Hello there! Get ready to explore an intriguing problem involving list manipulation and combinatorial logic using the capabilities of PHP. We will tackle the task of finding combinations in a given list whose sum is equivalent to a specified target value. Let's dive into the realm of PHP and number theory for an engaging journey.

The objective is to write a PHP function that accepts a list of distinct integers and a target sum as input. The goal is to identify exactly four numbers in the list that, when summed, equal this target. If there are multiple sets meeting this condition, your function should return any one of them. If no such quadruple exists, the function should return an empty array.

Consider this list as an example: [5, 15, 2, 7, 8, 4]. If your target sum is 24, a four-number set that adds up to this value could be [5, 7, 4, 8].

The input list will contain at least 4 and at most 1000 distinct integers. The input integers will be in the range from $-10^6$ to $10^6$. The target sum will also be within the range $-10^6$ to $10^6$. There is a time limit for the solution to evaluate within 3 seconds.

The simplest solution is the brute-force approach that iterates over every quadruple of numbers in the list. This has a complexity of $O(N^4)$.

If you imagine each PHP operation takes a small amount of time and you have a list of a thousand integers, performing $O(N^4)$ operations may result in a runtime far exceeding acceptable limits, possibly taking several hours. Therefore, this is not optimal.

To reduce the complexity to $O(N^3)$, you can iterate over only three elements and check if the target - element1 - element2 - element3 exists in the list using an array of seen sums. This optimization reduces the time significantly, but further efficiencies can be achieved.

With a solution complexity of $O(N^2)$ — the one we'll build in this lesson using PHP — the execution time becomes feasible, fitting well within the 3-second boundary.

These estimations highlight why optimized solutions are crucial. Our organized solution with time complexity $O(N^2)$ is efficient even for larger inputs, thanks to PHP's built-in functions for handling arrays effectively.

To solve this problem using PHP efficiently, we utilize an approach with a time complexity of $O(N^2)$, leveraging associative arrays for quick lookups.

Conceptual Breakdown:

1. Store Pair Sums: We'll initialize an associative array to track all pairs of numbers and their sums. This array will have sums as keys and the indices of the numbers forming these sums as values.

2. Finding Complement Pairs: For each pair of numbers, calculate the difference between the target sum and the current pair’s sum. This difference represents the required sum from another pair.

3. Verify Distinct Indices: Using our associative array, check for a pair whose sum equals this difference while ensuring that the indices do not overlap with the initial pair. If such pairs exist, return these four numbers.

Why This Works:

• Efficiency: PHP's associative arrays allow for rapid insertion and lookup operations, significantly speeding up the process compared to brute-force.
• Scalability: With a maximum limit of 1000 entries, this method ensures prompt execution well within the time limit.

Our initial approach is to set up an empty associative array. This associative array will be used to store sums of all pairs of numbers in the list as keys, with indices of the pair as their corresponding values.

php
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1<?php
2function findQuadSum($targetSum, $numbers) {
3    $length = count($numbers);
4    $sumMap = array();
5}
6?>

Next, populate the associative array. For every pair of integers in the list, calculate their sum and store it as a key in the array, using the indices of the pair as the values.

php
Copy to clipboard
1<?php
2function findQuadSum($targetSum, $numbers) {
3    $length = count($numbers);
4    $sumMap = array();
5    
6    for ($i = 0; $i < $length - 1; ++$i) {
7        for ($j = $i + 1; $j < $length; ++$j) {
8            $pairwiseSum = $numbers[$i] + $numbers[$j];
9            if (!isset($sumMap[$pairwiseSum])) {
10                $sumMap[$pairwiseSum] = array();
11            }
12            $sumMap[$pairwiseSum][] = array($i, $j);
13        }
14    }
15}
16?>

Finally, iterate through all pairs and, for each, calculate the difference between the target sum and the pair sum, searching for this difference in the associative array. For successful matches, verify that none of the indices overlap. If a suitable set is found, return the four numbers. If unsuccessful, return an empty array.

php
Copy to clipboard
1<?php
2function findQuadSum($targetSum, $numbers) {
3    $length = count($numbers);
4    $sumMap = array();
5    
6    // Step 2: Populate the associative array with sums of all pairs
7    for ($i = 0; $i < $length - 1; ++$i) {
8        for ($j = $i + 1; $j < $length; ++$j) {
9            $pairwiseSum = $numbers[$i] + $numbers[$j];
10            if (!isset($sumMap[$pairwiseSum])) {
11                $sumMap[$pairwiseSum] = array();
12            }
13            $sumMap[$pairwiseSum][] = array($i, $j);
14        }
15    }
16
17    // Step 3: Search for the complement pairs
18    foreach ($sumMap as $sum => $pairs1) {
19        $complement = $targetSum - $sum;
20        if (isset($sumMap[$complement])) {
21            $pairs2 = $sumMap[$complement];
22            foreach ($pairs1 as $pair1) {
23                foreach ($pairs2 as $pair2) {
24                    list($a, $b) = $pair1;
25                    list($c, $d) = $pair2;
26                    
27                    // Ensure all indices are distinct
28                    if ($a !== $c && $a !== $d && $b !== $c && $b !== $d) {
29                        return [$numbers[$a], $numbers[$b], $numbers[$c], $numbers[$d]];
30                    }
31                }
32            }
33        }
34    }
35
36    return [];
37}
38
39$numbers = [5, 15, 2, 7, 8, 4];
40$target = 24;
41print_r(findQuadSum($target, $numbers));
42// Array
43// (
44//     [0] => 5
45//     [1] => 7
46//     [2] => 8
47//     [3] => 4
48// )

Fantastic work! By completing this task, you've learned how to exploit PHP's powerful associative arrays to solve computational problems effectively. This skill of combining list handling, combinatorial logic, and efficient code writing is invaluable in your journey as a programmer.

Don't stop here; take your knowledge further by experimenting with different lists and target sums. Test yourself and enrich your understanding by tackling similar problems. Use this lesson as a guide and continue to explore the versatility of PHP. Happy coding!