Pairwise Comparison

Pairwise comparison refers to jointly analyzing or comparing pairs of objects or entities. This reveals insights not visible when considering items individually.

Some examples:

• Comparing pairs of test cases to find equivalent ones
• Pairwise similarity of documents for clustering
• Preference elicitation by comparing pairs of items

Java - Pairwise string similarity:

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double similarity(String s1, String s2) {
  // Calculate similarity metric
  return simScore; 
}

double pairwiseSimilarity(List<String> documents) {
  double total = 0.0;
  for (int i = 0; i < docs.size(); i++) {
    for (int j = i+1; j < docs.size(); j++) {
      double sim = similarity(docs.get(i), docs.get(j));
      total += sim;
    }
  }

  return total;
}

C++ - Pairwise point distance:

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double distance(Point p1, Point p2) {
  // Calculate distance
}

double totalDistance(const vector<Point>& points) {
  double total = 0.0;
  for (int i = 0; i < points.size(); i++) {
    for (int j = i+1; j < points.size(); j++) {
      total += distance(points[i], points[j]);
    }
  }
  return total;
}

Python - Pairwise match rate:

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def match(a, b):
  # Return match percentage
  return match_rate 

def pairwise_match(items):
  total = 0
  for i in range(len(items)):
    for j in range(i+1, len(items)):
      total += match(items[i], items[j])

  return total / (len(items) * (len(items) - 1) / 2)

Analyzing pairs reveals relationships and properties not visible individually. Useful across many domains.

Pairwise Comparison is a technique used to compare each element in a collection with every other element in the same collection. It is often utilized in sorting, searching, or evaluating specific conditions between elements. The pairwise comparison process involves taking all possible pairs of elements and applying a comparison operation like equality, less than, or greater than. It plays a crucial role in algorithms like bubble sort, where adjacent pairs are compared and swapped if needed.

Solution

Below, you’ll find code implementations for pairwise comparison, where all pairs of elements in an array are compared in Java, C++, and Python.

Java

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public class PairwiseComparison {
    public static void comparePairs(int[] arr) {
        for (int i = 0; i < arr.length; i++) {
            for (int j = i + 1; j < arr.length; j++) {
                // Compare arr[i] with arr[j]
                if (arr[i] > arr[j]) {
                    System.out.println(arr[i] + " is greater than " + arr[j]);
                } else {
                    System.out.println(arr[i] + " is not greater than " + arr[j]);
                }
            }
        }
    }

    public static void main(String[] args) {
        int[] arr = {3, 5, 2, 8};
        comparePairs(arr);
    }
}

C++

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#include <iostream>

void comparePairs(int arr[], int size) {
    for (int i = 0; i < size; i++) {
        for (int j = i + 1; j < size; j++) {
            // Compare arr[i] with arr[j]
            if (arr[i] > arr[j]) {
                std::cout << arr[i] << " is greater than " << arr[j] << std::endl;
            } else {
                std::cout << arr[i] << " is not greater than " << arr[j] << std::endl;
            }
        }
    }
}

int main() {
    int arr[] = {3, 5, 2, 8};
    int size = sizeof(arr) / sizeof(arr[0]);
    comparePairs(arr, size);
    return 0;
}

Python

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def compare_pairs(arr):
    for i in range(len(arr)):
        for j in range(i + 1, len(arr)):
            # Compare arr[i] with arr[j]
            if arr[i] > arr[j]:
                print(f"{arr[i]} is greater than {arr[j]}")
            else:
                print(f"{arr[i]} is not greater than {arr[j]}")

arr = [3, 5, 2, 8]
compare_pairs(arr)

Tabular Representation

Let’s consider an array of 4 elements [3, 5, 2, 8] and see how Pairwise Comparison works. The table below shows what happens at each iteration:

Here’s a table that specifically highlights the elements under comparison for each iteration, considering the array [3, 5, 2, 8]:

Explanation

• The “Elements in Array” column represents the original array in its entirety, as it does not change during the pairwise comparison process.
• The “Elements Under Comparison” column indicates the specific pair of elements being compared in each iteration.
• The “Comparison” column shows the result of the comparison between the two elements under consideration.
• This table gives a clear visualization of how each pair of elements is compared in the pairwise comparison process.

Key Insights

• The above table represents the iterations in the nested loops for pairwise comparison.
• In the first iteration, the 1st element is compared with the remaining elements (2nd, 3rd, and 4th).
• In the second iteration, the 2nd element is compared with the remaining elements (3rd and 4th).
• The process continues, comparing each element with all subsequent elements in the array.
• The result is a total of n * (n - 1) / 2 comparisons for an array of n elements, where n is the number of elements in the array.
• This process can be used to assess any particular property or relationship between elements, such as equality, less than, greater than, etc.

Key Takeaways

• Pairwise Comparison involves comparing each element with every other element in the collection.
• It can be implemented using nested loops where the outer loop selects one element, and the inner loop compares it with the remaining elements.
• This technique is foundational in many algorithms related to sorting and searching, and it has a time complexity of O(n^2) where n is the number of elements in the collection.

Pairwise comparison and adjacent element comparison are different concepts:

Pairwise Comparison

• In pairwise comparison, every element is compared with every other element in the collection.
• It involves comparing all possible pairs, which means that each element is compared with all the other elements (excluding itself).
• Pairwise comparison has a time complexity of O(n^2) where n is the number of elements.
• This is a broader comparison technique that is used in various algorithms and scenarios, including sorting algorithms like bubble sort.

Adjacent Element Comparison

• In adjacent element comparison, an element is only compared with its immediate neighboring elements.
• It involves comparing pairs of elements that are next to each other in the collection.
• Adjacent element comparison has a time complexity of O(n), as each element is compared only once with its neighbor.
• This technique is used in specific algorithms and scenarios, like checking if a collection is sorted or identifying local maxima and minima.

Key Takeaways

• Pairwise comparison compares every element with every other element, resulting in n * (n - 1) / 2 comparisons.
• Adjacent element comparison compares only neighboring elements, resulting in n - 1 comparisons for a collection of n elements.
• Pairwise comparison is more general and computationally intensive, while adjacent element comparison is specific to neighboring elements and is more efficient.