This guide explains Floyd-Warshall from first principles. The goal is to understand the problem it solves, why the algorithm works, how to implement it, and where it appears in real software systems.
What problem does it solve?
Floyd-Warshall belongs to the graph-algorithm toolbox. Before coding, define what the vertices represent, what an edge means, and what result the system actually needs.
The core idea in simple terms
برای هر رأس k بررسی میکنیم آیا عبور از k مسیر i تا j را کوتاهتر میکند.
Step-by-step process
- ماتریس فاصله را مقداردهی میکنیم.
- قطر اصلی صفر است.
- برای هر k، i و j رابطه min را اعمال میکنیم.
- ماتریس نهایی شامل فاصله همه جفتهاست.
JavaScript example
function floydWarshall(matrix) {
const d = matrix.map(r => [...r]), n = d.length;
for (let k=0;k<n;k++) for (let i=0;i<n;i++) for (let j=0;j<n;j++)
d[i][j] = Math.min(d[i][j], d[i][k] + d[k][j]);
return d;
}Real-world and workplace examples
- فاصله بین همه شعب
- Routing Matrix
- Reachability
- پیشمحاسبه مسیر Logistics
Time and space complexity
O(V³) time و O(V²) space
When should you use it?
Use it when the problem matches this condition: نیاز به all-pairs shortest paths با V متوسط.
When is it not a good fit?
It is usually not the best choice when: برای گراف بسیار بزرگ sparse معمولاً مناسب نیست..
Summary
Do not choose an algorithm by name alone. First identify whether the graph is directed or undirected, weighted or unweighted, and whether the goal is traversal, reachability, shortest path, connectivity, spanning structure, or combinatorial optimization.