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Explanation of DFS-Based Approach for Removing Stones

Overview

This repository contains the implementation of a Depth-First Search (DFS) approach to solve the problem of removing the maximum number of stones from a 2D plane while ensuring that at least one stone remains from each connected component.

The problem is solved in five different programming languages: C++, Java, JavaScript, Python, and Go. Below, you'll find a step-by-step explanation of the logic used in each implementation.

Step-by-Step Explanation

1. Understanding the Problem

  • We are given n stones on a 2D plane, each located at integer coordinates.
  • Two stones are connected if they share the same row or the same column.
  • We want to remove as many stones as possible such that at least one stone remains from each connected component.

2. Graph Representation

  • The problem can be modeled as a graph:
    • Each stone is represented as a node.
    • An edge is drawn between two nodes if the corresponding stones share the same row or column.
  • The goal is to count the number of connected components in this graph. Once we have the connected components, the maximum number of stones that can be removed is n - numComponents, where numComponents is the number of connected components.

3. Depth-First Search (DFS) Approach

  • Step 1: Initialize the Graph Structure

    • Create an adjacency list to represent the graph where each stone is connected to other stones that share the same row or column.

  • Step 2: Build the Graph

    • Iterate over all pairs of stones.
    • If two stones share the same row or column, add an edge between them in the adjacency list.

  • Step 3: DFS to Find Connected Components

    • Initialize a data structure (set, visited map, or boolean array) to keep track of visited nodes (stones).
    • Iterate through each stone:
      • If the stone hasn't been visited, start a DFS from that stone to explore all stones in its connected component.
      • After completing the DFS for a component, increment the component count.

  • Step 4: Calculate the Result

    • The maximum number of stones that can be removed is the total number of stones minus the number of connected components.

4. Implementation Details

C++ Implementation

  • Language Constructs:
    • vector for adjacency list.
    • unordered_set for tracking visited nodes.
  • DFS Function:
    • Recursively visits all neighbors of a node, marking them as visited.
  • Graph Building:
    • Nested loops to find and connect stones in the same row/column.

Java Implementation

  • Language Constructs:
    • List<List<Integer>> for adjacency list.
    • Set<Integer> for visited nodes.
  • DFS Function:
    • Similar recursive approach to traverse all connected stones.
  • Graph Building:
    • Uses ArrayList to maintain adjacency lists for each stone.

JavaScript Implementation

  • Language Constructs:
    • Array for adjacency list.
    • Set for visited nodes.
  • DFS Function:
    • Recursive function to explore connected stones.
  • Graph Building:
    • Uses arrays to represent the graph and connects stones based on shared row/column.

Python Implementation

  • Language Constructs:
    • List for adjacency list.
    • Set for visited nodes.
  • DFS Function:
    • Recursive DFS to visit connected components.
  • Graph Building:
    • Iterates over pairs of stones and builds the adjacency list.

Go Implementation

  • Language Constructs:
    • [][]int for adjacency list.
    • map[int]bool for tracking visited nodes.
  • DFS Function:
    • DFS implemented as a separate function, marking nodes as visited.
  • Graph Building:
    • Constructs the graph by connecting stones with shared rows/columns.

5. Final Notes

  • The DFS approach is efficient and works well with the problem constraints.
  • The main challenge lies in correctly identifying and connecting all stones that belong to the same connected component.
  • This method ensures that the maximum number of stones are removed while keeping the graph connected.