Data Structures & AlgorithmsGraph Types

Graph Types — Directed, Undirected, Weighted

LevelIntermediate

Duration60 mins

TopicGraph Types

3 / 4

Weighted Graphs — Edges with Costs

When Connections Have Value

In the real world, not all connections are created equal. The flight from New York to Los Angeles takes 5 hours, but LA to San Francisco takes only 1 hour. The highway has a higher speed limit but tolls; the back road is free but slower. A strong friendship differs from a casual acquaintance. These quantitative differences between edges is the defining characteristic of weighted graphs.

Weighted graphs extend our basic graph model by associating a numerical value—called a weight—with each edge. This simple addition dramatically expands the modeling power of graphs and fundamentally changes which algorithms we can and should use. The shortest path is no longer about fewest edges; it's about minimum total weight.

What You Will Learn

By the end of this page, you will understand the formal definition and various representations of weighted graphs, recognize their diverse real-world applications, grasp how weights fundamentally change algorithmic approaches, and appreciate the crucial distinction between positive, zero, and negative weights.

Formal Definition and Notation

A weighted graph is a graph where each edge has an associated numerical value called its weight, cost, length, or capacity, depending on the application context.

Formal Definition:

A weighted graph G is typically defined as a triple G = (V, E, w), where:

V is a finite set of vertices
E is a set of edges (directed or undirected)
w is a weight function w: E → ℝ that assigns a real number to each edge

For an edge e = (u, v) or e = {u, v}, we write w(e) or w(u, v) to denote its weight.

Alternative Notation:

Some authors define weighted graphs as G = (V, E) where each edge is a tuple containing its weight: e = (u, v, w). Both notations capture the same concept—edges with associated values.

Weight Semantics Vary by Application

Weights can represent distance (roads), time (flights), cost (pricing), capacity (network flow), probability (Markov chains), similarity (clustering), or any other quantitative relationship. The same graph structure with different weight interpretations yields different problem domains.

Weighted Path and Distance:

In a weighted graph, the weight of a path is the sum of weights of edges along the path:

w(P) = Σw(eᵢ) for all edges eᵢ in path P

The weighted shortest path from u to v is the path minimizing this sum. This differs fundamentally from unweighted graphs, where the shortest path simply minimizes edge count.

Example:

Consider weighted edges: A→B(3), A→C(1), C→B(1)

Direct path A→B has weight 3
Path A→C→B has weight 1 + 1 = 2
The shortest weighted path is A→C→B, even though it uses more edges

This is why BFS (which finds shortest edge-count paths) doesn't work for weighted graphs—we need algorithms like Dijkstra's that consider edge weights.

Weighted Graph Terminology
Term	Definition	Example
Weight Function	Mapping from edges to real numbers: w: E → ℝ	w(A,B) = 5 means edge A-B has weight 5
Path Weight	Sum of weights of edges in a path	Path A→B→C with w(A,B)=2, w(B,C)=3 has weight 5
Weighted Shortest Path	Path minimizing total weight (not edge count)	May use more edges if their total weight is less
Positive Weights	All edge weights > 0	Enables efficient algorithms like Dijkstra's
Non-negative Weights	All edge weights ≥ 0	Zero-weight edges are valid; Dijkstra's still works
Negative Weights	Some edge weights < 0	Requires Bellman-Ford; may have negative cycles
Negative Cycle	Cycle whose total weight is negative	Infinitely decreasing paths; no shortest path exists

Representation and Storage

Weighted graphs require modified data structures to store both connectivity and edge weights. The two primary representations—adjacency matrix and adjacency list—adapt naturally to weighted graphs.

Weighted Adjacency Matrix:

For a graph with n vertices, the adjacency matrix M is an n×n matrix where:

M[i][j] = w(i, j) if edge (i, j) exists
M[i][j] = ∞ (or a sentinel value) if edge (i, j) doesn't exist
M[i][i] = 0 (path from vertex to itself has zero weight)

Pros: O(1) edge weight lookup, simple implementation, good for dense graphs Cons: O(V²) space regardless of edge count, wasteful for sparse graphs

Weighted Adjacency List:

For each vertex, store a list of (neighbor, weight) pairs:

graph[u] = [(v₁, w₁), (v₂, w₂), ...]

Pros: O(V + E) space, efficient for sparse graphs, natural for traversals Cons: O(degree) edge lookup, slightly more complex than unweighted

weighted_graph_representation.py
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from typing import Dict, List, Tuple, Optional
import math
 
class WeightedGraphMatrix:
    """
    Weighted graph using adjacency matrix representation.
    
    Best for: Dense graphs, frequent edge weight lookups, 
              algorithms that iterate over all edges.
    """
    def __init__(self, vertices: List[str]):
        self.vertices = vertices
        self.vertex_to_idx = {v: i for i, v in enumerate(vertices)}
        n = len(vertices)
        # Initialize with infinity (no edge)
        self.matrix = [[math.inf] * n for _ in range(n)]
        # Distance to self is 0
        for i in range(n):
            self.matrix[i][i] = 0
    
    def add_edge(self, u: str, v: str, weight: float, directed: bool = True):
        i, j = self.vertex_to_idx[u], self.vertex_to_idx[v]
        self.matrix[i][j] = weight
        if not directed:
            self.matrix[j][i] = weight
    
    def get_weight(self, u: str, v: str) -> float:
        i, j = self.vertex_to_idx[u], self.vertex_to_idx[v]
        return self.matrix[i][j]
    
    def has_edge(self, u: str, v: str) -> bool:
        return self.get_weight(u, v) != math.inf
 
 
class WeightedGraphList:
    """
    Weighted graph using adjacency list representation.
    
    Best for: Sparse graphs, traversal algorithms, 
              memory-constrained environments.
    """
    def __init__(self):
        # graph[u] = [(v, weight), ...]
        self.graph: Dict[str, List[Tuple[str, float]]] = {}
    
    def add_vertex(self, v: str):
        if v not in self.graph:
            self.graph[v] = []
    
    def add_edge(self, u: str, v: str, weight: float, directed: bool = True):
        self.add_vertex(u)
        self.add_vertex(v)
        self.graph[u].append((v, weight))
        if not directed:
            self.graph[v].append((u, weight))
    
    def get_neighbors(self, u: str) -> List[Tuple[str, float]]:
        return self.graph.get(u, [])
    
    def get_weight(self, u: str, v: str) -> Optional[float]:
        for neighbor, weight in self.graph.get(u, []):
            if neighbor == v:
                return weight
        return None
    
    def get_all_edges(self) -> List[Tuple[str, str, float]]:
        edges = []
        for u, neighbors in self.graph.items():
            for v, w in neighbors:
                edges.append((u, v, w))
        return edges
 
 
# Example usage
print("=== Adjacency List (Sparse Graph) ===")
graph = WeightedGraphList()
graph.add_edge('A', 'B', 4)
graph.add_edge('A', 'C', 2)
graph.add_edge('B', 'C', 1)
graph.add_edge('B', 'D', 5)
graph.add_edge('C', 'D', 8)
graph.add_edge('C', 'E', 10)
graph.add_edge('D', 'E', 2)
 
print(f"Neighbors of A: {graph.get_neighbors('A')}")
# Output: [('B', 4), ('C', 2)]
 
print(f"Weight A->B: {graph.get_weight('A', 'B')}")
# Output: 4
 
print(f"All edges: {graph.get_all_edges()}")
 
 
print("\n=== Adjacency Matrix (Dense Graph) ===")
matrix_graph = WeightedGraphMatrix(['A', 'B', 'C', 'D'])
matrix_graph.add_edge('A', 'B', 1)
matrix_graph.add_edge('A', 'C', 4)
matrix_graph.add_edge('B', 'C', 2)
matrix_graph.add_edge('C', 'D', 3)
 
print(f"Weight A->C: {matrix_graph.get_weight('A', 'C')}")  # 4
print(f"Weight A->D: {matrix_graph.get_weight('A', 'D')}")  # inf (no edge)

Real-World Applications of Weighted Graphs

Weighted graphs are everywhere in practical applications. The quantitative nature of edge weights makes them indispensable for optimization problems and realistic modeling. Let's explore major application domains.

Major Application Domains

•Navigation and Routing — GPS systems use weighted graphs where intersections are vertices, roads are edges, and weights represent travel time, distance, or real-time traffic conditions. Dijkstra's algorithm or A* finds the fastest route. Weight meanings might combine distance, speed limits, and current congestion.
•Network Flow and Capacity — Internet routing, water distribution, and electrical grids model capacity as edge weights. How much data, water, or current can flow through each link? Maximum flow algorithms find optimal throughput; minimum-cost flow finds the cheapest way to move resources.
•Airline and Transportation Networks — Cities are vertices, flights/train routes are edges, and weights represent ticket prices, flight times, or carbon emissions. Finding the cheapest itinerary, the fastest connection, or the minimum-emission route are weighted shortest path problems.
•Telecommunications — Network latency, bandwidth, and reliability can all serve as edge weights. Quality-of-Service (QoS) routing seeks paths that minimize latency while meeting bandwidth requirements—a multi-objective weighted path problem.
•Social Network Analysis — Edge weights can represent interaction frequency, relationship strength, or influence. Weighted centrality measures identify truly influential nodes (not just those with many connections, but those with strong connections).
•Machine Learning and Clustering — K-nearest neighbors, spectral clustering, and graph neural networks often operate on weighted similarity graphs. Weights encode how similar two data points are, enabling structure discovery in high-dimensional data.

Weight Interpretations by Domain
Domain	Weight Represents	Optimization Goal	Algorithm
Navigation	Travel time / Distance	Shortest path	Dijkstra / A*
Network Flow	Capacity	Maximum throughput	Ford-Fulkerson
Airlines	Ticket price	Cheapest route	Dijkstra / Bellman-Ford
Infrastructure	Construction cost	Minimum spanning tree	Kruskal / Prim
Similarity Graphs	Distance / Dissimilarity	Clustering	MST-based clustering
Influence Networks	Interaction strength	Influence maximization	Weighted centrality

Weight Semantics Determine Algorithm Choice

When weights represent costs/distances, we minimize path weight. When weights represent similarity/affinity, we might maximize path weight or use weights inversely. Always clarify: 'Is a larger weight good or bad?' This determines whether you minimize or maximize, and whether algorithms apply directly or need adaptation.

Positive, Non-Negative, and Negative Weights

The sign of edge weights profoundly impacts which algorithms work and how we interpret results. This is one of the most important practical considerations when working with weighted graphs.

Positive Weights (w(e) > 0 for all e):

All edge weights are strictly positive. This is the most common and well-behaved case.

Dijkstra's algorithm works correctly: once a vertex is finalized, we've found its shortest path
Shortest paths are well-defined; they always exist (in connected components)
No risk of infinite loops or decreasing cycle issues

Non-Negative Weights (w(e) ≥ 0 for all e):

Zero-weight edges are allowed. Dijkstra's still works, but zero-weight edges mean multiple distinct paths may have the same total weight.

Negative Weights (some w(e) < 0):

Some edges have negative weight. This creates serious complications:

Dijkstra's Fails with Negative Weights

Dijkstra's algorithm assumes that once we've found the shortest path to a vertex, no future discovery will improve it. Negative edges violate this: we might find a 'longer' path initially, then later discover a negative edge that makes an alternative path shorter. Dijkstra's greedy choice becomes incorrect.

Handling Negative Weights:

Bellman-Ford Algorithm: Works correctly with negative weights. Runs in O(V × E) time—slower than Dijkstra but handles negative edges. Also detects negative cycles.
Johnson's Algorithm: For all-pairs shortest paths with negative weights. Reweights edges to eliminate negatives, then runs Dijkstra from each vertex.

Negative Cycles:

A negative cycle is a cycle whose total weight is negative. If a negative cycle exists:

Shortest paths through the cycle are undefined (we can keep going around, decreasing the path weight infinitely)
Bellman-Ford can detect negative cycles (if a |V|th relaxation still improves distances, a negative cycle exists)
We must decide: report 'no shortest path exists' or find paths that don't use the cycle

Positive Weights

•Dijkstra's algorithm: O((V+E) log V)
•A* with admissible heuristics
•Shortest paths always well-defined
•No cycle issues (all cycles positive)
•Most practical applications

Negative Weights

•Bellman-Ford algorithm: O(V × E)
•Must check for negative cycles
•Shortest path may not exist
•Dijkstra's gives wrong answers
•Currency arbitrage, game theory

shortest_path_algorithms.py
Python
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import heapq
from typing import Dict, List, Tuple, Optional
import math
 
def dijkstra(graph: Dict[str, List[Tuple[str, float]]], start: str) -> Dict[str, float]:
    """
    Dijkstra's algorithm for shortest paths from start vertex.
    
    REQUIREMENT: All edge weights must be non-negative!
    
    Time Complexity: O((V + E) log V) with binary heap
    Space Complexity: O(V)
    """
    distances = {v: math.inf for v in graph}
    distances[start] = 0
    priority_queue = [(0, start)]  # (distance, vertex)
    visited = set()
    
    while priority_queue:
        current_dist, current = heapq.heappop(priority_queue)
        
        if current in visited:
            continue
        visited.add(current)
        
        for neighbor, weight in graph.get(current, []):
            if neighbor not in visited:
                new_dist = current_dist + weight
                if new_dist < distances[neighbor]:
                    distances[neighbor] = new_dist
                    heapq.heappush(priority_queue, (new_dist, neighbor))
    
    return distances
 
 
def bellman_ford(
    graph: Dict[str, List[Tuple[str, float]]], 
    start: str
) -> Tuple[Dict[str, float], bool]:
    """
    Bellman-Ford algorithm for shortest paths.
    
    Handles negative edge weights. Detects negative cycles.
    
    Time Complexity: O(V × E)
    Space Complexity: O(V)
    
    Returns: (distances, has_negative_cycle)
    """
    # Get all vertices
    vertices = set(graph.keys())
    for neighbors in graph.values():
        for v, _ in neighbors:
            vertices.add(v)
    
    distances = {v: math.inf for v in vertices}
    distances[start] = 0
    
    # Relax all edges |V| - 1 times
    for _ in range(len(vertices) - 1):
        for u in graph:
            for v, weight in graph[u]:
                if distances[u] != math.inf and distances[u] + weight < distances[v]:
                    distances[v] = distances[u] + weight
    
    # Check for negative cycle: if we can still relax, cycle exists
    has_negative_cycle = False
    for u in graph:
        for v, weight in graph[u]:
            if distances[u] != math.inf and distances[u] + weight < distances[v]:
                has_negative_cycle = True
                break
    
    return distances, has_negative_cycle
 
 
# Example: Graph with positive weights (Dijkstra works)
positive_graph = {
    'A': [('B', 4), ('C', 2)],
    'B': [('C', 1), ('D', 5)],
    'C': [('D', 8), ('E', 10)],
    'D': [('E', 2)],
    'E': []
}
 
print("=== Dijkstra (Positive Weights) ===")
distances = dijkstra(positive_graph, 'A')
print(f"Shortest distances from A: {distances}")
# Output: {'A': 0, 'B': 4, 'C': 2, 'D': 9, 'E': 11}
 
 
# Example: Graph with negative weights (need Bellman-Ford)
negative_graph = {
    'A': [('B', 4), ('C', 2)],
    'B': [('D', 3)],
    'C': [('B', -3), ('D', 5)],  # Negative edge C->B
    'D': []
}
 
print("\n=== Bellman-Ford (Negative Weights) ===")
distances, has_cycle = bellman_ford(negative_graph, 'A')
print(f"Shortest distances from A: {distances}")
print(f"Has negative cycle: {has_cycle}")
# C->B is negative but no negative cycle exists
# A->C->B has total weight 2 + (-3) = -1, better than A->B (4)
 
 
# Example: Graph with negative cycle
negative_cycle_graph = {
    'A': [('B', 1)],
    'B': [('C', 2)],
    'C': [('A', -4)]  # Cycle A->B->C->A has weight 1+2-4 = -1
}
 
print("\n=== Bellman-Ford (Negative Cycle) ===")
distances, has_cycle = bellman_ford(negative_cycle_graph, 'A')
print(f"Has negative cycle: {has_cycle}")  # True

Minimum Spanning Trees

One of the most elegant problems on weighted graphs is finding a Minimum Spanning Tree (MST)—a subset of edges that connects all vertices with minimum total weight, without forming cycles.

Definition:

Given a connected, undirected, weighted graph G = (V, E, w), a spanning tree is a subgraph that:

Is a tree (connected and acyclic)
Contains all vertices of G
Contains exactly |V| - 1 edges

A minimum spanning tree is a spanning tree with minimum total edge weight.

Key Properties:

An MST may not be unique (multiple distinct MSTs with the same total weight may exist)
MST is meaningful only for undirected graphs (directed graphs have arborescences)
MST problems apply to connected graphs; for disconnected graphs, we find a minimum spanning forest

Cut Property

The cut property is key to MST algorithms: For any cut (partition of vertices into two sets), the minimum-weight edge crossing the cut is in some MST. Both Prim's and Kruskal's algorithms exploit this property, just in different ways.

Two Classic Algorithms:

Kruskal's Algorithm:

Sort all edges by weight
Process edges in ascending weight order
Add edge to MST if it doesn't create a cycle (use Union-Find to check)
Stop when |V| - 1 edges are added

Time Complexity: O(E log E) — dominated by sorting

Prim's Algorithm:

Start with any vertex in the MST
Repeatedly add the minimum-weight edge connecting MST vertices to non-MST vertices
Use a priority queue for efficient minimum edge selection
Stop when all vertices are in MST

Time Complexity: O(E log V) with binary heap

minimum_spanning_tree.py
Python
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from typing import Dict, List, Tuple
import heapq
 
class UnionFind:
    """Disjoint Set Union for cycle detection in Kruskal's."""
    def __init__(self, vertices):
        self.parent = {v: v for v in vertices}
        self.rank = {v: 0 for v in vertices}
    
    def find(self, x):
        if self.parent[x] != x:
            self.parent[x] = self.find(self.parent[x])
        return self.parent[x]
    
    def union(self, x, y) -> bool:
        px, py = self.find(x), self.find(y)
        if px == py:
            return False  # Already connected (would create cycle)
        if self.rank[px] < self.rank[py]:
            px, py = py, px
        self.parent[py] = px
        if self.rank[px] == self.rank[py]:
            self.rank[px] += 1
        return True
 
 
def kruskal_mst(vertices: List[str], edges: List[Tuple[str, str, float]]) -> Tuple[List[Tuple[str, str, float]], float]:
    """
    Kruskal's algorithm for Minimum Spanning Tree.
    
    Args:
        vertices: List of all vertices
        edges: List of (u, v, weight) tuples
    
    Returns:
        (MST edges, total MST weight)
    
    Time Complexity: O(E log E)
    """
    # Sort edges by weight
    sorted_edges = sorted(edges, key=lambda e: e[2])
    
    uf = UnionFind(vertices)
    mst_edges = []
    total_weight = 0
    
    for u, v, weight in sorted_edges:
        if uf.union(u, v):  # No cycle created
            mst_edges.append((u, v, weight))
            total_weight += weight
            if len(mst_edges) == len(vertices) - 1:
                break  # MST complete
    
    return mst_edges, total_weight
 
 
def prim_mst(graph: Dict[str, List[Tuple[str, float]]], start: str) -> Tuple[List[Tuple[str, str, float]], float]:
    """
    Prim's algorithm for Minimum Spanning Tree.
    
    Args:
        graph: Adjacency list with (neighbor, weight) pairs
        start: Starting vertex
    
    Returns:
        (MST edges, total MST weight)
    
    Time Complexity: O(E log V)
    """
    mst_edges = []
    total_weight = 0
    visited = {start}
    
    # Priority queue: (weight, from_vertex, to_vertex)
    edges = [(w, start, v) for v, w in graph.get(start, [])]
    heapq.heapify(edges)
    
    while edges and len(visited) < len(graph):
        weight, u, v = heapq.heappop(edges)
        
        if v in visited:
            continue
        
        visited.add(v)
        mst_edges.append((u, v, weight))
        total_weight += weight
        
        # Add edges from newly added vertex
        for neighbor, w in graph.get(v, []):
            if neighbor not in visited:
                heapq.heappush(edges, (w, v, neighbor))
    
    return mst_edges, total_weight
 
 
# Example: Undirected weighted graph
vertices = ['A', 'B', 'C', 'D', 'E']
edges = [
    ('A', 'B', 4), ('A', 'C', 2),
    ('B', 'C', 1), ('B', 'D', 5),
    ('C', 'D', 8), ('C', 'E', 10),
    ('D', 'E', 2)
]
 
# Kruskal's MST
mst_edges, total = kruskal_mst(vertices, edges)
print("=== Kruskal's MST ===")
print(f"MST Edges: {mst_edges}")
print(f"Total Weight: {total}")
 
# Build adjacency list for Prim's
graph = {v: [] for v in vertices}
for u, v, w in edges:
    graph[u].append((v, w))
    graph[v].append((u, w))  # Undirected
 
# Prim's MST
mst_edges, total = prim_mst(graph, 'A')
print("\n=== Prim's MST ===")
print(f"MST Edges: {mst_edges}")
print(f"Total Weight: {total}")

Practical Considerations

When working with weighted graphs in practice, several considerations beyond the core algorithms become important:

Weight Representation:

Integer vs Floating-Point: Integer weights are safer (no precision issues), but many applications need real-valued weights
Overflow: Very large weights or many-hop paths can cause integer overflow; use appropriate data types
Floating-Point Comparison: Be careful comparing floating-point distances; small errors accumulate
Sentinel Values: Use float('inf') or sys.maxsize for 'no edge' / unreachable distances

Multi-Edge and Self-Loop Handling:

Multiple edges between the same vertices: Keep the minimum weight? Sum them? Depends on semantics
Self-loops: Usually weight 0, but some applications assign non-zero weights

Composite Weights

Real-world edges often have multiple attributes (distance, time, cost). Solutions: (1) Combine into a single weight (e.g., weighted sum), (2) Use multi-objective optimization (Pareto-optimal paths), (3) Apply constraints (minimize time subject to cost ≤ budget). The choice depends on your problem requirements.

Dynamic Weights:

In many applications, weights change over time:

Traffic conditions update continuously
Prices fluctuate throughout the day
Network latency varies with load

Static algorithms (Dijkstra, Bellman-Ford) must recompute from scratch when weights change. For frequently changing weights, consider:

Incremental algorithms that update only affected paths
Time-varying graphs with weight functions w(e, t)
Approximate methods that trade accuracy for speed

Scale Considerations:

For very large graphs (millions of vertices):

Memory-efficient representations become critical
Hierarchical decomposition (contraction hierarchies) enables fast queries
Approximate algorithms may be necessary
Parallelization and distributed algorithms can help

Algorithm Selection Guide
Scenario	Recommended Algorithm	Notes
Single-source, positive weights	Dijkstra	O((V+E) log V), most common case
Single-source, negative weights	Bellman-Ford	O(V × E), detects negative cycles
All-pairs, dense graph	Floyd-Warshall	O(V³), simple implementation
All-pairs, sparse + negative	Johnson's	Runs Dijkstra from each vertex after reweighting
Minimum spanning tree	Kruskal or Prim	Choose based on graph density
Very large graphs	A* with heuristic	If goal-directed search applies

Summary: Weighted Graphs

We've explored weighted graphs comprehensively—from their formal definition to practical algorithms and real-world applications. Let's consolidate the key concepts:

Key Takeaways

•Weights add quantitative meaning to edges — Distances, costs, times, capacities—edge weights model real-world numerical relationships between entities.
•Path weight is the sum of edge weights — Shortest weighted path minimizes total weight, not edge count. This fundamentally changes algorithm choice.
•Weight sign matters critically — Positive weights enable Dijkstra; negative weights require Bellman-Ford. Negative cycles make shortest paths undefined.
•Representation choices — Adjacency matrix allows O(1) weight lookup; adjacency list saves space for sparse graphs. Include weights in either structure.
•Minimum Spanning Trees — Connect all vertices with minimum total edge weight. Kruskal and Prim are the classic O(E log V) algorithms.
•Practical considerations — Floating-point precision, composite weights, dynamic updates, and scale all influence real-world implementations.

What's Next:

We've covered directed graphs, undirected graphs, and weighted graphs. The final page explores mixed and special graph types—multigraphs, hypergraphs, bipartite graphs, and other specialized structures that extend the basic graph model for specific application domains.

Page Complete

You now have thorough understanding of weighted graphs: their formal definition, representation strategies, real-world applications, the critical importance of weight signs, and fundamental algorithms for shortest paths and minimum spanning trees. You can identify when problems require weighted modeling and choose appropriate algorithms.

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Data Structures & AlgorithmsGraph Types

Graph Types — Directed, Undirected, Weighted

LevelIntermediate

Duration60 mins

TopicGraph Types

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Weighted Graphs — Edges with Costs

When Connections Have Value

What You Will Learn

Formal Definition and Notation

A weighted graph is a graph where each edge has an associated numerical value called its weight, cost, length, or capacity, depending on the application context.

Formal Definition:

A weighted graph G is typically defined as a triple G = (V, E, w), where:

V is a finite set of vertices
E is a set of edges (directed or undirected)
w is a weight function w: E → ℝ that assigns a real number to each edge

For an edge e = (u, v) or e = {u, v}, we write w(e) or w(u, v) to denote its weight.

Alternative Notation:

Some authors define weighted graphs as G = (V, E) where each edge is a tuple containing its weight: e = (u, v, w). Both notations capture the same concept—edges with associated values.

Weight Semantics Vary by Application

Weighted Path and Distance:

In a weighted graph, the weight of a path is the sum of weights of edges along the path:

w(P) = Σw(eᵢ) for all edges eᵢ in path P

The weighted shortest path from u to v is the path minimizing this sum. This differs fundamentally from unweighted graphs, where the shortest path simply minimizes edge count.

Example:

Consider weighted edges: A→B(3), A→C(1), C→B(1)

Direct path A→B has weight 3
Path A→C→B has weight 1 + 1 = 2
The shortest weighted path is A→C→B, even though it uses more edges

This is why BFS (which finds shortest edge-count paths) doesn't work for weighted graphs—we need algorithms like Dijkstra's that consider edge weights.

Weighted Graph Terminology
Term	Definition	Example
Weight Function	Mapping from edges to real numbers: w: E → ℝ	w(A,B) = 5 means edge A-B has weight 5
Path Weight	Sum of weights of edges in a path	Path A→B→C with w(A,B)=2, w(B,C)=3 has weight 5
Weighted Shortest Path	Path minimizing total weight (not edge count)	May use more edges if their total weight is less
Positive Weights	All edge weights > 0	Enables efficient algorithms like Dijkstra's
Non-negative Weights	All edge weights ≥ 0	Zero-weight edges are valid; Dijkstra's still works
Negative Weights	Some edge weights < 0	Requires Bellman-Ford; may have negative cycles
Negative Cycle	Cycle whose total weight is negative	Infinitely decreasing paths; no shortest path exists

Representation and Storage

Weighted Adjacency Matrix:

For a graph with n vertices, the adjacency matrix M is an n×n matrix where:

M[i][j] = w(i, j) if edge (i, j) exists
M[i][j] = ∞ (or a sentinel value) if edge (i, j) doesn't exist
M[i][i] = 0 (path from vertex to itself has zero weight)

Pros: O(1) edge weight lookup, simple implementation, good for dense graphs Cons: O(V²) space regardless of edge count, wasteful for sparse graphs

Weighted Adjacency List:

For each vertex, store a list of (neighbor, weight) pairs:

graph[u] = [(v₁, w₁), (v₂, w₂), ...]

Pros: O(V + E) space, efficient for sparse graphs, natural for traversals Cons: O(degree) edge lookup, slightly more complex than unweighted

weighted_graph_representation.py
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from typing import Dict, List, Tuple, Optional
import math
 
class WeightedGraphMatrix:
    """
    Weighted graph using adjacency matrix representation.
    
    Best for: Dense graphs, frequent edge weight lookups, 
              algorithms that iterate over all edges.
    """
    def __init__(self, vertices: List[str]):
        self.vertices = vertices
        self.vertex_to_idx = {v: i for i, v in enumerate(vertices)}
        n = len(vertices)
        # Initialize with infinity (no edge)
        self.matrix = [[math.inf] * n for _ in range(n)]
        # Distance to self is 0
        for i in range(n):
            self.matrix[i][i] = 0
    
    def add_edge(self, u: str, v: str, weight: float, directed: bool = True):
        i, j = self.vertex_to_idx[u], self.vertex_to_idx[v]
        self.matrix[i][j] = weight
        if not directed:
            self.matrix[j][i] = weight
    
    def get_weight(self, u: str, v: str) -> float:
        i, j = self.vertex_to_idx[u], self.vertex_to_idx[v]
        return self.matrix[i][j]
    
    def has_edge(self, u: str, v: str) -> bool:
        return self.get_weight(u, v) != math.inf
 
 
class WeightedGraphList:
    """
    Weighted graph using adjacency list representation.
    
    Best for: Sparse graphs, traversal algorithms, 
              memory-constrained environments.
    """
    def __init__(self):
        # graph[u] = [(v, weight), ...]
        self.graph: Dict[str, List[Tuple[str, float]]] = {}
    
    def add_vertex(self, v: str):
        if v not in self.graph:
            self.graph[v] = []
    
    def add_edge(self, u: str, v: str, weight: float, directed: bool = True):
        self.add_vertex(u)
        self.add_vertex(v)
        self.graph[u].append((v, weight))
        if not directed:
            self.graph[v].append((u, weight))
    
    def get_neighbors(self, u: str) -> List[Tuple[str, float]]:
        return self.graph.get(u, [])
    
    def get_weight(self, u: str, v: str) -> Optional[float]:
        for neighbor, weight in self.graph.get(u, []):
            if neighbor == v:
                return weight
        return None
    
    def get_all_edges(self) -> List[Tuple[str, str, float]]:
        edges = []
        for u, neighbors in self.graph.items():
            for v, w in neighbors:
                edges.append((u, v, w))
        return edges
 
 
# Example usage
print("=== Adjacency List (Sparse Graph) ===")
graph = WeightedGraphList()
graph.add_edge('A', 'B', 4)
graph.add_edge('A', 'C', 2)
graph.add_edge('B', 'C', 1)
graph.add_edge('B', 'D', 5)
graph.add_edge('C', 'D', 8)
graph.add_edge('C', 'E', 10)
graph.add_edge('D', 'E', 2)
 
print(f"Neighbors of A: {graph.get_neighbors('A')}")
# Output: [('B', 4), ('C', 2)]
 
print(f"Weight A->B: {graph.get_weight('A', 'B')}")
# Output: 4
 
print(f"All edges: {graph.get_all_edges()}")
 
 
print("\n=== Adjacency Matrix (Dense Graph) ===")
matrix_graph = WeightedGraphMatrix(['A', 'B', 'C', 'D'])
matrix_graph.add_edge('A', 'B', 1)
matrix_graph.add_edge('A', 'C', 4)
matrix_graph.add_edge('B', 'C', 2)
matrix_graph.add_edge('C', 'D', 3)
 
print(f"Weight A->C: {matrix_graph.get_weight('A', 'C')}")  # 4
print(f"Weight A->D: {matrix_graph.get_weight('A', 'D')}")  # inf (no edge)

Real-World Applications of Weighted Graphs

Major Application Domains

•Navigation and Routing — GPS systems use weighted graphs where intersections are vertices, roads are edges, and weights represent travel time, distance, or real-time traffic conditions. Dijkstra's algorithm or A* finds the fastest route. Weight meanings might combine distance, speed limits, and current congestion.
•Network Flow and Capacity — Internet routing, water distribution, and electrical grids model capacity as edge weights. How much data, water, or current can flow through each link? Maximum flow algorithms find optimal throughput; minimum-cost flow finds the cheapest way to move resources.
•Airline and Transportation Networks — Cities are vertices, flights/train routes are edges, and weights represent ticket prices, flight times, or carbon emissions. Finding the cheapest itinerary, the fastest connection, or the minimum-emission route are weighted shortest path problems.
•Telecommunications — Network latency, bandwidth, and reliability can all serve as edge weights. Quality-of-Service (QoS) routing seeks paths that minimize latency while meeting bandwidth requirements—a multi-objective weighted path problem.
•Social Network Analysis — Edge weights can represent interaction frequency, relationship strength, or influence. Weighted centrality measures identify truly influential nodes (not just those with many connections, but those with strong connections).
•Machine Learning and Clustering — K-nearest neighbors, spectral clustering, and graph neural networks often operate on weighted similarity graphs. Weights encode how similar two data points are, enabling structure discovery in high-dimensional data.

Weight Interpretations by Domain
Domain	Weight Represents	Optimization Goal	Algorithm
Navigation	Travel time / Distance	Shortest path	Dijkstra / A*
Network Flow	Capacity	Maximum throughput	Ford-Fulkerson
Airlines	Ticket price	Cheapest route	Dijkstra / Bellman-Ford
Infrastructure	Construction cost	Minimum spanning tree	Kruskal / Prim
Similarity Graphs	Distance / Dissimilarity	Clustering	MST-based clustering
Influence Networks	Interaction strength	Influence maximization	Weighted centrality

Weight Semantics Determine Algorithm Choice

Positive, Non-Negative, and Negative Weights

The sign of edge weights profoundly impacts which algorithms work and how we interpret results. This is one of the most important practical considerations when working with weighted graphs.

Positive Weights (w(e) > 0 for all e):

All edge weights are strictly positive. This is the most common and well-behaved case.

Dijkstra's algorithm works correctly: once a vertex is finalized, we've found its shortest path
Shortest paths are well-defined; they always exist (in connected components)
No risk of infinite loops or decreasing cycle issues

Non-Negative Weights (w(e) ≥ 0 for all e):

Zero-weight edges are allowed. Dijkstra's still works, but zero-weight edges mean multiple distinct paths may have the same total weight.

Negative Weights (some w(e) < 0):

Some edges have negative weight. This creates serious complications:

Dijkstra's Fails with Negative Weights

Handling Negative Weights:

Bellman-Ford Algorithm: Works correctly with negative weights. Runs in O(V × E) time—slower than Dijkstra but handles negative edges. Also detects negative cycles.
Johnson's Algorithm: For all-pairs shortest paths with negative weights. Reweights edges to eliminate negatives, then runs Dijkstra from each vertex.

Negative Cycles:

A negative cycle is a cycle whose total weight is negative. If a negative cycle exists:

Shortest paths through the cycle are undefined (we can keep going around, decreasing the path weight infinitely)
Bellman-Ford can detect negative cycles (if a |V|th relaxation still improves distances, a negative cycle exists)
We must decide: report 'no shortest path exists' or find paths that don't use the cycle

Positive Weights

•Dijkstra's algorithm: O((V+E) log V)
•A* with admissible heuristics
•Shortest paths always well-defined
•No cycle issues (all cycles positive)
•Most practical applications

Negative Weights

•Bellman-Ford algorithm: O(V × E)
•Must check for negative cycles
•Shortest path may not exist
•Dijkstra's gives wrong answers
•Currency arbitrage, game theory

shortest_path_algorithms.py
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import heapq
from typing import Dict, List, Tuple, Optional
import math
 
def dijkstra(graph: Dict[str, List[Tuple[str, float]]], start: str) -> Dict[str, float]:
    """
    Dijkstra's algorithm for shortest paths from start vertex.
    
    REQUIREMENT: All edge weights must be non-negative!
    
    Time Complexity: O((V + E) log V) with binary heap
    Space Complexity: O(V)
    """
    distances = {v: math.inf for v in graph}
    distances[start] = 0
    priority_queue = [(0, start)]  # (distance, vertex)
    visited = set()
    
    while priority_queue:
        current_dist, current = heapq.heappop(priority_queue)
        
        if current in visited:
            continue
        visited.add(current)
        
        for neighbor, weight in graph.get(current, []):
            if neighbor not in visited:
                new_dist = current_dist + weight
                if new_dist < distances[neighbor]:
                    distances[neighbor] = new_dist
                    heapq.heappush(priority_queue, (new_dist, neighbor))
    
    return distances
 
 
def bellman_ford(
    graph: Dict[str, List[Tuple[str, float]]], 
    start: str
) -> Tuple[Dict[str, float], bool]:
    """
    Bellman-Ford algorithm for shortest paths.
    
    Handles negative edge weights. Detects negative cycles.
    
    Time Complexity: O(V × E)
    Space Complexity: O(V)
    
    Returns: (distances, has_negative_cycle)
    """
    # Get all vertices
    vertices = set(graph.keys())
    for neighbors in graph.values():
        for v, _ in neighbors:
            vertices.add(v)
    
    distances = {v: math.inf for v in vertices}
    distances[start] = 0
    
    # Relax all edges |V| - 1 times
    for _ in range(len(vertices) - 1):
        for u in graph:
            for v, weight in graph[u]:
                if distances[u] != math.inf and distances[u] + weight < distances[v]:
                    distances[v] = distances[u] + weight
    
    # Check for negative cycle: if we can still relax, cycle exists
    has_negative_cycle = False
    for u in graph:
        for v, weight in graph[u]:
            if distances[u] != math.inf and distances[u] + weight < distances[v]:
                has_negative_cycle = True
                break
    
    return distances, has_negative_cycle
 
 
# Example: Graph with positive weights (Dijkstra works)
positive_graph = {
    'A': [('B', 4), ('C', 2)],
    'B': [('C', 1), ('D', 5)],
    'C': [('D', 8), ('E', 10)],
    'D': [('E', 2)],
    'E': []
}
 
print("=== Dijkstra (Positive Weights) ===")
distances = dijkstra(positive_graph, 'A')
print(f"Shortest distances from A: {distances}")
# Output: {'A': 0, 'B': 4, 'C': 2, 'D': 9, 'E': 11}
 
 
# Example: Graph with negative weights (need Bellman-Ford)
negative_graph = {
    'A': [('B', 4), ('C', 2)],
    'B': [('D', 3)],
    'C': [('B', -3), ('D', 5)],  # Negative edge C->B
    'D': []
}
 
print("\n=== Bellman-Ford (Negative Weights) ===")
distances, has_cycle = bellman_ford(negative_graph, 'A')
print(f"Shortest distances from A: {distances}")
print(f"Has negative cycle: {has_cycle}")
# C->B is negative but no negative cycle exists
# A->C->B has total weight 2 + (-3) = -1, better than A->B (4)
 
 
# Example: Graph with negative cycle
negative_cycle_graph = {
    'A': [('B', 1)],
    'B': [('C', 2)],
    'C': [('A', -4)]  # Cycle A->B->C->A has weight 1+2-4 = -1
}
 
print("\n=== Bellman-Ford (Negative Cycle) ===")
distances, has_cycle = bellman_ford(negative_cycle_graph, 'A')
print(f"Has negative cycle: {has_cycle}")  # True

Minimum Spanning Trees

One of the most elegant problems on weighted graphs is finding a Minimum Spanning Tree (MST)—a subset of edges that connects all vertices with minimum total weight, without forming cycles.

Definition:

Given a connected, undirected, weighted graph G = (V, E, w), a spanning tree is a subgraph that:

Is a tree (connected and acyclic)
Contains all vertices of G
Contains exactly |V| - 1 edges

A minimum spanning tree is a spanning tree with minimum total edge weight.

Key Properties:

An MST may not be unique (multiple distinct MSTs with the same total weight may exist)
MST is meaningful only for undirected graphs (directed graphs have arborescences)
MST problems apply to connected graphs; for disconnected graphs, we find a minimum spanning forest

Cut Property

Two Classic Algorithms:

Kruskal's Algorithm:

Sort all edges by weight
Process edges in ascending weight order
Add edge to MST if it doesn't create a cycle (use Union-Find to check)
Stop when |V| - 1 edges are added

Time Complexity: O(E log E) — dominated by sorting

Prim's Algorithm:

Start with any vertex in the MST
Repeatedly add the minimum-weight edge connecting MST vertices to non-MST vertices
Use a priority queue for efficient minimum edge selection
Stop when all vertices are in MST

Time Complexity: O(E log V) with binary heap

minimum_spanning_tree.py
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from typing import Dict, List, Tuple
import heapq
 
class UnionFind:
    """Disjoint Set Union for cycle detection in Kruskal's."""
    def __init__(self, vertices):
        self.parent = {v: v for v in vertices}
        self.rank = {v: 0 for v in vertices}
    
    def find(self, x):
        if self.parent[x] != x:
            self.parent[x] = self.find(self.parent[x])
        return self.parent[x]
    
    def union(self, x, y) -> bool:
        px, py = self.find(x), self.find(y)
        if px == py:
            return False  # Already connected (would create cycle)
        if self.rank[px] < self.rank[py]:
            px, py = py, px
        self.parent[py] = px
        if self.rank[px] == self.rank[py]:
            self.rank[px] += 1
        return True
 
 
def kruskal_mst(vertices: List[str], edges: List[Tuple[str, str, float]]) -> Tuple[List[Tuple[str, str, float]], float]:
    """
    Kruskal's algorithm for Minimum Spanning Tree.
    
    Args:
        vertices: List of all vertices
        edges: List of (u, v, weight) tuples
    
    Returns:
        (MST edges, total MST weight)
    
    Time Complexity: O(E log E)
    """
    # Sort edges by weight
    sorted_edges = sorted(edges, key=lambda e: e[2])
    
    uf = UnionFind(vertices)
    mst_edges = []
    total_weight = 0
    
    for u, v, weight in sorted_edges:
        if uf.union(u, v):  # No cycle created
            mst_edges.append((u, v, weight))
            total_weight += weight
            if len(mst_edges) == len(vertices) - 1:
                break  # MST complete
    
    return mst_edges, total_weight
 
 
def prim_mst(graph: Dict[str, List[Tuple[str, float]]], start: str) -> Tuple[List[Tuple[str, str, float]], float]:
    """
    Prim's algorithm for Minimum Spanning Tree.
    
    Args:
        graph: Adjacency list with (neighbor, weight) pairs
        start: Starting vertex
    
    Returns:
        (MST edges, total MST weight)
    
    Time Complexity: O(E log V)
    """
    mst_edges = []
    total_weight = 0
    visited = {start}
    
    # Priority queue: (weight, from_vertex, to_vertex)
    edges = [(w, start, v) for v, w in graph.get(start, [])]
    heapq.heapify(edges)
    
    while edges and len(visited) < len(graph):
        weight, u, v = heapq.heappop(edges)
        
        if v in visited:
            continue
        
        visited.add(v)
        mst_edges.append((u, v, weight))
        total_weight += weight
        
        # Add edges from newly added vertex
        for neighbor, w in graph.get(v, []):
            if neighbor not in visited:
                heapq.heappush(edges, (w, v, neighbor))
    
    return mst_edges, total_weight
 
 
# Example: Undirected weighted graph
vertices = ['A', 'B', 'C', 'D', 'E']
edges = [
    ('A', 'B', 4), ('A', 'C', 2),
    ('B', 'C', 1), ('B', 'D', 5),
    ('C', 'D', 8), ('C', 'E', 10),
    ('D', 'E', 2)
]
 
# Kruskal's MST
mst_edges, total = kruskal_mst(vertices, edges)
print("=== Kruskal's MST ===")
print(f"MST Edges: {mst_edges}")
print(f"Total Weight: {total}")
 
# Build adjacency list for Prim's
graph = {v: [] for v in vertices}
for u, v, w in edges:
    graph[u].append((v, w))
    graph[v].append((u, w))  # Undirected
 
# Prim's MST
mst_edges, total = prim_mst(graph, 'A')
print("\n=== Prim's MST ===")
print(f"MST Edges: {mst_edges}")
print(f"Total Weight: {total}")

Practical Considerations

When working with weighted graphs in practice, several considerations beyond the core algorithms become important:

Weight Representation:

Integer vs Floating-Point: Integer weights are safer (no precision issues), but many applications need real-valued weights
Overflow: Very large weights or many-hop paths can cause integer overflow; use appropriate data types
Floating-Point Comparison: Be careful comparing floating-point distances; small errors accumulate
Sentinel Values: Use float('inf') or sys.maxsize for 'no edge' / unreachable distances

Multi-Edge and Self-Loop Handling:

Multiple edges between the same vertices: Keep the minimum weight? Sum them? Depends on semantics
Self-loops: Usually weight 0, but some applications assign non-zero weights

Composite Weights

Dynamic Weights:

In many applications, weights change over time:

Traffic conditions update continuously
Prices fluctuate throughout the day
Network latency varies with load

Static algorithms (Dijkstra, Bellman-Ford) must recompute from scratch when weights change. For frequently changing weights, consider:

Incremental algorithms that update only affected paths
Time-varying graphs with weight functions w(e, t)
Approximate methods that trade accuracy for speed

Scale Considerations:

For very large graphs (millions of vertices):

Memory-efficient representations become critical
Hierarchical decomposition (contraction hierarchies) enables fast queries
Approximate algorithms may be necessary
Parallelization and distributed algorithms can help

Algorithm Selection Guide
Scenario	Recommended Algorithm	Notes
Single-source, positive weights	Dijkstra	O((V+E) log V), most common case
Single-source, negative weights	Bellman-Ford	O(V × E), detects negative cycles
All-pairs, dense graph	Floyd-Warshall	O(V³), simple implementation
All-pairs, sparse + negative	Johnson's	Runs Dijkstra from each vertex after reweighting
Minimum spanning tree	Kruskal or Prim	Choose based on graph density
Very large graphs	A* with heuristic	If goal-directed search applies

Summary: Weighted Graphs

We've explored weighted graphs comprehensively—from their formal definition to practical algorithms and real-world applications. Let's consolidate the key concepts:

Key Takeaways

•Weights add quantitative meaning to edges — Distances, costs, times, capacities—edge weights model real-world numerical relationships between entities.
•Path weight is the sum of edge weights — Shortest weighted path minimizes total weight, not edge count. This fundamentally changes algorithm choice.
•Weight sign matters critically — Positive weights enable Dijkstra; negative weights require Bellman-Ford. Negative cycles make shortest paths undefined.
•Representation choices — Adjacency matrix allows O(1) weight lookup; adjacency list saves space for sparse graphs. Include weights in either structure.
•Minimum Spanning Trees — Connect all vertices with minimum total edge weight. Kruskal and Prim are the classic O(E log V) algorithms.
•Practical considerations — Floating-point precision, composite weights, dynamic updates, and scale all influence real-world implementations.

What's Next:

Page Complete

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