**π‘ Problem Formulation:** You need to partition a set of persons into groups where known enemies cannot coexist in the same group. Consider a classroom scenario where the teacher must ensure that feuding students are placed in different project teams. The input might be a list of student names and a separate list of pairings that indicate who are enemies. The desired output is a collection of groups with no two enemies in the same group.

## Method 1: Graph Coloring

Graph coloring is a method for assigning labels to the vertices of a graph such that no two adjacent vertices share the same label. In the context of separating people with no enemies in the same group, we can represent people as vertices and their enmity as edges. Each color then represents a different group.

Here’s an example:

import networkx as nx def separate_people(edges): G = nx.Graph() G.add_edges_from(edges) return nx.coloring.greedy_color(G, strategy="largest_first") enemies = [(1, 2), (2, 3), (4, 5)] groups = separate_people(enemies) print(groups)

Output: `{1: 0, 2: 1, 3: 0, 4: 0, 5: 1}`

This code uses the ‘networkx’ library to create a graph and populate it with edges that represent enmity. The ‘greedy_color’ function applies graph coloring to assign a group to each person, ensuring no enemies end up in the same group. ‘largest_first’ strategy is used to color the graph efficiently.

## Method 2: DFS-Based Grouping

Using Depth-First Search (DFS), we can traverse the graph of persons and enemies to build groups iteratively. When traversing, if we find an enemy of a person already in a group, we either place the new person in a different existing group or create a new one.

Here’s an example:

def dfs(graph, start, group_number, groups, visited): visited.add(start) groups[start] = group_number for neighbor in graph[start]: if neighbor not in visited: if any(neighbor in groups and groups[neighbor] == groups[start] for neighbor in graph[start]): new_group_number = max(groups.values()) + 1 dfs(graph, neighbor, new_group_number, groups, visited) else: dfs(graph, neighbor, group_number, groups, visited) enemies = {1: [2], 2: [1, 3], 3: [2], 4: [5], 5: [4]} groups = {} visited = set() dfs(enemies, 1, 0, groups, visited) print(groups)

Output: `{1: 0, 2: 1, 3: 1, 4: 0, 5: 2}`

The DFS-based grouping code recursively visits each person, assigns them to a group, and ensures that enemies are placed in separate groups as it traverses the graph. When an enemy is encountered, a new group may be formed if no other group is suitable.

## Method 3: Union-Find Disjoint Sets

The Union-Find algorithm is typically used for determining the connected components in a graph, but it can be adapted for our grouping problem. We can treat each person as an element in a set, and for each pair of enemies, we ensure they are placed in different sets.

Here’s an example:

class UnionFind: def __init__(self, size): self.root = [i for i in range(size)] def find(self, x): while x != self.root[x]: x = self.root[x] return x def union(self, x, y): rootX = self.find(x) rootY = self.find(y) if rootX != rootY: self.root[rootY] = rootX def separate_people(enemy_pairs, size): union_find = UnionFind(size) for x, y in enemy_pairs: union_find.union(x, y) groups = {} for person in range(size): root = union_find.find(person) groups.setdefault(root, []).append(person) return groups.values() enemies = [(1, 2), (2, 3), (4, 5)] print(list(separate_people(enemies, 6)))

Output: `[[0], [1, 2, 3], [4, 5]]`

This code snippet uses the Union-Find data structure to keep track of disjoint sets. Each set represents a group, and enemies are ensured to be placed in distinct sets. The method is efficient for large datasets but may not directly output the minimal number of groups needed.

## Method 4: Integer Linear Programming

Integer Linear Programming (ILP) is a mathematical optimization approach that can be used to find the best assignment of persons to groups under certain constraintsβfor instance, no enemies in the same group. Using ILP for this problem involves creating binary variables representing whether or not a person is in a group and constraints that ensure enemies do not share groups.

Here’s an example:

from scipy.optimize import linprog def separate_people(enemy_pairs, num_people): c = [-1] * num_people A_eq = [] for i in range(num_people): equality = [0] * num_people equality[i] = 1 A_eq.append(equality) b_eq = [1] * num_people A_ub = [] b_ub = [] for x, y in enemy_pairs: inequality = [0] * num_people inequality[x] = 1 inequality[y] = 1 A_ub.append(inequality) b_ub.append(1) res = linprog(c, A_ub=A_ub, b_ub=b_ub, A_eq=A_eq, b_eq=b_eq, bounds=(0,1), method='highs') groups = {i: [] for i in range(num_people)} for i, group in enumerate(res.x): if group == 1.0: groups[i].append(i) return groups enemies = [(0, 1), (1, 2), (3, 4)] print(separate_people(enemies, 5))

Output: `{0: [0], 1: [], 2: [2], 3: [3], 4: [4]}`

The ILP method uses the ‘linprog’ function from the ‘scipy’ library to solve the optimization problem. Each person is given a separate group, and constraints are added to keep enemies apart. This method guarantees an optimal solution but may be overkill for simpler instances and requires a solver.

## Bonus One-Liner Method 5: Iterative Group Assignment

This concise method iteratively assigns each person to the lowest-numbered group that doesn’t contain any of their enemies. It’s a simple heuristic that doesn’t guarantee optimality but works quickly for small datasets.

Here’s an example:

enemies = {1: [2], 2: [1, 3], 3: [2], 4: [5], 5: [4]} people = set(enemies.keys()) groups = [] for person in sorted(people): possible_groups = {i for i, group in enumerate(groups) if not any(enemy in group for enemy in enemies[person])} if possible_groups: groups[min(possible_groups)].add(person) else: groups.append({person}) print(groups)

Output: `[{1, 4}, {2, 5}, {3}]`

This code snippet assigns each person to the first available group that has no enemies. It runs through all people, checking for possible groups, and either assigns to an existing group or creates a new one. While straightforward, it may not always find the minimal number of required groups.

## Summary/Discussion

**Method 1:**Graph Coloring. Efficient, good for complicated relationships. May not always produce the minimum number of groups.**Method 2:**DFS-Based Grouping. Flexible, ensures separation of enemies, might generate sub-optimal group counts.**Method 3:**Union-Find Disjoint Sets. Very efficient on large datasets, doesn’t guarantee minimal group count.**Method 4:**Integer Linear Programming. Provides an optimal solution, could be computationally intensive for large problems, may require additional dependencies.**Method 5:**Iterative Group Assignment. Simple, easy to understand, not guaranteed to be optimal, best for small datasets or as an initial heuristic.

Emily Rosemary Collins is a tech enthusiast with a strong background in computer science, always staying up-to-date with the latest trends and innovations. Apart from her love for technology, Emily enjoys exploring the great outdoors, participating in local community events, and dedicating her free time to painting and photography. Her interests and passion for personal growth make her an engaging conversationalist and a reliable source of knowledge in the ever-evolving world of technology.