🔍 Graph & Grid Algorithm Library

A clean, dependency-free Python library for common BFS, DFS, and grid traversal algorithms. Designed as a reliable toolkit for competitive programming, interview prep, and educational use.

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📥 Installation

pip install graph-algs

Then import directly from the top-level package — no need to reference internal subfolders:

from graph_lib import shortest_path, bfs, lee_algorithm, has_cycle_directed

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📦 Modules

graph_algorithms/
├── bfs/
│   ├── bfs_general_lib.py      # BFS traversal and shortest paths
│   ├── bfs_undirected_lib.py   # Connectivity for undirected graphs
│   ├── bfs_directed_lib.py     # Weak connectivity for directed graphs
│   └── grid_bfs_lib.py         # Grid BFS — islands, pathfinding, distance problems
└── dfs/
    ├── dfs_general_lib.py      # DFS traversal
    ├── dfs_undirected_lib.py   # Cycle detection for undirected graphs
    └── dfs_directed_lib.py     # Cycle detection for directed graphs

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⚙️ Implementation Conventions

Understanding these conventions is essential for using the library correctly.

Error Handling — Exceptions

All functions raise exceptions on invalid or degenerate input rather than returning a sentinel value:

• ValueError — empty graph or grid ({}, []), invalid board size (n <= 0)
• KeyError — start/end node not present in the graph, out-of-bounds grid coordinates, start or end cell being a wall

bfs({}, 'A')                        # → ValueError: Graph is empty!
bfs(graph, 'Z')                     # → KeyError: Vertex Z not in graph!
lee_algorithm(grid, (0,0), (9,9))   # → KeyError: (0, 0) or (9, 9) cell not in grid!
count_islands([])                   # → ValueError: Grid is empty!

No Path / No Result — None

When inputs are structurally valid but the goal is unreachable, functions return None:

shortest_path(graph, 'A', 'Z')       # → None  (Z unreachable from A)
lee_algorithm(grid, (0,0), (3,3))    # → None  (blocked)
rotting_oranges(grid)                # → None  (some fresh oranges can never rot)

Input Validation Scope

The library performs only semantic validation — it checks whether inputs make sense for the algorithm, not whether they conform to a type contract.

✅ Checked:

• Whether the graph or grid is empty
• Whether start/end nodes exist in the graph
• Whether grid coordinates are in bounds
• Whether start/end cells are traversable (value 0)
• Whether board size is positive (knight moves)

❌ Not checked:

• Whether graph is actually a dict
• Whether grid rows are all the same length
• Whether node values are hashable
• Whether start/end are tuples vs. lists in grid functions

If you pass malformed data (e.g., a list instead of a dict, or a grid with jagged rows), you will get a native Python exception.

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📖 API Reference

bfs/bfs_general_lib.py

Basic Traversal

bfs(graph, start, visited=None) → set

Returns the set of all nodes reachable from start. Accepts an optional visited set to continue a traversal across multiple calls.

bfs_order(graph, start, visited=None) → list

Returns nodes in BFS visit order. Accepts an optional shared visited set.

bfs_count(graph, start, visited=None) → int

Returns the count of nodes reachable from start. Accepts an optional shared visited set.

bfs_distances(graph, start) → dict

Returns a dictionary mapping each reachable node to its shortest distance (in number of edges) from start.

Shortest Paths

shortest_path(graph, start, end) → list | None

Returns the shortest path as a node list. Returns [start] if start == end. Returns None if no path exists.

shortest_path_len(graph, start, end) → int | None

Returns the length (number of edges) of the shortest path. Returns 0 if start == end. Returns None if no path exists.

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bfs/bfs_undirected_lib.py

is_connected_undirected(graph) → bool

Returns True if the undirected graph is connected.

count_components_undirected(graph) → int

Returns the number of connected components.

get_components_order_undirected(graph) → list[list]

Returns a list of components, each as an ordered BFS traversal list.

largest_component_size_undirected(graph) → int

Returns the size of the largest component.

shortest_path_len_undirected(graph, start, end) → int | None

Faster shortest path length using bidirectional BFS. Requires both start and end to be present in the graph. Returns None if no path exists.

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bfs/bfs_directed_lib.py

Connectivity functions for directed graphs use weak connectivity — the graph is normalised to undirected via normalize() before traversal.

normalize(graph) → dict

Converts a directed graph into an undirected one by adding reverse edges. Used internally; safe to call directly.

is_connected_directed(graph) → bool

Returns True if the directed graph is weakly connected.

count_components_directed(graph) → int

Returns the number of weakly connected components.

get_components_order_directed(graph) → list[list]

Returns a list of weakly connected components, each as an ordered BFS traversal list.

largest_component_size_directed(graph) → int

Returns the size of the largest weakly connected component.

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dfs/dfs_general_lib.py

dfs(graph, start, visited=None) → set

Recursive DFS. Returns the set of all visited nodes. Accepts an optional shared visited set.

dfs_order(graph, start, visited=None) → list

Iterative DFS using an explicit stack. Returns nodes in DFS visit order. Accepts an optional shared visited set.

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dfs/dfs_undirected_lib.py

has_cycle_undirected(graph) → bool

Detects cycles in an undirected graph using parent tracking.

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dfs/dfs_directed_lib.py

has_cycle_directed(graph) → bool

Detects cycles in a directed graph using the white/gray/black coloring method.

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bfs/grid_bfs_lib.py

Grid convention: 0 = traversable cell, 1 = wall/obstacle (except dist_to_nearest_one and rotting_oranges which use domain-specific values).

Island Problems

count_islands(grid) → int

Counts the number of connected regions of 0-cells (4-directional).

get_islands_order(grid) → list[list[tuple]]

Returns all islands as lists of (row, col) coordinates in BFS order.

largest_island_area(grid) → int

Returns the cell count of the largest island.

Pathfinding (Lee Algorithm)

lee_algorithm(grid, start, end) → list[tuple] | None

Finds the shortest path in a grid from start to end. Returns path as a list of (row, col) tuples. Returns None if unreachable.

lee_algorithm_len(grid, start, end) → int | None

Returns the length of the shortest grid path. Returns 0 if start == end. Returns None if unreachable.

Distance & Simulation

dist_to_nearest_one(grid) → list[list[int]]

Multi-source BFS. Returns a grid where each cell contains its distance to the nearest 1-cell. Cells that are 1 have distance 0.

rotting_oranges(grid) → int | None

Simulates rotting spread (LeetCode 994). Grid values: 0 = empty, 1 = fresh orange, 2 = rotten orange. Returns minutes until all oranges rot, or None if impossible.

Knight Moves

knight_moves(n, start, end) → list[tuple] | None

Finds the shortest path for a chess knight on an n×n board. Returns path as (row, col) tuples. Returns None if unreachable.

knight_moves_len(n, start, end) → int | None

Returns the minimum number of moves for a knight to travel from start to end. Returns None if unreachable.

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🗺️ Graph Format

All graph functions expect an adjacency list as a Python dict.

Implicit sink nodes in directed graphs

In a directed graph, nodes that have no outgoing edges don't have to appear as explicit keys — they can exist only as values in other nodes' neighbour lists. This is supported via graph.get(node, []) instead of graph[node] everywhere neighbours are iterated, so a missing key is safely treated as having no outgoing edges.

# ✅ Both forms are equivalent and accepted

# Explicit — sink nodes listed with empty neighbour lists
graph = {
    'A': ['B', 'C'],
    'B': ['D'],
    'C': [],       # sink, explicitly listed
    'D': []        # sink, explicitly listed
}

# Compact — sink nodes omitted as keys entirely
graph = {
    'A': ['B', 'C'],
    'B': ['D'],
    # C and D are implied by appearing in neighbour lists
}

⚠️ Exception: functions that validate start by checking if start not in graph will raise KeyError if a sink node is passed as start and is not an explicit key. If you need to start traversal from a sink, include it explicitly with an empty list.

For undirected graphs, all edges must still be listed in both directions:

graph = {
    1: [2, 3],
    2: [1, 4],
    3: [1],
    4: [2]
}

Nodes can be any hashable type (strings, integers, tuples, etc.).

🗺️ Grid Format

Grid functions expect a 2D list of integers:

grid = [
    [0, 0, 1, 0],
    [1, 0, 0, 0],
    [0, 0, 1, 1],
    [0, 0, 0, 0],
]

Coordinates are always (row, col). All movement is 4-directional (up, down, left, right).

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💡 Usage Examples

from graph_lib import (
    shortest_path, bfs_distances, count_components_undirected, count_components_directed,
    dfs_order, has_cycle_directed, has_cycle_undirected,
    lee_algorithm, rotting_oranges
)

# Shortest path in an undirected graph
graph = {1: [2, 3], 2: [1, 4], 3: [1], 4: [2]}
print(shortest_path(graph, 1, 4))  # → [1, 2, 4]

# Get distances from a start node
graph = {'A': ['B', 'C'], 'B': ['D'], 'C': [], 'D': []}
print(bfs_distances(graph, 'A'))  # → {'A': 0, 'B': 1, 'C': 1, 'D': 2}

# Count weakly connected components in a directed graph
dgraph = {'A': ['B'], 'B': [], 'C': ['D'], 'D': []}
print(count_components_directed(dgraph))  # → 2

# Count connected components in an undirected graph
ugraph = {1: [2], 2: [1], 3: [4], 4: [3]}
print(count_components_undirected(ugraph))  # → 2

# Cycle detection
print(has_cycle_directed({'A': ['B'], 'B': ['A']}))        # → True
print(has_cycle_undirected({1: [2], 2: [1, 3], 3: [2]}))  # → False

# Grid shortest path
grid = [[0, 0, 0], [1, 1, 0], [0, 0, 0]]
print(lee_algorithm(grid, (0, 0), (2, 2)))  # → [(0,0), (0,1), (0,2), (1,2), (2,2)]

# Rotting oranges
grid = [[2, 1, 1], [1, 1, 0], [0, 1, 1]]
print(rotting_oranges(grid))  # → 4

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📌 Quick Reference: Return Values

Situation	Return Value
Invalid / empty input	ValueError or KeyError
Valid input, goal unreachable	None
start == end (path functions)	[start] or 0
Success	Result value or list (dict)

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