Function Generating Problem

A function generating problem involves constructing a function that satisfies a given property or optimizes an objective. The goal is to synthesize the desired function.

Java - Random number generator:

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double[] random(int n) {
  double[] result = new double[n];
  Random rand = new Random();

  for (int i = 0; i < n; i++) {
    result[i] = rand.nextDouble(); 
  }

  return result;
}

C++ - Sorting network generator:

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vector<pair<int, int>> sortNet(int n) {
  vector<pair<int,int>> net;
  
  // Logic to generate sorting network
  // with n channels  

  return net;
}

Python - Regular expression generator:

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import re

def regex_gen(text):
  pattern = ""

  # Logic to analyze text and
  # generate desired regex

  regex = re.compile(pattern)
  return regex

Key aspects are:

  • Specifying desired properties of output function
  • Designing procedure to construct function satisfying properties
  • Validating generated function

Function generation has applications in machine learning, optimization, satisfiability problems, etc.

In computer science and mathematics, the function generating problem revolves around finding a function or an equation that can produce a particular sequence of numbers or satisfy certain conditions. It could involve using generating functions to solve problems in combinatorics, sequence enumeration, or algorithmic complexity. The goal is often to simplify complex problems into algebraic expressions that are easier to manage and solve.

Example

Let’s consider a problem where we want to find the number of ways to make change for a given amount ( N ) using coins of denominations ( c_1, c_2, \ldots, c_m ). We can represent this as a generating function problem and find the solution using dynamic programming.

Java

Here’s a Java example:

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public class CoinChange {
    public static int changeWays(int N, int[] coins) {
        int[] dp = new int[N + 1];
        dp[0] = 1;

        for (int coin : coins) {
            for (int i = coin; i <= N; i++) {
                dp[i] += dp[i - coin];
            }
        }

        return dp[N];
    }

    public static void main(String[] args) {
        int[] coins = {1, 2, 5};
        System.out.println(changeWays(5, coins));
    }
}

C++

In C++:

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#include <iostream>
#include <vector>
using namespace std;

int changeWays(int N, vector<int> coins) {
    vector<int> dp(N + 1, 0);
    dp[0] = 1;

    for (int coin : coins) {
        for (int i = coin; i <= N; i++) {
            dp[i] += dp[i - coin];
        }
    }

    return dp[N];
}

int main() {
    vector<int> coins = {1, 2, 5};
    cout << changeWays(5, coins) << endl;
    return 0;
}

Python

Python version:

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def changeWays(N, coins):
    dp = [0] * (N + 1)
    dp[0] = 1

    for coin in coins:
        for i in range(coin, N + 1):
            dp[i] += dp[i - coin]

    return dp[N]

coins = [1, 2, 5]
print(changeWays(5, coins))

Each code snippet solves the problem by initializing a dynamic programming array dp where dp[i] will contain the number of ways to make change for amount ( i ). It then iterates through the coin denominations and updates the dp array. Finally, it returns the value stored in dp[N], which is the number of ways to make change for ( N ).