In the process of preparing for coding tests, solving various algorithm problems is very important. In this post, we will examine the problem of building a bridge according to given conditions. Bridge construction is a type of optimization problem, where the goal is to develop an algorithm that builds a bridge as efficiently as possible while meeting specified conditions.
Problem Description
You want to build a bridge across a wide river. The bridge is made of wooden planks with a specified range, and each wooden plank can support a unique load. To support the bridge, the maximum load that each plank can sustain must not be exceeded. Additionally, the total length of the bridge must be at least a given value, and it must be built at the minimum cost.
Input
- First line: Length of the bridge L (1 ≤ L ≤ 1000)
- Second line: Number of wooden planks N (1 ≤ N ≤ 100)
- Third line: An integer array W of length N representing the maximum load each wooden plank can support (1 ≤ W[i] ≤ 1000)
Output
Print the minimum number of wooden planks that can be used. If no possible combination exists, print -1.
Solution Process
To solve this problem, we need to explore all possible combinations of the given wooden planks based on their load to construct the bridge. To approach the problem more systematically, we aim to resolve it through the following steps.
1. Problem Analysis
To build the bridge, we need to decide how to select the planks to match the total bridge length L while also considering the load of each plank. This ultimately boils down to a combination problem.
2. Considering Combinations
Using combinations from the given N wooden planks, we need to generate all combinations that can create a bridge length of at least L, and check the load of these combinations to find valid options.
3. Implementation Method
To aid understanding, I will show the implementation in C#. Here, I will try all combinations using recursive calls.
using System;
using System.Linq;
class Program
{
static int[] woodWeights;
static int minPlanks = int.MaxValue;
static void Main(string[] args)
{
int L = int.Parse(Console.ReadLine());
int N = int.Parse(Console.ReadLine());
woodWeights = Console.ReadLine().Split(' ').Select(int.Parse).ToArray();
MakeBridge(0, 0, 0);
Console.WriteLine(minPlanks == int.MaxValue ? -1 : minPlanks);
}
static void MakeBridge(int index, int totalLength, int count)
{
if (totalLength >= L)
{
minPlanks = Math.Min(minPlanks, count);
return;
}
for (int i = index; i < woodWeights.Length; i++)
{
MakeBridge(i + 1, totalLength + woodWeights[i], count + 1);
}
}
}
4. Code Explanation
The above code recursively checks the combinations of each wooden plank to find meaningful ways to build the bridge. The MakeBridge function starts from a given index, selects each wooden plank, and explores all possible combinations through recursive calls. Ultimately, when the length of the bridge is at least L, it compares the number of planks and updates the minimum value.
5. Time Complexity
The worst-case time complexity of this algorithm is O(2^N). This is because it attempts all combinations from N wooden planks. As the number of wooden planks increases, the time complexity grows further.
6. Additional Optimization
This problem has a reasonable range for N, so we can enhance performance by adopting specific conditions to prune the search space in the bridge-building problem. For instance, if the load of the planks selected so far already exceeds L, further recursive calls are unnecessary, thus optimizing performance.
Conclusion
The bridge construction problem is an interesting algorithmic challenge where we must find a solution that satisfies given constraints through various combinations. Therefore, there is a high probability of encountering similar problems in actual job interviews. Optimizing time and constructing efficient algorithms are crucial during the problem-solving process. You should practice more problems based on the algorithms understood from this post.
References
- Algorithm Problem Solving Strategy
- Complete Analysis of Programming Interviews