Summary of "Top 5 Dynamic Programming Patterns for Coding Interviews - For Beginners"

Main ideas and concepts (5 dynamic programming patterns)

Setup / purpose

• The speaker introduces a Google spreadsheet containing:
  • Five DP categories/patterns
  • Sample problems for each category
  • Links to prior “video solutions” for many of the samples
• The goal is to understand fundamental DP patterns to solve harder DP problems later.
• Mentions overlap between categories: some problems can fit multiple patterns.

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1) Fibonacci Numbers (1D DP, bottom-up)

Definition / recurrence

• Let the Fibonacci function be f(n).
• Base cases:
  • f(0) = 0
  • f(1) = 1
• Recurrence for n ≥ 2:
  • f(n) = f(n−1) + f(n−2)

Why it’s a DP problem

• A naive recursion forms a decision tree where subproblems repeat (e.g., computing f(4) appears multiple times inside the tree for f(6)).
• Dynamic programming avoids recomputation by reusing previously computed results.

Bottom-up approach

• Compute from smaller base cases up to the target.
• Example behavior:
  • Compute f(2) using f(0) and f(1)
  • Compute f(3) using the last two computed values
  • Continue until reaching f(6)

Space optimization

• Although it’s conceptually a 1D array of size n, you only need the last two values:
  • Keep f(n−1) and f(n−2)
• Older values can be discarded since future results depend only on the two most recent ones.

Typical problems mentioned

• Climbing Stairs is cited as falling into the Fibonacci pattern.

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2) 0/1 Knapsack (2D DP)

Problem framing (subset-sum style)

• Given a set of items/coins and a target sum (example target: 5).
• Key question emphasized:
  • Is it possible to reach the target sum? (boolean DP)

Why it’s “0/1”

• Each coin can be used either:
  • 0 times, or
  • 1 time
• Not an infinite number of times.

Brute force inefficiency

• For n coins, each coin has 2 choices (use or not use), giving 2ⁿ combinations.

Subproblem structure

• Split the decision based on whether to use a particular coin:
  • If you use the current coin, the target decreases (e.g., from 5 to 4)
  • If you skip it, the target stays the same but you move to the next coins

DP grid dimensions

• One dimension: target value (from 0 up to the original target)
• Another dimension: coin index / allowed set
  • e.g., “we can use coins from some position to the end” (or equivalently, first k coins)

Base cases

• If target = 0, return true (sum of zero is always achievable: using nothing).
• If the target becomes negative, it is treated as invalid/false (as described in the explanation).

Transition idea (boolean feasibility)

• For a DP cell representing:
  • “Can we achieve target T using the allowed set of coins?”
• Check:
  • Use coin i → move to reduced target with remaining allowed coins
  • Skip coin i → keep target, reduce the allowed coin set accordingly

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3) Unbounded Knapsack (2D DP)

Problem framing (coin change style)

• Determine feasibility (and relates to coin change).
• Difference from 0/1:
  • Each coin can be used infinitely many times.

Relationship to 0/1 knapsack

• The same kind of 2D DP grid applies.
• What changes is the dependency direction / transition:
  • In unbounded knapsack, choosing a coin does not remove it from availability.

Transition direction intuition

• When computing a cell, the dependencies imply:
  • Use coin again → move in a way that allows staying with the same coin conceptually (e.g., staying in the same “row” while reducing the target)
  • Skip coin → move to the next coin option (conceptually move “down”)

Bottom-up filling order

• Dependencies are arranged so the speaker describes computing:
  • from the bottom, and from already-computed neighbors (bottom and right)
• This matches the typical bottom-up tabulation idea used in other 2D DP problems.

Typical problem mentioned

• Coin Change is explicitly referenced.

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4) Longest Common Subsequence (LCS) (2D DP)

Subsequence concept

• A subsequence:
  • preserves relative order
  • does not need to be contiguous
• For each character, you decide whether to include it or not.
• Order matters (subsequence formation is explained conceptually).

LCS definition

• Given two strings (example: ABC and ACE):
  • Find the longest subsequence common to both.

Recursive/decision logic

• Compare characters starting from the beginning:
  • If characters match:
    • take the match and reduce to the remaining suffixes of both strings
  • If characters do not match:
    • branch into two subproblems:
      • skip from the first string
      • or skip from the second string

DP grid dimensions

• One dimension: positions in string 1
• Other dimension: positions in string 2
• Grid size is proportional to the product of string lengths.

Base cases

• If both suffixes are empty (reached ends of both strings), LCS length is 0.
• The speaker emphasizes that correct DP bases correspond to empty-string comparisons.

Transition / movement in grid

• For a matched cell:
  • move diagonally (advance both strings)
• For a non-matched cell:
  • consider moves corresponding to:
    • skip from string 1 (e.g., move down)
    • skip from string 2 (e.g., move right)
• Bottom-up recommendation:
  • compute from the bottom-right toward the top-left so dependencies are ready.

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5) Palindrome DP pattern (expand-around-center + memoization)

Goal

• Use DP to avoid repeated work in palindrome-related problems (e.g., counting palindromes).

Why naive checking is expensive

• Checking whether a substring is a palindrome by comparing ends takes O(n) per substring.
• Repeating this across many substrings causes major inefficiency.

DP “trick”: expand outward using known palindrome status

• Instead of repeatedly rebuilding and re-checking from scratch:
  • Start with palindromes of minimal length
  • Then expand outward by adding one character to the left and one to the right
• Key observation:
  • If the inner substring is already known to be a palindrome and the new outer characters match, then the expanded substring is also a palindrome.
• This makes palindrome verification for expanded substrings O(1) after the inner result is known.

Odd-length palindromes

• Start with substrings of length 1 (always palindromes).
• Then expand outward by adding two characters total each time:
  • producing odd-length palindromes.

Even-length palindromes

• Start with substrings of length 2.
• If the two characters match, expand outward similarly.

Impact

• This “expand outward with DP” technique underlies many palindrome DP problems efficiently.
• Mentions multiple problems in the spreadsheet that require this method.

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Overall lessons / meta-points

• DP patterns rely on:
  • identifying overlapping subproblems
  • choosing an appropriate DP state representation
  • often using bottom-up tabulation (and sometimes space optimization)
• Categories can overlap (a single problem may match multiple patterns).
• The spreadsheet serves as a curated learning path with linked prior solutions.

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Speakers / sources

• Speaker: YouTube video narrator/instructor (no name provided in the subtitles)
• Source referenced: The instructor’s Google spreadsheet (linked in the description per subtitles) and the instructor’s previous YouTube videos (e.g., climbing stairs, coin change, LCS)
• Website/platform referenced: LeetCode

Category

Educational

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