Hello, Codeforces community! I wanted to write an educational and necessary blog post for the community. Because as you guys can see, my contribution is bad (-1) because of my stupid comment on another blog.
We've all been there.
You open a problem, try greedy, get WA. Try brute force, get TLE. Stare at the constraints for 10 minutes, then finally whisper:
"Wait... is this DP?"
Dynamic Programming is one of the most powerful tools in competitive programming. But here's the thing: DP isn't about memorizing 100 formulas. It's about learning what to remember.
- Think, Don't Memorize Imagine you're climbing stairs. You can take either 1 or 2 steps at a time.
To reach step (i), you must come from step (i-1) or (i-2).
So:
[ dp[i]=dp[i-1]+dp[i-2] ]
That's your first DP recurrence.
The real trick wasn't the formula. It was asking:
"What smaller problems can help me solve this one?"
- The Three Questions That Matter
Whenever you suspect DP, ask yourself:
State: What does dp[i] actually mean?
Transition: How can I calculate it using previous states?
Base case: Where does everything begin?
If you can't explain these three things, don't rush to code.
A clear state is often half the solution.
- From Beginner to Advanced
DP starts with simple problems, but it doesn't stop there.
Beginner: Fibonacci, climbing stairs, grid paths.
Intermediate: Knapsack, LIS, LCS.
Advanced: Tree DP, bitmask DP, digit DP.
Boss level: Divide and conquer optimization, convex hull trick, and other optimization techniques.
The interesting part? They all share the same fundamental idea: solve smaller states and reuse their answers.
Sometimes the hardest part isn't writing the recurrence. It's finding the right state.
- Why Your DP Gets WA
Three classic mistakes:
Your state doesn't mean what you think it means.
Your transition misses a possible choice.
Your iteration order accidentally reuses information that shouldn't be reused.
And please, don't optimize a recurrence you haven't proved correct yet. A fast wrong answer is still a wrong answer.
- A Challenge for You
You have an array:
[ [2,7,9,3,1] ]
Find the maximum sum of a subsequence such that no two selected elements are adjacent.
The answer is 12, by choosing (2,9,1).
Now the real questions:
What should dp[i] represent?
What's the transition?
Can you solve it in (O(n)) time and (O(1)) extra space?
Drop your approach in the comments. Let's see how many different explanations lead to the same solution.
Final thought
The difference between someone who struggles with DP and someone who solves it confidently isn't necessarily talent. It's practice at recognizing states and understanding why transitions work.
Stop collecting DP templates. Start collecting DP ideas.
May your transitions be correct and your verdicts be Accepted.
SORRY IF THE BLOG HAD ANY PROBLEMS! (I'm a newbie; the only reason I've opened this account is that I recently did!)
