Minimum spanning tree certificate check

minimum-spanning-tree-certificate-check · version 1.0.0 · Numbers & math · free, no key needed

Check a claimed minimum spanning tree: structure, exact weight, then the cycle/cut optimality test with a swap witness

Use when you need to: verify minimum spanning tree · check mst optimality · is this the minimum spanning tree.

Decide before calling

Read the versioned contract and the supported scope below. Reuse minimum-spanning-tree-certificate-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 minimum-spanning-tree-certificate-check@1.0.0 for verify minimum spanning tree. 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 minimum spanning tree
• check mst optimality
• is this the minimum spanning tree
• mst cycle property check
• mst cut property certificate
• find cheaper swap for spanning tree
• validate kruskal or prim result

Not supported

• computing a minimum spanning tree from scratch
• minimum spanning forests of disconnected graphs
• directed minimum arborescences (Chu-Liu/Edmonds)
• bottleneck or degree-constrained spanning trees

Behavior

• Order: structural stage (missing_edge, duplicate_edge, cycle_edge with a cycle witness, disconnected) in tree_edge_ids order; then, when claimed_weight is not null, the exact tree weight must equal it (weight_mismatch); then the cycle condition over non-tree edges in graph.edges order.
• Cycle condition: for every non-tree edge e = {u, v} with u != v, w(e) must be at least the heaviest weight on the unique tree path from u to v; the first failure is cycle_property_violated with index = position in graph.edges, expected = heaviest tree-path weight, actual = w(e), witness_edge_ids = the tree path from u to v.
• The condition holds for all non-tree edges exactly when the tree is a minimum spanning tree (cycle property, equivalently each tree edge is light for its fundamental cut, CLRS Theorem 21.1). Swapping e for the first heaviest witness edge lowers the weight. Ties are allowed, so every alternative minimum tree passes.
• claimed_weight is a canonical integer string of at most 40 digits or null (no weight claim); tree_weight is the computed sum when the structure is valid, otherwise null. Arithmetic is exact.
• Undirected edges {id, u, v, weight}; parallel edges and self-loops allowed (non-tree self-loops never violate the condition); IDs compared code unit for code unit. Graph errors throw invalid_input; size limits throw limit_exceeded first.
• Reference: CLRS 4th ed. (2022) chapter 21; reference_version "clrs-4e-2022-ch21-minimum-spanning-trees".

Input

• graph (object, required)
• tree_edge_ids (array of string, required): max items 2000; each min length 1; each max length 64
• claimed_weight (string or null, required): pattern ^(0|-?[1-9][0-9]*)$

Output

• valid (boolean, required)
• reason (one of null, "missing_edge", "duplicate_edge", "cycle_edge", "disconnected", "weight_mismatch", "cycle_property_violated", required)
• first_discrepancy (object or null, required)
• tree_weight (string or null, required): pattern ^(0|-?[1-9][0-9]*)$
• reference_version (constant "clrs-4e-2022-ch21-minimum-spanning-trees", required)

Limits

• max vertices: 500
• max edges: 2000
• max tree edges: 2000
• max id bytes: 64
• max integer digits: 40

Example

Request input:

{
  "graph": {
    "vertices": [
      "a",
      "b",
      "c",
      "d"
    ],
    "edges": [
      {
        "id": "ab",
        "u": "a",
        "v": "b",
        "weight": "1"
      },
      {
        "id": "bc",
        "u": "b",
        "v": "c",
        "weight": "2"
      },
      {
        "id": "cd",
        "u": "c",
        "v": "d",
        "weight": "3"
      },
      {
        "id": "da",
        "u": "d",
        "v": "a",
        "weight": "4"
      },
      {
        "id": "ac",
        "u": "a",
        "v": "c",
        "weight": "5"
      }
    ]
  },
  "tree_edge_ids": [
    "ab",
    "bc",
    "cd"
  ],
  "claimed_weight": "6"
}

Response:

{
  "result": {
    "valid": true,
    "reason": null,
    "first_discrepancy": null,
    "tree_weight": "6",
    "reference_version": "clrs-4e-2022-ch21-minimum-spanning-trees"
  }
}

How to call it

MCP

Connect https://computefirst.net/mcp (setup), then call execute with:

{
  "id": "minimum-spanning-tree-certificate-check",
  "version": "1.0.0",
  "input": {
    "graph": {
      "vertices": [
        "a",
        "b",
        "c",
        "d"
      ],
      "edges": [
        {
          "id": "ab",
          "u": "a",
          "v": "b",
          "weight": "1"
        },
        {
          "id": "bc",
          "u": "b",
          "v": "c",
          "weight": "2"
        },
        {
          "id": "cd",
          "u": "c",
          "v": "d",
          "weight": "3"
        },
        {
          "id": "da",
          "u": "d",
          "v": "a",
          "weight": "4"
        },
        {
          "id": "ac",
          "u": "a",
          "v": "c",
          "weight": "5"
        }
      ]
    },
    "tree_edge_ids": [
      "ab",
      "bc",
      "cd"
    ],
    "claimed_weight": "6"
  }
}

HTTP (no key)

curl -X POST https://computefirst.net/v1/tools/minimum-spanning-tree-certificate-check/versions/1.0.0/execute \
  -H "Content-Type: application/json" \
  -d '{"graph":{"vertices":["a","b","c","d"],"edges":[{"id":"ab","u":"a","v":"b","weight":"1"},{"id":"bc","u":"b","v":"c","weight":"2"},{"id":"cd","u":"c","v":"d","weight":"3"},{"id":"da","u":"d","v":"a","weight":"4"},{"id":"ac","u":"a","v":"c","weight":"5"}]},"tree_edge_ids":["ab","bc","cd"],"claimed_weight":"6"}'

The machine-readable contract is at /v1/tools/minimum-spanning-tree-certificate-check/versions/1.0.0.

CLI

node cli.mjs run minimum-spanning-tree-certificate-check 1.0.0 --input input.json --base-url https://computefirst.net

Get the client at /clients/cli/.

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