Roadmap for DSA in c++

Roadmap for Dynamic Programming in C++: A Comprehensive Guide

Dynamic Programming (DP) is a powerful technique used to solve optimization problems, which can be broken down into smaller, overlapping subproblems. If you're looking to improve your problem-solving skills and tackle complex algorithms, DP is an essential concept to master. In this guide, we'll explore a roadmap for learning Dynamic Programming in C++, focusing on key concepts, techniques, and resources.

Prerequisites

Before diving into Dynamic Programming, it's essential to have a solid understanding of the following topics:

• Basic C++ syntax and data structures (arrays, vectors, linked lists, etc.)
• Recursion and its applications
• Time and space complexity analysis

Key Concepts

Overlapping Subproblems: DP is useful when the same subproblem is encountered multiple times. By storing and reusing the results, we can avoid redundant computation and improve efficiency.

Memoization and Tabulation: Memoization is a top-down approach, where we store the results of subproblems in a lookup table. Tabulation, on the other hand, is a bottom-up approach, where we build up the solution incrementally, starting with the base cases and moving towards the final solution.

2D and 1D arrays: DP solutions often use 2D arrays to store memoized results in the top-down approach and to compute and store solutions in the bottom-up approach. Understanding how to use these arrays efficiently is crucial.

Techniques

Identifying Overlapping Subproblems: To apply DP, you need to recognize whether a problem has overlapping subproblems. Consider breaking down the problem into smaller subproblems, and check if they appear multiple times.

Choosing the Right DP Approach: Decide whether a top-down or bottom-up approach is more suitable for the problem at hand. Memoization might be easier to implement but can lead to longer execution times due to function calls. Tabulation eliminates function calls but requires a clear understanding of base cases and how to incrementally build up the solution.

Reducing Space Complexity: Some DP problems can be solved with a space complexity lower than O(n^2), where n is the size of the input. Identify opportunities to reduce space complexity, such as using iterative approaches instead of recursion, or reusing memory by using 1D arrays.

Resources

Here are some resources to help you learn Dynamic Programming in C++:

• Dynamic Programming video lecture from the Algorithmic Toolbox course on Coursera
• GeeksforGeeks Dynamic Programming articles
• CP Algorithms - a comprehensive resource for competitive programming, including Dynamic Programming

Practical Examples

To solidify your understanding, here are some classic Dynamic Programming problems to practice in C++:

• Fibonacci Numbers: Implement a DP solution for computing Fibonacci numbers with a space complexity of O(1)

  int fib(int n) {
  if (n <= 1)
  return n;
  int prevPrev = 0, prev = 1, curr;
  for (int i = 2; i <= n; i++) {
  curr = prev + prevPrev;
  prevPrev = prev;
  prev = curr;
  }
  return curr;
  }
  

• Longest Common Subsequence: Given two sequences, find the longest common subsequence using both top-down and bottom-up approaches

  // Top-down
  int lcsMemo(string x, string y, int n, int m) {
  if (n == 0 || m == 0)
  return 0;
  if (x[n - 1] == y[m - 1])
  return 1 + lcsMemo(x, y, n - 1, m - 1);
  return max(lcsMemo(x, y, n - 1, m), lcsMemo(x, y, n, m - 1));
  }
  
  // Bottom-up
  int lcsTab(string x, string y) {
  int n = x.length(), m = y.length();
  int dp[n + 1][m + 1];
  for (int i = 0; i <= n; i++) {
  for (int j = 0; j <= m; j++) {
  if (i == 0 || j == 0)
  dp[i][j] = 0;
  else if (x[i - 1] == y[j - 1])
  dp[i][j] = dp[i - 1][j - 1] + 1;
  else
  dp[i][j] = max(dp[i - 1][j], dp[i][j - 1]);
  }
  }
  return dp[n][m];
  }
  

• 0/1 Knapsack: Given a set of items, each with a weight and a value, determine the items to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible

  // Top-down
  int knapSackMemo(int W, vector wt, vector val, int n) {
  if (n == 0 || W == 0)
  return 0;
  if (wt[n - 1] > W)
  return knapSackMemo(W, wt, val, n - 1);
  return max(knapSackMemo(W, wt, val, n - 1),
  val[n - 1] + knapSackMemo(W - wt[n - 1], wt, val, n - 1));
  }
  
  // Bottom-up
  int knapSackTab(int W, vector wt, vector val) {
  int n = wt.size();
  int K[n + 1][W + 1];
  for (int i = 0; i <= n; i++) {
  for (int w = 0; w <= W; w++) {
  if (i == 0 || w == 0)
  K[i][w] = 0;
  else if (wt[i - 1] <= w)
  K[i][w] = max(val[i - 1] + K[i - 1][w - wt[i - 1]], K[i - 1][w]);
  else
  K[i][w] = K[i - 1][w];
  }
  }
  return K[n][W];
  }
  

Conclusion

Dynamic Programming is a powerful technique that can help you tackle complex algorithms and improve your problem-solving skills. By understanding the key concepts, techniques, and resources in this roadmap, and practicing with practical examples, you'll be well on your way to mastering DP in C++.