Quartz 4

Bellman-Ford Algorithm - Edge Relaxation Pattern

4 min read

Bellman-Ford Algorithm - Edge Relaxation Pattern

The Fundamental Approach

Bellman-Ford’s strategy is deceptively simple: relax all edges, repeatedly, until distances stabilize.

This differs from other shortest path algorithms in its treatment of edges as the primary unit of computation, rather than vertices.

Edge-Centric vs Vertex-Centric

Bellman-Ford (Edge-Centric)

# Process ALL edges in each iteration
for iteration in range(V - 1):
    for each edge (u, v, weight):
        relax(u, v, weight)

Dijkstra (Vertex-Centric)

# Process edges from ONE vertex at a time
while priority_queue:
    u = extract_min()
    for each edge (u, v, weight):
        relax(u, v, weight)

This difference is fundamental:

  • Dijkstra: “Which vertex should I explore next?”
  • Bellman-Ford: “Let me check all edges again”

Why This Works: The Inductive Proof

Define distance_k[v] = shortest path to v using ≤ k edges

Base case: distance_0[source] = 0, all others = ∞

Inductive step: In iteration k, we compute:

distance_k[v] = min(distance_{k-1}[v], 
                    min over all u: distance_{k-1}[u] + weight(u,v))

This is exactly what the Shortest Path Relaxation Operation does!

After k iterations, we have the shortest paths using at most k edges. After V-1 iterations, we have all shortest simple paths (since simple paths have at most V-1 edges).

The Order-Independence Property

A remarkable property: edge order doesn’t matter for correctness, only efficiency.

You can relax edges in any order:

# All of these are correct
edges_shuffled = random.shuffle(edges)
edges_reversed = reversed(edges)
edges_sorted = sorted(edges, key=lambda e: e.weight)
 
# All produce the same final distances

Why? Because we’re computing the same fixed point. The order only affects:

  1. How quickly we reach convergence
  2. Intermediate distance values (but not final)

Early Termination Optimization

We can stop if no relaxation succeeds in an iteration:

for iteration in range(V - 1):
    changed = False
    
    for u, v, weight in edges:
        if distance[u] + weight < distance[v]:
            distance[v] = distance[u] + weight
            predecessor[v] = u
            changed = True
    
    if not changed:
        break  # Converged early!

This optimization is crucial in practice - many graphs converge in far fewer than V-1 iterations.

Comparison with Dynamic Programming

Bellman-Ford is essentially dynamic programming on the graph structure:

State: dp[k][v] = shortest path to v using ≤ k edges

Transition:

dp[k][v] = min(dp[k-1][v],                    # don't use edge k
               min_{u→v} dp[k-1][u] + w(u,v)) # use edge k

We can optimize space from O(V²) to O(V) by reusing the same array:

# Space-optimized: reuse the same distance array
distance = [float('inf')] * V
distance[source] = 0
 
for _ in range(V - 1):
    for u, v, weight in edges:
        distance[v] = min(distance[v], distance[u] + weight)

Edge List Representation Advantage

Bellman-Ford naturally works with edge list representation:

edges = [
    (0, 1, 4),   # (source, destination, weight)
    (0, 2, 2),
    (1, 2, 1),
    (2, 3, 5),
]

This is often more compact than adjacency lists for sparse graphs and makes the “process all edges” pattern straightforward.

Parallelization Potential

The edge-centric approach enables parallelization:

# In each iteration, all edge relaxations are independent
# They only read from distance array of previous iteration
for iteration in range(V - 1):
    parallel_for (u, v, weight) in edges:
        new_distance = distance[u] + weight
        atomic_min(distance[v], new_distance)

This is harder to parallelize in vertex-centric algorithms like Dijkstra due to the priority queue bottleneck.

When to Use Edge Relaxation

The edge relaxation pattern is ideal when:

  1. Negative weights exist (no greedy shortcut available)
  2. Simple implementation is preferred (no complex data structures)
  3. Edge list representation is natural for the problem
  4. Parallelization might be needed

The Philosophical Difference

Edge relaxation represents a different computational philosophy:

  • Dijkstra: Be smart about order, finalize early
  • Bellman-Ford: Be exhaustive, guarantee correctness

This mirrors a broader pattern in algorithm design:

  • Greedy (Dijkstra) is fast but requires special structure
  • Dynamic programming (Bellman-Ford) is robust but potentially slower

Reconstruction: The correctness follows from the inductive argument on path length. Order-independence follows from the fact that we’re computing a fixed point (the shortest path distances) that is unique.

Graph View

Backlinks