Algorithms: Graph Algorithms

Algorithms: Graph Algorithms

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Algorithms: Graph Algorithms

Kosaraju Strongly Connected Components in Java

Strongly connected components group vertices that can reach each other in both directions..

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// Kosaraju Strongly Connected Components in Java
//
// PURPOSE
// Strongly connected components group vertices that can reach each other in both directions.
//
// METHOD
// First perform iterative DFS to record finishing order. Traverse vertices in reverse
// finishing order using incoming edges, counting each newly discovered component.
//
// COMPLEXITY
// O(n squared) time for the dense matrix and O(n) auxiliary space.
//
// CONTRACT
// Flattened n by n directed 0/1 matrix. Returns number of components; empty graph returns
// zero.
// 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: 2.
// 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 scc_count(int[] graph, int n) {
        java.util.Objects.requireNonNull(graph, "graph");
        alg_require(((n >= 0) && (n <= 1000000)));
        alg_require((graph.length == Math.multiplyExact(n, n)));
        int[] seen = new int[alg_size(Math.addExact(n, 1))];
        int[] stack = new int[alg_size(Math.addExact(n, 1))];
        int[] next_vertex = new int[alg_size(Math.addExact(n, 1))];
        int[] order = new int[alg_size(Math.addExact(n, 1))];
        int source = 0;
        int top = 0;
        int finished = 0;
        int u = 0;
        int v = 0;
        int count = 0;
        while ((source < n)) {
            if ((seen[source] == 0)) {
                stack[0] = source;
                top = 1;
                seen[source] = 1;
                while ((top > 0)) {
                    u = stack[Math.subtractExact(top, 1)];
                    if ((next_vertex[u] == n)) {
                        order[finished] = u;
                        finished = Math.addExact(finished, 1);
                        top = Math.subtractExact(top, 1);
                    } else {
                        v = next_vertex[u];
                        next_vertex[u] = Math.addExact(next_vertex[u], 1);
                        if (((graph[Math.addExact(Math.multiplyExact(u, n), v)] != 0) && (seen[v] == 0))) {
                            seen[v] = 1;
                            stack[top] = v;
                            top = Math.addExact(top, 1);
                        }
                    }
                }
            }
            source = Math.addExact(source, 1);
        }
        source = 0;
        while ((source < n)) {
            seen[source] = 0;
            source = Math.addExact(source, 1);
        }
        while ((finished > 0)) {
            finished = Math.subtractExact(finished, 1);
            source = order[finished];
            if ((seen[source] == 0)) {
                count = Math.addExact(count, 1);
                stack[0] = source;
                top = 1;
                seen[source] = 1;
                while ((top > 0)) {
                    top = Math.subtractExact(top, 1);
                    u = stack[top];
                    v = 0;
                    while ((v < n)) {
                        if (((graph[Math.addExact(Math.multiplyExact(v, n), u)] != 0) && (seen[v] == 0))) {
                            seen[v] = 1;
                            stack[top] = v;
                            top = Math.addExact(top, 1);
                        }
                        v = Math.addExact(v, 1);
                    }
                }
            }
        }
        return count;
    }

    public static void main(String[] args) {
        int[] graph = new int[] {0, 1, 0, 1, 0, 1, 0, 0, 0};
        int n = 3;
        int result = scc_count(graph, n);
        System.out.println(result);
    }
}
Nov 15, 2026 Java
Algorithms: Graph Algorithms

Kosaraju Strongly Connected Components in JavaScript

Strongly connected components group vertices that can reach each other in both directions..

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"use strict";
// Kosaraju Strongly Connected Components in JavaScript
//
// PURPOSE
// Strongly connected components group vertices that can reach each other in both directions.
//
// METHOD
// First perform iterative DFS to record finishing order. Traverse vertices in reverse
// finishing order using incoming edges, counting each newly discovered component.
//
// COMPLEXITY
// O(n squared) time for the dense matrix and O(n) auxiliary space.
//
// CONTRACT
// Flattened n by n directed 0/1 matrix. Returns number of components; empty graph returns
// zero.
// 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
// Node.js 18+: Int32Array inputs and workspaces provide compact integer storage. Automatic
// lengths and checked arithmetic reject values outside signed 32-bit. Invalid inputs or
// overflow throw RangeError.
//
// EXAMPLE
// Expected return: 2.
// Run: node example.js
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

function alg_require(valid) { if (!valid) throw new RangeError("invalid algorithm input"); }
function alg_checked(v) { if (!Number.isInteger(v) || v < -2147483648 || v > 2147483647) throw new RangeError("32-bit result overflow"); return v; }
function alg_add(a,b) { return alg_checked(a+b); }
function alg_sub(a,b) { return alg_checked(a-b); }
function alg_mul(a,b) { return alg_checked(a*b); }
function alg_size(n) { alg_require(Number.isInteger(n) && n>=0 && n<=4000000); return n; }

function scc_count(graph, n) {
    alg_require(graph instanceof Int32Array);
    alg_require(Number.isInteger(n) && n >= -2147483648 && n <= 2147483647);
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require((graph.length == alg_mul(n, n)));
    let seen = new Int32Array(alg_size(alg_add(n, 1)));
    let stack = new Int32Array(alg_size(alg_add(n, 1)));
    let next_vertex = new Int32Array(alg_size(alg_add(n, 1)));
    let order = new Int32Array(alg_size(alg_add(n, 1)));
    let source = 0;
    let top = 0;
    let finished = 0;
    let u = 0;
    let v = 0;
    let count = 0;
    while ((source < n)) {
        if ((seen[source] == 0)) {
            stack[0] = source;
            top = 1;
            seen[source] = 1;
            while ((top > 0)) {
                u = stack[alg_sub(top, 1)];
                if ((next_vertex[u] == n)) {
                    order[finished] = u;
                    finished = alg_add(finished, 1);
                    top = alg_sub(top, 1);
                } else {
                    v = next_vertex[u];
                    next_vertex[u] = alg_add(next_vertex[u], 1);
                    if (((graph[alg_add(alg_mul(u, n), v)] != 0) && (seen[v] == 0))) {
                        seen[v] = 1;
                        stack[top] = v;
                        top = alg_add(top, 1);
                    }
                }
            }
        }
        source = alg_add(source, 1);
    }
    source = 0;
    while ((source < n)) {
        seen[source] = 0;
        source = alg_add(source, 1);
    }
    while ((finished > 0)) {
        finished = alg_sub(finished, 1);
        source = order[finished];
        if ((seen[source] == 0)) {
            count = alg_add(count, 1);
            stack[0] = source;
            top = 1;
            seen[source] = 1;
            while ((top > 0)) {
                top = alg_sub(top, 1);
                u = stack[top];
                v = 0;
                while ((v < n)) {
                    if (((graph[alg_add(alg_mul(v, n), u)] != 0) && (seen[v] == 0))) {
                        seen[v] = 1;
                        stack[top] = v;
                        top = alg_add(top, 1);
                    }
                    v = alg_add(v, 1);
                }
            }
        }
    }
    return count;
}

let graph = new Int32Array([0, 1, 0, 1, 0, 1, 0, 0, 0]);
let n = 3;
let result = scc_count(graph, n);
console.log(result);
Nov 15, 2026 javascript
Algorithms: Graph Algorithms

Kosaraju Strongly Connected Components in Python

Strongly connected components group vertices that can reach each other in both directions..

python 
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from bisect import bisect_left

# Kosaraju Strongly Connected Components in Python
#
# PURPOSE
# Strongly connected components group vertices that can reach each other in both directions.
#
# METHOD
# First perform iterative DFS to record finishing order. Traverse vertices in reverse
# finishing order using incoming edges, counting each newly discovered component.
#
# COMPLEXITY
# O(n squared) time for the dense matrix and O(n) auxiliary space.
#
# CONTRACT
# Flattened n by n directed 0/1 matrix. Returns number of components; empty graph returns
# zero.
# 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
# Python 3.9+: automatic lengths, native lists, arbitrary-precision arithmetic, and
# bisect_left for LIS. Invalid scalar/shape inputs raise ValueError. Integer element types
# and sorted/tree invariants remain caller preconditions.
#
# EXAMPLE
# Expected return: 2.
# Run: python example.py
# Tests for this exact source artifact run before publication. The accompanying source
# archive includes unit cases, runtime results, and source hashes.

def alg_require(condition):
    if not condition:
        raise ValueError("invalid algorithm input; see contract")
def alg_size(n):
    alg_require(0 <= n <= 4000000)
    return n

def scc_count(graph: list[int], n: int) -> int:
    alg_require(isinstance(graph, (list, tuple)) and len(graph) <= 1000000)
    alg_require(type(n) is int and -2147483648 <= n <= 2147483647)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(len(graph) == n * n)
    seen: list[int] = [0] * alg_size(n + 1)
    stack: list[int] = [0] * alg_size(n + 1)
    next_vertex: list[int] = [0] * alg_size(n + 1)
    order: list[int] = [0] * alg_size(n + 1)
    source: int = 0
    top: int = 0
    finished: int = 0
    u: int = 0
    v: int = 0
    count: int = 0
    while source < n:
        if seen[source] == 0:
            stack[0] = source
            top = 1
            seen[source] = 1
            while top > 0:
                u = stack[top - 1]
                if next_vertex[u] == n:
                    order[finished] = u
                    finished += 1
                    top -= 1
                else:
                    v = next_vertex[u]
                    next_vertex[u] += 1
                    if graph[u * n + v] != 0 and seen[v] == 0:
                        seen[v] = 1
                        stack[top] = v
                        top += 1
        source += 1
    source = 0
    while source < n:
        seen[source] = 0
        source += 1
    while finished > 0:
        finished -= 1
        source = order[finished]
        if seen[source] == 0:
            count += 1
            stack[0] = source
            top = 1
            seen[source] = 1
            while top > 0:
                top -= 1
                u = stack[top]
                v = 0
                while v < n:
                    if graph[v * n + u] != 0 and seen[v] == 0:
                        seen[v] = 1
                        stack[top] = v
                        top += 1
                    v += 1
    return count

graph = [0, 1, 0, 1, 0, 1, 0, 0, 0]
n = 3
result = scc_count(graph, n)
print(result)
Nov 15, 2026 python
Algorithms: Graph Algorithms

Kosaraju Strongly Connected Components in C

Strongly connected components group vertices that can reach each other in both directions..

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#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <limits.h>
#include <errno.h>
// Kosaraju Strongly Connected Components in C
//
// PURPOSE
// Strongly connected components group vertices that can reach each other in both directions.
//
// METHOD
// First perform iterative DFS to record finishing order. Traverse vertices in reverse
// finishing order using incoming edges, counting each newly discovered component.
//
// COMPLEXITY
// O(n squared) time for the dense matrix and O(n) auxiliary space.
//
// CONTRACT
// Flattened n by n directed 0/1 matrix. Returns number of components; empty graph returns
// zero.
// 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
// C99: checked calloc/free workspace, const input pointers, explicit size_t lengths. Errors
// return INT_MIN with errno=EINVAL, ERANGE, or ENOMEM. Check errno before using a result.
// Input/output arrays must not alias. No variable-length stack arrays.
//
// EXAMPLE
// Expected return: 2.
// Run: gcc -std=c99 -O2 example.c -o example && ./example
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

static int alg_min(int a, int b) { return a < b ? a : b; }
static int alg_max(int a, int b) { return a > b ? a : b; }
static int alg_checked(int64_t v, int *error) {
    if (v < INT_MIN || v > INT_MAX) { *error = 1; return 0; }
    return (int)v;
}
static int alg_add(int a, int b, int *e) { return alg_checked((int64_t)a+b,e); }
static int alg_sub(int a, int b, int *e) { return alg_checked((int64_t)a-b,e); }
static int alg_mul(int a, int b, int *e) { return alg_checked((int64_t)a*b,e); }

int scc_count(const int *graph, size_t graph_length, int n) {
    int alg_error = 0;
    int alg_result = INT_MIN;
    errno = 0;
    int *seen = NULL;
    int *stack = NULL;
    int *next_vertex = NULL;
    int *order = NULL;
    if (graph_length > 1000000 || (graph_length && !graph)) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((n >= 0) && (n <= 1000000)))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((int)graph_length == alg_mul(n, n, &alg_error)))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    {
        int alg_length = alg_add(n, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_length < 0 || alg_length > 4000000) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
        seen = calloc((size_t)(alg_length ? alg_length : 1), sizeof(*seen));
        if (!seen) { errno = ENOMEM; alg_result = INT_MIN; goto alg_cleanup; }
    }
    {
        int alg_length = alg_add(n, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_length < 0 || alg_length > 4000000) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
        stack = calloc((size_t)(alg_length ? alg_length : 1), sizeof(*stack));
        if (!stack) { errno = ENOMEM; alg_result = INT_MIN; goto alg_cleanup; }
    }
    {
        int alg_length = alg_add(n, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_length < 0 || alg_length > 4000000) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
        next_vertex = calloc((size_t)(alg_length ? alg_length : 1), sizeof(*next_vertex));
        if (!next_vertex) { errno = ENOMEM; alg_result = INT_MIN; goto alg_cleanup; }
    }
    {
        int alg_length = alg_add(n, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_length < 0 || alg_length > 4000000) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
        order = calloc((size_t)(alg_length ? alg_length : 1), sizeof(*order));
        if (!order) { errno = ENOMEM; alg_result = INT_MIN; goto alg_cleanup; }
    }
    int source = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int top = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int finished = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int u = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int v = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int count = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    while (1) {
        int alg_condition = (source < n);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        int alg_condition_1 = (seen[source] == 0);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_condition_1) {
            stack[0] = source;
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            top = 1;
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            seen[source] = 1;
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            while (1) {
                int alg_condition = (top > 0);
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                if (!alg_condition) break;
                u = stack[alg_sub(top, 1, &alg_error)];
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                int alg_condition_2 = (next_vertex[u] == n);
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                if (alg_condition_2) {
                    order[finished] = u;
                    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                    finished = alg_add(finished, 1, &alg_error);
                    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                    top = alg_sub(top, 1, &alg_error);
                    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                } else {
                    v = next_vertex[u];
                    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                    next_vertex[u] = alg_add(next_vertex[u], 1, &alg_error);
                    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                    int alg_condition_3 = ((graph[alg_add(alg_mul(u, n, &alg_error), v, &alg_error)] != 0) && (seen[v] == 0));
                    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                    if (alg_condition_3) {
                        seen[v] = 1;
                        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                        stack[top] = v;
                        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                        top = alg_add(top, 1, &alg_error);
                        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                    }
                }
            }
        }
        source = alg_add(source, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    source = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    while (1) {
        int alg_condition = (source < n);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        seen[source] = 0;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        source = alg_add(source, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    while (1) {
        int alg_condition = (finished > 0);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        finished = alg_sub(finished, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        source = order[finished];
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        int alg_condition_4 = (seen[source] == 0);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_condition_4) {
            count = alg_add(count, 1, &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            stack[0] = source;
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            top = 1;
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            seen[source] = 1;
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            while (1) {
                int alg_condition = (top > 0);
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                if (!alg_condition) break;
                top = alg_sub(top, 1, &alg_error);
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                u = stack[top];
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                v = 0;
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                while (1) {
                    int alg_condition = (v < n);
                    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                    if (!alg_condition) break;
                    int alg_condition_5 = ((graph[alg_add(alg_mul(v, n, &alg_error), u, &alg_error)] != 0) && (seen[v] == 0));
                    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                    if (alg_condition_5) {
                        seen[v] = 1;
                        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                        stack[top] = v;
                        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                        top = alg_add(top, 1, &alg_error);
                        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                    }
                    v = alg_add(v, 1, &alg_error);
                    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                }
            }
        }
    }
    alg_result = count;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    goto alg_cleanup;
alg_cleanup:
    free(seen);
    free(stack);
    free(next_vertex);
    free(order);
    return alg_result;
}

int main(void) {
    int graph[] = {0, 1, 0, 1, 0, 1, 0, 0, 0};
    int n = 3;
    int result = scc_count(graph, 9, n);
    printf("%d\n", result);
    if (errno) { perror("algorithm"); return 1; }
    return 0;
}
Nov 15, 2026 C
Algorithms: Graph Algorithms

Kosaraju Strongly Connected Components in C++

Strongly connected components group vertices that can reach each other in both directions..

cpp 
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#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// Kosaraju Strongly Connected Components in C++
//
// PURPOSE
// Strongly connected components group vertices that can reach each other in both directions.
//
// METHOD
// First perform iterative DFS to record finishing order. Traverse vertices in reverse
// finishing order using incoming edges, counting each newly discovered component.
//
// COMPLEXITY
// O(n squared) time for the dense matrix and O(n) auxiliary space.
//
// CONTRACT
// Flattened n by n directed 0/1 matrix. Returns number of components; empty graph returns
// zero.
// 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
// C++17: const vector references for inputs, mutable references for outputs, RAII workspace,
// standard lower_bound for LIS. Invalid inputs throw invalid_argument; overflow throws
// overflow_error; allocation failure propagates bad_alloc.
//
// EXAMPLE
// Expected return: 2.
// Run: g++ -std=c++17 -O2 example.cpp -o example && ./example
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

static int alg_min(int a, int b) { return a < b ? a : b; }
static int alg_max(int a, int b) { return a > b ? a : b; }
static void alg_require(bool valid) { if (!valid) throw std::invalid_argument("invalid algorithm input"); }
static int alg_checked(int64_t v) { if (v < INT32_MIN || v > INT32_MAX) throw std::overflow_error("32-bit result overflow"); return (int)v; }
static int alg_add(int a,int b) { return alg_checked((int64_t)a+b); }
static int alg_sub(int a,int b) { return alg_checked((int64_t)a-b); }
static int alg_mul(int a,int b) { return alg_checked((int64_t)a*b); }
static size_t alg_size(int n) { alg_require(n>=0 && n<=4000000); return (size_t)n; }

int scc_count(const std::vector<int>& graph, int n) {
    alg_require(graph.size() <= 1000000);
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require(((int)graph.size() == alg_mul(n, n)));
    std::vector<int> seen(alg_size(alg_add(n, 1)), 0);
    std::vector<int> stack(alg_size(alg_add(n, 1)), 0);
    std::vector<int> next_vertex(alg_size(alg_add(n, 1)), 0);
    std::vector<int> order(alg_size(alg_add(n, 1)), 0);
    int source = 0;
    int top = 0;
    int finished = 0;
    int u = 0;
    int v = 0;
    int count = 0;
    while ((source < n)) {
        if ((seen[source] == 0)) {
            stack[0] = source;
            top = 1;
            seen[source] = 1;
            while ((top > 0)) {
                u = stack[alg_sub(top, 1)];
                if ((next_vertex[u] == n)) {
                    order[finished] = u;
                    finished = alg_add(finished, 1);
                    top = alg_sub(top, 1);
                } else {
                    v = next_vertex[u];
                    next_vertex[u] = alg_add(next_vertex[u], 1);
                    if (((graph[alg_add(alg_mul(u, n), v)] != 0) && (seen[v] == 0))) {
                        seen[v] = 1;
                        stack[top] = v;
                        top = alg_add(top, 1);
                    }
                }
            }
        }
        source = alg_add(source, 1);
    }
    source = 0;
    while ((source < n)) {
        seen[source] = 0;
        source = alg_add(source, 1);
    }
    while ((finished > 0)) {
        finished = alg_sub(finished, 1);
        source = order[finished];
        if ((seen[source] == 0)) {
            count = alg_add(count, 1);
            stack[0] = source;
            top = 1;
            seen[source] = 1;
            while ((top > 0)) {
                top = alg_sub(top, 1);
                u = stack[top];
                v = 0;
                while ((v < n)) {
                    if (((graph[alg_add(alg_mul(v, n), u)] != 0) && (seen[v] == 0))) {
                        seen[v] = 1;
                        stack[top] = v;
                        top = alg_add(top, 1);
                    }
                    v = alg_add(v, 1);
                }
            }
        }
    }
    return count;
}

int main(void) {
    std::vector<int> graph = {0, 1, 0, 1, 0, 1, 0, 0, 0};
    int n = 3;
    int result = scc_count(graph, n);
    printf("%d\n", result);
    return 0;
}
Nov 15, 2026 C++