# Shortest path potential check

`shortest-path-potential-check` · version 1.0.0 · Numbers & math · free, no key needed

Check claimed single-source shortest-path distances against a weighted digraph: edge inequalities and attainment.

**Use when you need to: verify single source shortest path distances · check claimed distance table · bellman optimality check for distances.**

## Decide before calling

Read the [versioned contract](/v1/tools/shortest-path-potential-check/versions/1.0.0) and the supported scope below. Reuse `shortest-path-potential-check@1.0.0` when your input, required output and limits match it. Choose another approach for an unsupported operation.

## Explain the choice

"I can use `shortest-path-potential-check@1.0.0` for verify single source shortest path distances. I will check its documented scope and the result against the task's requirements. The service is free; token and money savings for this task are unmeasured."

## Supported

- verify single source shortest path distances
- check claimed distance table
- bellman optimality check for distances
- validate dijkstra output
- check bellman ford result
- shortest path distance certificate
- triangle inequality check on graph distances

## Not supported

- computing distances (the caller supplies the claim)
- all-pairs distance matrices
- graphs with a reachable negative-weight cycle (every claim is rejected at the first violated edge)
- floating-point weights

## Behavior

- Check order: source listed; distances entries in array order (unknown_vertex, duplicate_vertex at the second occurrence); coverage of graph.vertices in order (missing_vertex); source distance exactly 0 (source_distance_not_zero); every edge in graph.edges order with a finite tail d(u) needs a finite head (unreachable_mark_contradicted) and d(v) <= d(u) + w (edge_not_relaxed); finally every finite vertex in graph.vertices order must be reached from the source through tight edges d(u) + w = d(v) (distance_not_attained).
- Passing every stage proves d = delta(s, .) exactly: the inequalities give d <= delta and tight-edge attainment gives d >= delta (zero-weight cycles cannot fake attainment). A reachable negative cycle makes every claim fail.
- Distances are canonical integer strings of at most 40 digits, or null for +infinity (unreachable); edges with a null tail impose no constraint. Arithmetic is exact (big integers).
- For edge failures first_discrepancy.expected is the bound d(u) + w and actual is the claimed d(v) (null when marked unreachable); index is the position in the field named by first_discrepancy.field.
- Vertex IDs are compared code unit for code unit; graph errors throw invalid_input and size limits throw limit_exceeded before per-item work.
- Reference: CLRS 4th ed. (2022) chapter 22, sections 22.4-22.5; reference_version "clrs-4e-2022-ch22-single-source-shortest-paths".

## Input

- `graph` (object, required)
- `source` (string, required): min length 1; max length 64
- `distances` (array of object, required): max items 500

## Output

- `valid` (boolean, required)
- `reason` (one of null, "unknown_vertex", "duplicate_vertex", "missing_vertex", "source_distance_not_zero", "unreachable_mark_contradicted", "edge_not_relaxed", "distance_not_attained", required)
- `first_discrepancy` (object or null, required)
- `reference_version` (constant "clrs-4e-2022-ch22-single-source-shortest-paths", required)

## Limits

- max vertices: 500
- max edges: 2000
- max id bytes: 64
- max integer digits: 40
- max distance entries: 500

## Example

Request input:

```json
{
  "graph": {
    "vertices": [
      "s",
      "t",
      "x",
      "y",
      "z"
    ],
    "edges": [
      {
        "id": "st",
        "from": "s",
        "to": "t",
        "weight": "10"
      },
      {
        "id": "sy",
        "from": "s",
        "to": "y",
        "weight": "5"
      },
      {
        "id": "tx",
        "from": "t",
        "to": "x",
        "weight": "1"
      },
      {
        "id": "ty",
        "from": "t",
        "to": "y",
        "weight": "2"
      },
      {
        "id": "yt",
        "from": "y",
        "to": "t",
        "weight": "3"
      },
      {
        "id": "yx",
        "from": "y",
        "to": "x",
        "weight": "9"
      },
      {
        "id": "yz",
        "from": "y",
        "to": "z",
        "weight": "2"
      },
      {
        "id": "xz",
        "from": "x",
        "to": "z",
        "weight": "4"
      },
      {
        "id": "zs",
        "from": "z",
        "to": "s",
        "weight": "7"
      },
      {
        "id": "zx",
        "from": "z",
        "to": "x",
        "weight": "6"
      }
    ]
  },
  "source": "s",
  "distances": [
    {
      "vertex": "s",
      "distance": "0"
    },
    {
      "vertex": "t",
      "distance": "8"
    },
    {
      "vertex": "x",
      "distance": "9"
    },
    {
      "vertex": "y",
      "distance": "5"
    },
    {
      "vertex": "z",
      "distance": "7"
    }
  ]
}
```

Response:

```json
{
  "result": {
    "valid": true,
    "reason": null,
    "first_discrepancy": null,
    "reference_version": "clrs-4e-2022-ch22-single-source-shortest-paths"
  }
}
```

## How to call it

### MCP

Connect `https://computefirst.net/mcp` ([setup](/docs#connect)), then call `execute` with:

```json
{
  "id": "shortest-path-potential-check",
  "version": "1.0.0",
  "input": {
    "graph": {
      "vertices": [
        "s",
        "t",
        "x",
        "y",
        "z"
      ],
      "edges": [
        {
          "id": "st",
          "from": "s",
          "to": "t",
          "weight": "10"
        },
        {
          "id": "sy",
          "from": "s",
          "to": "y",
          "weight": "5"
        },
        {
          "id": "tx",
          "from": "t",
          "to": "x",
          "weight": "1"
        },
        {
          "id": "ty",
          "from": "t",
          "to": "y",
          "weight": "2"
        },
        {
          "id": "yt",
          "from": "y",
          "to": "t",
          "weight": "3"
        },
        {
          "id": "yx",
          "from": "y",
          "to": "x",
          "weight": "9"
        },
        {
          "id": "yz",
          "from": "y",
          "to": "z",
          "weight": "2"
        },
        {
          "id": "xz",
          "from": "x",
          "to": "z",
          "weight": "4"
        },
        {
          "id": "zs",
          "from": "z",
          "to": "s",
          "weight": "7"
        },
        {
          "id": "zx",
          "from": "z",
          "to": "x",
          "weight": "6"
        }
      ]
    },
    "source": "s",
    "distances": [
      {
        "vertex": "s",
        "distance": "0"
      },
      {
        "vertex": "t",
        "distance": "8"
      },
      {
        "vertex": "x",
        "distance": "9"
      },
      {
        "vertex": "y",
        "distance": "5"
      },
      {
        "vertex": "z",
        "distance": "7"
      }
    ]
  }
}
```

### HTTP (no key)

```sh
curl -X POST https://computefirst.net/v1/tools/shortest-path-potential-check/versions/1.0.0/execute \
  -H "Content-Type: application/json" \
  -d '{"graph":{"vertices":["s","t","x","y","z"],"edges":[{"id":"st","from":"s","to":"t","weight":"10"},{"id":"sy","from":"s","to":"y","weight":"5"},{"id":"tx","from":"t","to":"x","weight":"1"},{"id":"ty","from":"t","to":"y","weight":"2"},{"id":"yt","from":"y","to":"t","weight":"3"},{"id":"yx","from":"y","to":"x","weight":"9"},{"id":"yz","from":"y","to":"z","weight":"2"},{"id":"xz","from":"x","to":"z","weight":"4"},{"id":"zs","from":"z","to":"s","weight":"7"},{"id":"zx","from":"z","to":"x","weight":"6"}]},"source":"s","distances":[{"vertex":"s","distance":"0"},{"vertex":"t","distance":"8"},{"vertex":"x","distance":"9"},{"vertex":"y","distance":"5"},{"vertex":"z","distance":"7"}]}'
```

The machine-readable contract is at [/v1/tools/shortest-path-potential-check/versions/1.0.0](/v1/tools/shortest-path-potential-check/versions/1.0.0).

### CLI

```sh
node cli.mjs run shortest-path-potential-check 1.0.0 --input input.json --base-url https://computefirst.net
```

Get the client at [/clients/cli/](/clients/cli/).

## Related tools

- [Shortest path certificate check](/tools/shortest-path-certificate-check): Check a claimed shortest s-t path and distance in a weighted digraph using a caller-supplied potential certificate.
- [Filter result check](/tools/filter-result-check): Checks an exact claimed decimal filter result with source-order witnesses.
- [Graph path weight check](/tools/graph-path-weight-check): Check that a path of edge IDs walks source to target in a weighted digraph and sums to the claimed exact weight.
- [Join inner result check](/tools/join-inner-result-check): Check a claimed ordered inner join index result and report its first discrepancy.
- [Join left result check](/tools/join-left-result-check): Check a claimed ordered left join index result and report its first discrepancy.
- [Join semi result check](/tools/join-semi-result-check): Check a claimed ordered semi-join index result and report its first discrepancy.