Greedy Algorithms: Maximum T-Distance Subset

Problem Description

Imagine you’re setting up sensors in a network, and you need to place them far enough apart to avoid interference. Each potential location is marked by a coordinate on a number line, and you want to place as many sensors as possible while ensuring that no two sensors are closer than a certain distance T from each other. This is an instance of the maximum T-distance subset problem.

Problem Constraints

1. You have an array A of n positive integers representing potential sensor locations
2. You need to select a subset S of these locations
3. For any two locations x and y in S, |x - y| ≥ T (the T-distance property)
4. The subset S should be as large as possible while maintaining the T-distance property

Example

Consider this scenario with 7 potential locations and a minimum distance T = 3:

Understanding the Solution

Why Greedy Works

The T-distance subset problem is perfect for a greedy approach. Here’s why:

1. Optimal Substructure: Once we place a sensor, the problem reduces to placing sensors in the remaining valid positions
2. Greedy Choice Property: Taking the leftmost valid position at each step leads to an optimal solution
3. Local Optimality: Making locally optimal choices (earliest valid position) leads to a globally optimal solution

The Greedy Strategy

Our approach can be broken down into these key steps:

1. Sort the array in ascending order to make distances easy to check
2. Start with the smallest number (leftmost position)
3. For each subsequent position:
   • Check if it’s at least T distance from the last selected position
   • If yes, select it and update our last selected position
   • If no, skip it and continue

Example Walkthrough

Let’s solve our example step by step:

1. First, sort the array:

   A (sorted): [3, 5, 7, 9, 10, 12, 25]
   

2. Start with 3 (smallest number):

   • Add 3 to our subset
   • Next valid position must be ≥ (3 + 3) = 6

3. Looking for next position (≥ 6):

   • Skip 5 (too close)
   • 7 works! Add it
   • Next valid position must be ≥ (7 + 3) = 10

4. Looking for next position (≥ 10):

   • Skip 9 (too close)
   • 10 works! Add it
   • Next valid position must be ≥ (10 + 3) = 13

And so on…

The final result is S = 25, which is a maximum-sized subset where all pairs are at least 3 units apart.

Solution Implementation

FIND-MAX-T-DISTANCE-SUBSET(A, n, T)
    # First sort the array to make it easier to process
    MERGE-SORT(A, 1, n)
    
    # Initialize result array S and its size
    Let S be a new array
    s_size = 1
    
    # Always include the first (smallest) element
    S[1] = A[1]
    
    # Last element we included in our subset
    last_included = A[1]
    
    # Try to add each subsequent element
    for i = 2 to n
        # If current element maintains T-distance with last included element
        if A[i] - last_included ≥ T
            s_size = s_size + 1
            S[s_size] = A[i]
            last_included = A[i]
        endif
    endfor
    
    return S, s_size

Complexity Analysis

Time Complexity $O(n * log(n))$

• Sorting the array: O(n log n)
• Single pass through sorted array: O(n)

Space Complexity $O(n)$

• O(n) to store the result array
• O(n) for the sorted array if we don’t modify input