Constructive Algorithms

In competitive programming, constructive problems are quite common. These challenges ask you to design an example or a solution that fulfills particular requirements, often based on some inherent pattern. In many cases, despite an exponential growth in the problem size, there is a pattern that lets you quickly generate the required answer. This forces you to analyze how increasing the problem size affects your solution and whether these effects can be generalized. For example, when designing dynamic programming methods, you might need to examine how transitions between states influence the overall result.

Characteristics

One striking feature of constructive problems is their high degree of freedom. In many cases, there exist multiple ways to build a valid solution, and usually, there is one relatively simple construction that meets the conditions. At first glance, it might seem that the relaxed requirements make the problem easier, but the same flexibility can leave you without a clear direction for your approach.

Another important aspect is the variety and flexibility of the problem formats. There isn’t a one-size-fits-all strategy or common pattern that applies to every constructive problem. Instead, you must often rely on creative insight and problem-specific observations to craft your solution.

Examples

Below are a few examples that illustrate common ideas and techniques in solving constructive problems. It’s recommended that you try to think through each problem before looking at the provided ideas.

Example 1

Problem: For any given positive integer $n$, construct three integers $x, y, z$ such that

Solution Idea:
One simple approach is to choose the triplet as $x = n, \quad y = n+1, \quad z = n(n+1).$
A brief justification: Plugging these values into the equation will satisfy the condition for any integer $n > 1$. (Note that when $n = 1$, the numbers $n+1$ and $n(n+1)$ become equal, which violates the requirement of having three distinct numbers.)

The inspiration for such a construction often comes from analyzing simple examples and letting your intuition guide you toward the right pattern.

Example 2

Problem:
There are two tasks involving a permutation of the set $\{1, 2, \ldots, n\}$.

Task 1:
Decide whether it is possible to construct a permutation so that the sum of the first $k$ elements (the prefix sums) are all distinct modulo $n$.

Task 2:
Decide whether it is possible to construct a permutation so that the product of the first $k$ elements (the prefix products) are all distinct modulo $n$.

Solution Ideas:

For Task 1, note the following:

• When $n$ is odd, it turns out that no valid permutation exists.
• When $n$ is even, one viable construction is to place $n$ at the beginning, followed by a carefully arranged order such as $n, 1, n-2, 3, \dots$.
  The reasoning is that if $n$ did not appear as the first element, the prefix sums before and after its appearance would force a collision when taken modulo $n$. One approach is to consider the differences between successive prefix sums. By designing a sequence where the differences modulo $n$ follow a pattern like

you can ensure that each prefix sum is unique modulo $n$.

For Task 2, the challenge is a bit different:

• When $n$ is a composite number (except for the special case when $n = 4$), the construction is impossible. This is because there exist factors $p$ and $q$ (with $p,q < n$) such that $p \times q \equiv 0 \pmod{n}$. Once both appear in the permutation, the product of the prefix becomes $0$ modulo $n$, causing repetition.
• However, when $n$ is a prime (or when $n = 4$), a valid permutation exists. One such construction is to use a sequence that starts with $1$ (which must be first, else two consecutive prefix products would be identical) and ends with $n$ (since after $n$ the product will be $0$ modulo $n$). Analyzing several examples suggests a pattern where each number (apart from the first and last) is derived from the multiplicative inverses in the set $\{1, 2, \dots, n-2\}$. With this reasoning, one can prove that the entire permutation meets the required property.

Example 3

Problem:
Given an integer $N$, construct a simple, connected undirected graph with $N$ nodes (numbered from $1$ to $N$). The graph must have the following property: there exists an integer $S$ such that for every node, the sum of the labels of all its adjacent nodes equals $S$. It is guaranteed that a solution exists for the input data.

Solution Idea:
One strategy is to look at small values of $n$ (like $3, 4, 5$) and spot the pattern. A constructive method is to build a complete multipartite graph with $k$ parts, ensuring that each part has the same total or “balance” of node labels. For instance, if $n$ is even, one can pair nodes in a symmetric way (for example, pair $1$ with $n$, $2$ with $n-1$, and so on) so that the sum of the labels for each pair is the same. When $n$ is odd, you can isolate the node with the highest number and then pair up the remaining nodes similarly. This structured pairing guarantees that the graph remains connected and that all nodes have the same sum $S$ of adjacent node labels when viewed appropriately.

Example 4

Problem:
Imagine recalling a classic 0/1 knapsack problem from your early days in competitive programming. The problem is as follows:

Given $n$ items with volumes $v_1, v_2, \dots, v_n$, determine the number of ways to select some of them (or potentially none) such that the total volume is exactly $w$. Since the number of ways can be enormous, the answer should be given modulo a number $P$.

In a twist inspired by memory, only the values of $w$ and $P$ are provided in the sample input, and the exact values of $n$ and $v$ (the item volumes) are left unspecified. Your task is to construct a consistent test case that matches the provided sample results.

Solution Idea:
This problem is among the most flexible constructive problems because the degrees of freedom in choosing the test data are very high. An insightful observation is that you can force the actual number of valid selections to be less than $P$ by tailoring the item volumes. One effective strategy is as follows:

• Include a large number of items with a small volume (for example, items of volume 1).
• Also, include a few items with volumes greater than $w/2$. These “big” items can only be selected at most once due to their size, and each one contributes a specific count of valid combinations (expressible via a combinatorial function like

where $x$ is the volume of a big item).

Using dynamic programming, you can precompute a function to match the minimal number of large items required for every possible case. In practice, you only need to consider a relatively small range (for example, up to $20$ items) to construct such test cases. This creative blend of many small items and a few large items leads to a test case that fits the sample input while fulfilling the problem constraint.

These examples illustrate that, while constructive problems can sometimes seem open-ended due to their flexibility, careful observation and a mix of combinatorial ideas often lead to elegant solutions. By examining small cases, identifying patterns, and understanding the strict conditions imposed by the problems, you can often find a viable construction that meets all requirements. Happy problem solving!