# Graph Shortest Path (Dijkstra Step-By-Step) Dijkstra with a full per-iteration trace: settled nodes, relaxations, distance table, and the final path; undirected or directed, negative weights rejected, unreachable targets reported. > Canonical page: https://elysiatools.com/en/tools/graph-shortest-path - **Category:** Math & Numbers - **Keywords:** dijkstra, shortest path, graph, weighted graph, distance table, relaxation, pathfinding, network, algorithm, discrete math ## Overview The Graph Shortest Path tool computes the optimal route between nodes using Dijkstra's algorithm and provides a complete step-by-step trace of every iteration. It logs settled nodes, edge relaxations, distance updates, and the final reconstructed path for both directed and undirected weighted graphs. ## Inputs - **Edges (one per line: from, to, weight)** (textarea): One edge per line: two node names (1–8 letters/digits) and a non-negative weight. - **Source node** (text): e.g. A - **Target node** (text): e.g. F - **Treat edges as directed** (checkbox): When checked, each edge only leads from the first node to the second; otherwise the graph is undirected. - **Decimal Places** (number) ## When to use - Verifying manual Dijkstra calculations and distance tables for discrete mathematics or algorithm homework. - Tracing step-by-step edge relaxation behavior to understand why a greedy route fails in complex networks. - Finding the exact shortest distance and path between two vertices in weighted directed or undirected graphs. ## How it works - Enter graph edges one per line as source, destination, and non-negative weight (e.g., 'A B 4'). - Specify the source and target node identifiers, and choose whether edges should be treated as directed. - The tool initializes distances, greedily settles the node with the minimum tentative distance, and relaxes outgoing edges. - View the generated per-iteration log, settled distance states, and the final shortest path reconstruction. ## Use cases - Computer science students checking homework traces and edge relaxation steps for Dijkstra exam problems. - Network engineers evaluating minimum latency or hop costs across simple routed network topologies. - Instructors generating step-by-step textbook-style pathfinding traces for discrete math lectures. ## Frequently asked questions ### Can this tool handle graphs with negative edge weights? No. Dijkstra's algorithm requires non-negative edge weights; negative values are rejected. ### How are ties broken when multiple nodes have the same tentative distance? Ties are resolved deterministically using the lexicographically smallest node identifier. ### What happens if the target node is unreachable from the source? The execution log traces all reachable settled nodes and reports the target as unreachable. ### Does the tool support directed graphs? Yes. Enable the 'Treat edges as directed' checkbox to ensure edges only travel from source to destination. ### Can I use decimal numbers for edge weights? Yes. Decimal weights are supported, and the decimal precision of the output can be customized. ## Related tools - [Audio to Text Transcriber (AI)](https://elysiatools.com/en/tools/audio-to-text-transcriber): Transcribe speech from audio (wav/mp3/m4a/flac/ogg/webm/aac) to text, SRT, VTT or JSON using the grok-stt AI model. 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