What are some advanced dynamic programming problems I can practice?

What are some advanced dynamic programming problems I can practice?

Here are a few advanced dynamic programming problems you can try: 1. Longest Increasing Subsequence 2. Edit Distance 3. 0/1 Knapsack Problem 4. Matrix Chain Multiplication 5. Word Break Problem.

Can you explain the Longest Increasing Subsequence problem?

Sure! The Longest Increasing Subsequence (LIS) problem involves finding the longest subsequence in a sequence of numbers where the subsequence's elements are in increasing order. For example, in the sequence [10, 22, 9, 33, 21, 50, 41, 60, 80], the LIS is [10, 22, 33, 50, 60, 80] with a length of 6.

What is the time complexity of solving the LIS problem?

The time complexity of the LIS problem can be O(n^2) using dynamic programming. However, it can be optimized to O(n log n) using binary search with a helper array.

Can you provide a code example for the optimized LIS solution?

Certainly! Here's a Python code snippet for the optimized LIS solution:

from bisect import bisect_left

def length_of_LIS(nums):
    sub = []
    for x in nums:
        i = bisect_left(sub, x)
        if i == len(sub):
            sub.append(x)
        else:
            sub[i] = x
    return len(sub)

# Example usage:
print(length_of_LIS([10, 22, 9, 33, 21, 50, 41, 60, 80]))  # Output: 6