Ultrametric Distance

Ultrametric Distance Skill

Status: ✅ Production Ready Trit: -1 (MINUS - validator/constrainer) Principle: d(x,z) ≤ max(d(x,y), d(y,z)) — Strong Triangle Inequality

Overview

Ultrametric Distance provides non-Archimedean distance functions where the strong triangle inequality holds. Essential for:

1. Hierarchical clustering: Natural tree structures emerge
2. p-adic analysis: Number-theoretic computations
3. Phylogenetic trees: Evolution distance metrics
4. Version control: Commit ancestry distances

Core Property

Ultrametric Inequality:
  d(x, z) ≤ max(d(x, y), d(y, z))

  Unlike Euclidean: d(x,z) ≤ d(x,y) + d(y,z)
  Ultrametric is STRONGER: max instead of sum

Key Insight

In ultrametric space, ALL triangles are isoceles with the unequal side being the shortest.

Python Implementation

import math
from typing import List, Tuple, Callable

def ultrametric_distance(x: List[float], y: List[float]) -> float:
    """Compute ultrametric (sup-norm) distance."""
    return max(abs(a - b) for a, b in zip(x, y))

def p_adic_valuation(n: int, p: int) -> int:
    """Compute p-adic valuation v_p(n) = max k such that p^k | n."""
    if n == 0:
        return float('inf')
    v = 0
    while n % p == 0:
        n //= p
        v += 1
    return v

def p_adic_distance(x: int, y: int, p: int) -> float:
    """
    Compute p-adic distance: d_p(x,y) = p^(-v_p(x-y))
    
    Properties:
    - d_p(x,x) = 0
    - d_p(x,y) = d_p(y,x)
    - d_p(x,z) ≤ max(d_p(x,y), d_p(y,z))  # Ultrametric!
    """
    if x == y:
        return 0.0
    v = p_adic_valuation(abs(x - y), p)
    return p ** (-v)

def verify_ultrametric(d: Callable, points: List) -> dict:
    """Verify that distance function satisfies ultrametric inequality."""
    violations = []
    for i, x in enumerate(points):
        for j, y in enumerate(points):
            for k, z in enumerate(points):
                dxz = d(x, z)
                dxy = d(x, y)
                dyz = d(y, z)
                if dxz > max(dxy, dyz) + 1e-10:
                    violations.append({
                        'x': x, 'y': y, 'z': z,
                        'd(x,z)': dxz,
                        'max(d(x,y),d(y,z))': max(dxy, dyz)
                    })
    return {
        'is_ultrametric': len(violations) == 0,
        'violations': violations[:5],
        'total_violations': len(violations)
    }

Hierarchical Clustering (UPGMA)

def ultrametric_upgma(distance_matrix: List[List[float]]) -> dict:
    """
    Build ultrametric tree via UPGMA clustering.
    Returns dendrogram as nested dict.
    """
    n = len(distance_matrix)
    clusters = [{i} for i in range(n)]
    heights = [0.0] * n
    tree = {i: {'leaf': i, 'height': 0} for i in range(n)}
    
    while len(clusters) > 1:
        # Find closest pair
        min_dist = float('inf')
        merge_i, merge_j = 0, 1
        
        for i in range(len(clusters)):
            for j in range(i + 1, len(clusters)):
                # Average linkage distance
                d = sum(distance_matrix[a][b] 
                       for a in clusters[i] 
                       for b in clusters[j]) / (len(clusters[i]) * len(clusters[j]))
                if d < min_dist:
                    min_dist, merge_i, merge_j = d, i, j
        
        # Merge clusters
        new_cluster = clusters[merge_i] | clusters[merge_j]
        new_height = min_dist / 2
        new_node = {
            'left': tree[merge_i],
            'right': tree[merge_j],
            'height': new_height,
            'members': list(new_cluster)
        }
        
        # Update
        tree[merge_i] = new_node
        del tree[merge_j]
        clusters[merge_i] = new_cluster
        del clusters[merge_j]
    
    return tree[0]

Git Commit Distance

def commit_ultrametric_distance(repo, commit_a: str, commit_b: str) -> int:
    """
    Ultrametric distance between commits = depth to common ancestor.
    
    d(A, B) = depth(merge_base(A, B))
    
    Satisfies ultrametric: branching creates natural hierarchy.
    """
    import subprocess
    
    # Find merge base
    merge_base = subprocess.check_output(
        ['git', 'merge-base', commit_a, commit_b],
        cwd=repo
    ).decode().strip()
    
    # Count commits from merge base to root
    depth = int(subprocess.check_output(
        ['git', 'rev-list', '--count', merge_base],
        cwd=repo
    ).decode().strip())
    
    return depth

Julia Implementation

module UltrametricDistance

"""
    p_adic_distance(x::Int, y::Int, p::Int) -> Float64

Compute p-adic distance between integers.
"""
function p_adic_distance(x::Int, y::Int, p::Int)
    x == y && return 0.0
    diff = abs(x - y)
    v = 0
    while diff % p == 0
        diff ÷= p
        v += 1
    end
    return Float64(p)^(-v)
end

"""
    ultrametric_ball(center, radius, points, d)

Return all points within ultrametric ball.
Note: In ultrametric space, every point in the ball is a center!
"""
function ultrametric_ball(center, radius, points, d)
    filter(p -> d(center, p) ≤ radius, points)
end

"""
    is_ultrametric(d, points) -> Bool

Verify ultrametric inequality for all triples.
"""
function is_ultrametric(d, points)
    for x in points, y in points, z in points
        d(x, z) > max(d(x, y), d(y, z)) && return false
    end
    return true
end

end # module

Integration with GF(3)

Ultrametric distances map naturally to trits:

def distance_to_trit(d: float, thresholds: Tuple[float, float] = (0.33, 0.66)) -> int:
    """
    Map ultrametric distance to trit.
    
    Close (d < 0.33)   → +1 (PLUS, same cluster)
    Medium (0.33-0.66) →  0 (ERGODIC, sibling clusters)  
    Far (d > 0.66)     → -1 (MINUS, distant branches)
    """
    if d < thresholds[0]:
        return 1
    elif d < thresholds[1]:
        return 0
    else:
        return -1

Commands

# Verify p-adic distances
python -c "from ultrametric import p_adic_distance; print(p_adic_distance(12, 20, 2))"

# Build UPGMA tree
python -m ultrametric.upgma --input distances.csv --output tree.json

# Git commit distance
git-ultrametric HEAD~5 main

Properties

Skill Name: ultrametric-distance Type: Distance Metric / Clustering Trit: -1 (MINUS) Use Case: Hierarchical validation, tree construction, version ancestry