Question

Design and implement an algorithm that gets as input a list of \(k\) integer values \(N_{1}, N_{2}, \ldots, N_{k}\) as well as a special value SUM. Your algorithm must locate a pair of values in the list \(N\) that sum to the value SUM. For example, if your list of values is \(3,8,13,2,17,18,10\), and the value of SUM is 20, then your algorithm would output either of the two values \((2,18)\) or \((3,17)\). If your algorithm cannot find any pair of values that sum to the value SUM, then it should print the message 'Sorry, there is no such pair of values'.

Short answer

Use a set to track numbers; iterate through list, check for complement in set.

Step-by-step solution

Initialize Variables

First, we need to create a set (let's call it 'seen') to keep track of the numbers we have encountered so far. This will help us determine if a complement exists for any given number in the list.

Iterate Through the List

Loop through each number in the list. For each number, calculate the difference between the special SUM value and the current number. This difference represents the needed complement that, when paired with the current number, will equal SUM.

Check for Complement

During each iteration, check if the calculated difference (complement) is already present in the 'seen' set. If it is, then we have found a pair of numbers that add up to SUM.

Output the Result

If a pair is found, return this pair as the solution. If the loop finishes without finding any such pair, then output the message 'Sorry, there is no such pair of values.'

Update the Set and Continue

After checking for the complement, add the current number to the 'seen' set to use in future iterations. This ensures that all numbers processed so far are available for complement checks.

Key concepts

Understanding the Pair Sum Problem

The pair sum problem is a classic exercise in algorithm design that requires finding two numbers in a given list that together add up to a specified sum, commonly referred to as the `SUM`. The concept behind this problem is finding such combinations efficiently. Instead of checking every possible pair blindly, we use tactics that notably reduce the number of operations needed. For instance, given a list such as \(3, 8, 13, 2, 17, 18, 10\) and a SUM value of 20, the goal is to determine pairs like \(2, 18\) or \(3,17\) that satisfy the condition. Solving this effectively requires a clear understanding of various algorithm concepts such as iteration and complement detection.

The Role of the Set Data Structure

In this problem, we employ a `set` data structure to help streamline the process of finding the required number pairs. The set, named `seen` in our algorithm, serves to keep track of all numbers encountered as we iterate through the list. The primary advantage of using a set is its quick lookup time which essentially makes checking for the existence of elements much faster than other data structures such as lists. When we need to find out if a particular number, specifically the complement of the current number, has already been seen, the set allows us to do this in constant time. Thus, using a set optimizes the efficiency of the algorithm and reduces unnecessary computations.

Iterating Through the List of Numbers

Iteration is key to solving the pair sum problem. We perform a loop over each element within the list of numbers. During each iteration, a specific number is checked along with its potential complement that together sums up to the specified SUM value. This is done by computing the difference between the SUM and the current number. By iterating through the numbers consecutively, we analyze each one in the context of having already seen numbers that could potentially form a valid pair. This method ensures every number is considered in both settings—whether it can be a part of a valid pair or if it could help form one with another seen number.

Identifying and Utilizing Complementary Values

To successfully find a pair that sums up to the desired value, identifying complementary values is crucial. Complementary values refer to the adjustments needed to the current number to reach the specified SUM. For any number being examined, its complement is calculated by subtracting it from the SUM. This potential complementary value is what we then look for in our `seen` set. If such a complement exists in the set, it indicates that a valid pair has been found. The algorithm then outputs these two numbers as the solution. This mechanism of checking complements essentially forms the crux of the pair sum solution, drastically reducing the need for nested iterations.