1. Introduction
Poj 1703 "Find them, Catch them" is a problem that involves the use of a data structure called disjoint-set data structure or Union-Find data structure. In this problem, we are given a list of pairs of integers where each pair represents a relationship between two people. The task is to determine if the given relationships are consistent or contradictory.
2. Union-Find Data Structure
The Union-Find data structure is a data structure that provides efficient operations for maintaining a partition of a set into disjoint subsets. It supports two main operations:
2.1 Union
The Union operation takes two elements and merges the sets containing these elements into a single set. It updates the parent of one set to the root of the other set. This operation is performed to merge two subsets into a single subset.
2.2 Find
The Find operation returns the representative element (root) of the set containing a given element. It is used to find the root of the subset to which an element belongs.
2.3 Implementation
The Union-Find data structure can be implemented using an array to store the parent of each element. Initially, each element is its own parent. The array is initialized with -1 for each element to indicate that it is the root of its own set.
def find(parent, i):
if parent[i] == -1:
return i
return find(parent, parent[i])
def union(parent, x, y):
x_root = find(parent, x)
y_root = find(parent, y)
parent[x_root] = y_root
3. Solution Approach
To solve the problem, we can follow the approach of initializing each element as a separate set and merge the sets based on the given relationships (pairs of integers). After merging all the sets, we can check if any contradiction is present, which implies that two elements belong to the same set but are also in a different set. If a contradiction is detected, the relationships are inconsistent; otherwise, they are consistent.
3.1 Algorithm
Create an array to store the parent of each element and initialize it with -1.
For each pair in the given relationships:
Find the parent of each element in the pair.
If the elements have the same parent (belong to the same set) and the relationship is contradictory, return "Inconsistent".
Otherwise, merge the sets that contain the elements.
After processing all the relationships, if no contradiction is found, return "Consistent".
3.2 Implementation
def find(parent, i):
if parent[i] == -1:
return i
return find(parent, parent[i])
def union(parent, x, y):
x_root = find(parent, x)
y_root = find(parent, y)
parent[x_root] = y_root
def are_relationships_consistent(n, relationships):
parent = [-1] * (2 * n)
for pair in relationships:
x, y, relationship = pair
x -= 1
y -= 1
if relationship == "D":
if find(parent, x) == find(parent, y):
return "Inconsistent"
union(parent, x, y)
return "Consistent"
4. Time and Space Complexity
The time complexity of the solution is O(N+M*logN), where N is the number of elements and M is the number of relationships. The find operation has a time complexity of O(logN) since it traverses the parents of an element up to the root. The union operation has a time complexity of O(1) since it updates the parent directly.
The space complexity of the solution is O(N), where N is the number of elements. The parent array is used to store the parent of each element, requiring O(N) space.
5. Conclusion
The Poj 1703 "Find them, Catch them" problem is a good example of applying the Union-Find data structure to solve a problem that involves checking the consistency of relationships. It demonstrates the usefulness of the Union-Find data structure in efficiently handling disjoint sets and providing operations to merge and find elements. By following the outlined solution approach and implementing the required functions, we can determine the consistency of the given relationships effectively.