Algorithms: Graph Algorithms

Algorithms: Graph Algorithms

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Algorithms: Graph Algorithms

DAG Shortest Path in JavaScript

A directed acyclic graph permits shortest-path relaxation in topological order even when weights are negative..

javascript 
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"use strict";
// DAG Shortest Path in JavaScript
//
// PURPOSE
// A directed acyclic graph permits shortest-path relaxation in topological order even when
// weights are negative.
//
// METHOD
// Use Kahn ordering and relax each outgoing edge when removing its source. Detect a cycle if
// not all vertices are processed.
//
// COMPLEXITY
// O(n squared) time with a dense matrix and O(n) auxiliary space.
//
// CONTRACT
// Flattened matrix uses INF=1000000000 for missing edges. Diagonal must be INF (no self-
// loops). Finite sums stay strictly between -INF and INF. Returns INF if unreachable, -INF
// if cyclic.
// 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 dag_shortest_path(graph, n, source, target) {
    alg_require(graph instanceof Int32Array);
    alg_require(Number.isInteger(n) && n >= -2147483648 && n <= 2147483647);
    alg_require(Number.isInteger(source) && source >= -2147483648 && source <= 2147483647);
    alg_require(Number.isInteger(target) && target >= -2147483648 && target <= 2147483647);
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require((graph.length == alg_mul(n, n)));
    alg_require(((n >= 1) && (source >= 0) && (source < n)));
    alg_require(((target >= 0) && (target < n)));
    let degree = new Int32Array(alg_size(n));
    let distance = new Int32Array(alg_size(n));
    let queue = new Int32Array(alg_size(n));
    let head = 0;
    let tail = 0;
    let u = 0;
    let v = 0;
    while ((u < n)) {
        distance[u] = 1000000000;
        v = 0;
        while ((v < n)) {
            if ((graph[alg_add(alg_mul(u, n), v)] != 1000000000)) {
                degree[v] = alg_add(degree[v], 1);
            }
            v = alg_add(v, 1);
        }
        u = alg_add(u, 1);
    }
    distance[source] = 0;
    v = 0;
    while ((v < n)) {
        if ((degree[v] == 0)) {
            queue[tail] = v;
            tail = alg_add(tail, 1);
        }
        v = alg_add(v, 1);
    }
    while ((head < tail)) {
        u = queue[head];
        head = alg_add(head, 1);
        v = 0;
        while ((v < n)) {
            if ((graph[alg_add(alg_mul(u, n), v)] != 1000000000)) {
                if ((distance[u] != 1000000000)) {
                    distance[v] = Math.min(distance[v], alg_add(distance[u], graph[alg_add(alg_mul(u, n), v)]));
                }
                degree[v] = alg_sub(degree[v], 1);
                if ((degree[v] == 0)) {
                    queue[tail] = v;
                    tail = alg_add(tail, 1);
                }
            }
            v = alg_add(v, 1);
        }
    }
    if ((tail != n)) {
        return alg_sub(0, 1000000000);
    }
    return distance[target];
}

let graph = new Int32Array([1000000000, 4, 10, 1000000000, 1000000000, -2, 1000000000, 1000000000, 1000000000]);
let n = 3;
let source = 0;
let target = 2;
let result = dag_shortest_path(graph, n, source, target);
console.log(result);
Nov 15, 2026 javascript
Algorithms: Graph Algorithms

DAG Shortest Path in Python

A directed acyclic graph permits shortest-path relaxation in topological order even when weights are negative..

python 
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from bisect import bisect_left

# DAG Shortest Path in Python
#
# PURPOSE
# A directed acyclic graph permits shortest-path relaxation in topological order even when
# weights are negative.
#
# METHOD
# Use Kahn ordering and relax each outgoing edge when removing its source. Detect a cycle if
# not all vertices are processed.
#
# COMPLEXITY
# O(n squared) time with a dense matrix and O(n) auxiliary space.
#
# CONTRACT
# Flattened matrix uses INF=1000000000 for missing edges. Diagonal must be INF (no self-
# loops). Finite sums stay strictly between -INF and INF. Returns INF if unreachable, -INF
# if cyclic.
# 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 dag_shortest_path(graph: list[int], n: int, source: int, target: int) -> int:
    alg_require(isinstance(graph, (list, tuple)) and len(graph) <= 1000000)
    alg_require(type(n) is int and -2147483648 <= n <= 2147483647)
    alg_require(type(source) is int and -2147483648 <= source <= 2147483647)
    alg_require(type(target) is int and -2147483648 <= target <= 2147483647)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(len(graph) == n * n)
    alg_require(n >= 1 and source >= 0 and (source < n))
    alg_require(target >= 0 and target < n)
    degree: list[int] = [0] * alg_size(n)
    distance: list[int] = [0] * alg_size(n)
    queue: list[int] = [0] * alg_size(n)
    head: int = 0
    tail: int = 0
    u: int = 0
    v: int = 0
    while u < n:
        distance[u] = 1000000000
        v = 0
        while v < n:
            if graph[u * n + v] != 1000000000:
                degree[v] += 1
            v += 1
        u += 1
    distance[source] = 0
    v = 0
    while v < n:
        if degree[v] == 0:
            queue[tail] = v
            tail += 1
        v += 1
    while head < tail:
        u = queue[head]
        head += 1
        v = 0
        while v < n:
            if graph[u * n + v] != 1000000000:
                if distance[u] != 1000000000:
                    distance[v] = min(distance[v], distance[u] + graph[u * n + v])
                degree[v] -= 1
                if degree[v] == 0:
                    queue[tail] = v
                    tail += 1
            v += 1
    if tail != n:
        return -1000000000
    return distance[target]

graph = [1000000000, 4, 10, 1000000000, 1000000000, -2, 1000000000, 1000000000, 1000000000]
n = 3
source = 0
target = 2
result = dag_shortest_path(graph, n, source, target)
print(result)
Nov 15, 2026 python
Algorithms: Graph Algorithms

DAG Shortest Path in C

A directed acyclic graph permits shortest-path relaxation in topological order even when weights are negative..

c 
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#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <limits.h>
#include <errno.h>
// DAG Shortest Path in C
//
// PURPOSE
// A directed acyclic graph permits shortest-path relaxation in topological order even when
// weights are negative.
//
// METHOD
// Use Kahn ordering and relax each outgoing edge when removing its source. Detect a cycle if
// not all vertices are processed.
//
// COMPLEXITY
// O(n squared) time with a dense matrix and O(n) auxiliary space.
//
// CONTRACT
// Flattened matrix uses INF=1000000000 for missing edges. Diagonal must be INF (no self-
// loops). Finite sums stay strictly between -INF and INF. Returns INF if unreachable, -INF
// if cyclic.
// 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 dag_shortest_path(const int *graph, size_t graph_length, int n, int source, int target) {
    int alg_error = 0;
    int alg_result = INT_MIN;
    errno = 0;
    int *degree = NULL;
    int *distance = NULL;
    int *queue = 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; }
    if (!(((n >= 1) && (source >= 0) && (source < n)))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((target >= 0) && (target < n)))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    {
        int alg_length = n;
        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; }
        degree = calloc((size_t)(alg_length ? alg_length : 1), sizeof(*degree));
        if (!degree) { errno = ENOMEM; alg_result = INT_MIN; goto alg_cleanup; }
    }
    {
        int alg_length = n;
        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; }
        distance = calloc((size_t)(alg_length ? alg_length : 1), sizeof(*distance));
        if (!distance) { errno = ENOMEM; alg_result = INT_MIN; goto alg_cleanup; }
    }
    {
        int alg_length = n;
        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; }
        queue = calloc((size_t)(alg_length ? alg_length : 1), sizeof(*queue));
        if (!queue) { errno = ENOMEM; alg_result = INT_MIN; goto alg_cleanup; }
    }
    int head = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int tail = 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; }
    while (1) {
        int alg_condition = (u < n);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        distance[u] = 1000000000;
        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_1 = (graph[alg_add(alg_mul(u, n, &alg_error), v, &alg_error)] != 1000000000);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            if (alg_condition_1) {
                degree[v] = alg_add(degree[v], 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; }
        }
        u = alg_add(u, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    distance[source] = 0;
    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_2 = (degree[v] == 0);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_condition_2) {
            queue[tail] = v;
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            tail = alg_add(tail, 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; }
    }
    while (1) {
        int alg_condition = (head < tail);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        u = queue[head];
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        head = alg_add(head, 1, &alg_error);
        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_3 = (graph[alg_add(alg_mul(u, n, &alg_error), v, &alg_error)] != 1000000000);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            if (alg_condition_3) {
                int alg_condition_4 = (distance[u] != 1000000000);
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                if (alg_condition_4) {
                    distance[v] = alg_min(distance[v], alg_add(distance[u], graph[alg_add(alg_mul(u, n, &alg_error), v, &alg_error)], &alg_error));
                    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                }
                degree[v] = alg_sub(degree[v], 1, &alg_error);
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                int alg_condition_5 = (degree[v] == 0);
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                if (alg_condition_5) {
                    queue[tail] = v;
                    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                    tail = alg_add(tail, 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; }
        }
    }
    int alg_condition_6 = (tail != n);
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_condition_6) {
        alg_result = alg_sub(0, 1000000000, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        goto alg_cleanup;
    }
    alg_result = distance[target];
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    goto alg_cleanup;
alg_cleanup:
    free(degree);
    free(distance);
    free(queue);
    return alg_result;
}

int main(void) {
    int graph[] = {1000000000, 4, 10, 1000000000, 1000000000, -2, 1000000000, 1000000000, 1000000000};
    int n = 3;
    int source = 0;
    int target = 2;
    int result = dag_shortest_path(graph, 9, n, source, target);
    printf("%d\n", result);
    if (errno) { perror("algorithm"); return 1; }
    return 0;
}
Nov 15, 2026 C
Algorithms: Graph Algorithms

DAG Shortest Path in C++

A directed acyclic graph permits shortest-path relaxation in topological order even when weights are negative..

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#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// DAG Shortest Path in C++
//
// PURPOSE
// A directed acyclic graph permits shortest-path relaxation in topological order even when
// weights are negative.
//
// METHOD
// Use Kahn ordering and relax each outgoing edge when removing its source. Detect a cycle if
// not all vertices are processed.
//
// COMPLEXITY
// O(n squared) time with a dense matrix and O(n) auxiliary space.
//
// CONTRACT
// Flattened matrix uses INF=1000000000 for missing edges. Diagonal must be INF (no self-
// loops). Finite sums stay strictly between -INF and INF. Returns INF if unreachable, -INF
// if cyclic.
// 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 dag_shortest_path(const std::vector<int>& graph, int n, int source, int target) {
    alg_require(graph.size() <= 1000000);
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require(((int)graph.size() == alg_mul(n, n)));
    alg_require(((n >= 1) && (source >= 0) && (source < n)));
    alg_require(((target >= 0) && (target < n)));
    std::vector<int> degree(alg_size(n), 0);
    std::vector<int> distance(alg_size(n), 0);
    std::vector<int> queue(alg_size(n), 0);
    int head = 0;
    int tail = 0;
    int u = 0;
    int v = 0;
    while ((u < n)) {
        distance[u] = 1000000000;
        v = 0;
        while ((v < n)) {
            if ((graph[alg_add(alg_mul(u, n), v)] != 1000000000)) {
                degree[v] = alg_add(degree[v], 1);
            }
            v = alg_add(v, 1);
        }
        u = alg_add(u, 1);
    }
    distance[source] = 0;
    v = 0;
    while ((v < n)) {
        if ((degree[v] == 0)) {
            queue[tail] = v;
            tail = alg_add(tail, 1);
        }
        v = alg_add(v, 1);
    }
    while ((head < tail)) {
        u = queue[head];
        head = alg_add(head, 1);
        v = 0;
        while ((v < n)) {
            if ((graph[alg_add(alg_mul(u, n), v)] != 1000000000)) {
                if ((distance[u] != 1000000000)) {
                    distance[v] = alg_min(distance[v], alg_add(distance[u], graph[alg_add(alg_mul(u, n), v)]));
                }
                degree[v] = alg_sub(degree[v], 1);
                if ((degree[v] == 0)) {
                    queue[tail] = v;
                    tail = alg_add(tail, 1);
                }
            }
            v = alg_add(v, 1);
        }
    }
    if ((tail != n)) {
        return alg_sub(0, 1000000000);
    }
    return distance[target];
}

int main(void) {
    std::vector<int> graph = {1000000000, 4, 10, 1000000000, 1000000000, -2, 1000000000, 1000000000, 1000000000};
    int n = 3;
    int source = 0;
    int target = 2;
    int result = dag_shortest_path(graph, n, source, target);
    printf("%d\n", result);
    return 0;
}
Nov 15, 2026 C++
Algorithms: Graph Algorithms

DAG Shortest Path in C#

A directed acyclic graph permits shortest-path relaxation in topological order even when weights are negative..

csharp 
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using System;
// DAG Shortest Path in C#
//
// PURPOSE
// A directed acyclic graph permits shortest-path relaxation in topological order even when
// weights are negative.
//
// METHOD
// Use Kahn ordering and relax each outgoing edge when removing its source. Detect a cycle if
// not all vertices are processed.
//
// COMPLEXITY
// O(n squared) time with a dense matrix and O(n) auxiliary space.
//
// CONTRACT
// Flattened matrix uses INF=1000000000 for missing edges. Diagonal must be INF (no self-
// loops). Finite sums stay strictly between -INF and INF. Returns INF if unreachable, -INF
// if cyclic.
// 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# / .NET 10: ReadOnlySpan<int> inputs, Span<int> outputs, primitive workspace arrays,
// Array.BinarySearch for LIS, and checked integer arithmetic. Invalid arguments and overflow
// throw exceptions.
//
// EXAMPLE
// Expected return: 2.
// Run: Use Program.cs in a net10.0 console project; dotnet run.
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

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

    static int dag_shortest_path(ReadOnlySpan<int> graph, int n, int source, int target) {
        checked {
            alg_require(((n >= 0) && (n <= 1000000)));
            alg_require((graph.Length == (n * n)));
            alg_require(((n >= 1) && (source >= 0) && (source < n)));
            alg_require(((target >= 0) && (target < n)));
            int[] degree = new int[alg_size(n)];
            int[] distance = new int[alg_size(n)];
            int[] queue = new int[alg_size(n)];
            int head = 0;
            int tail = 0;
            int u = 0;
            int v = 0;
            while ((u < n)) {
                distance[u] = 1000000000;
                v = 0;
                while ((v < n)) {
                    if ((graph[((u * n) + v)] != 1000000000)) {
                        degree[v] = (degree[v] + 1);
                    }
                    v = (v + 1);
                }
                u = (u + 1);
            }
            distance[source] = 0;
            v = 0;
            while ((v < n)) {
                if ((degree[v] == 0)) {
                    queue[tail] = v;
                    tail = (tail + 1);
                }
                v = (v + 1);
            }
            while ((head < tail)) {
                u = queue[head];
                head = (head + 1);
                v = 0;
                while ((v < n)) {
                    if ((graph[((u * n) + v)] != 1000000000)) {
                        if ((distance[u] != 1000000000)) {
                            distance[v] = Math.Min(distance[v], (distance[u] + graph[((u * n) + v)]));
                        }
                        degree[v] = (degree[v] - 1);
                        if ((degree[v] == 0)) {
                            queue[tail] = v;
                            tail = (tail + 1);
                        }
                    }
                    v = (v + 1);
                }
            }
            if ((tail != n)) {
                return (-1000000000);
            }
            return distance[target];
        }
    }

    static void Main() {
        int[] graph = new int[] {1000000000, 4, 10, 1000000000, 1000000000, -2, 1000000000, 1000000000, 1000000000};
        int n = 3;
        int source = 0;
        int target = 2;
        int result = dag_shortest_path(graph, n, source, target);
        Console.WriteLine(result);
    }
}
Nov 15, 2026 csharp