parent[v];
            }
            if ((u != v)) {
                if ((size[u] < size[v])) {
                    temp = u;
                    u = v;
                    v = temp;
                }
                parent[v] = u;
                size[u] = Math.addExact(size[u], size[v]);
                accepted = Math.addExact(accepted, 1);
                total = Math.addExact(total, weight[order[edge]]);
            }
            edge = Math.addExact(edge, 1);
        }
        if ((accepted != Math.subtractExact(n, 1))) {
            return Math.negateExact(1);
        }
        return total;
    }

    public static void main(String[] args) {
        int[] from_vertex = new int[] {0, 1, 0};
        int[] to_vertex = new int[] {1, 2, 2};
        int[] weight = new int[] {4, 2, 10};
        int n = 3;
        int result = kruskal(from_vertex, to_vertex, weight, n);
        System.out.println(result);
    }
}