Dynamic Programming for Interviews: From Zero to Hero

Dynamic programming is the most feared topic in coding interviews — and also one of the most frequently tested. But here's the good news: DP is a skill that can be learned systematically. This guide will take you from zero understanding to confidently solving dynamic programming interview problems.

What Is Dynamic Programming?

Dynamic programming is an optimization technique that solves complex problems by breaking them into smaller overlapping subproblems. It works when a problem has two properties:

1. Optimal Substructure — The optimal solution contains optimal solutions to subproblems.
2. Overlapping Subproblems — The same subproblems are solved multiple times.

Without DP, a naive recursive solution recomputes the same subproblems exponentially. DP stores results (memoization) or builds solutions iteratively (tabulation) to avoid redundant work.

Memoization vs Tabulation

There are two approaches to implementing DP:

Top-Down (Memoization)

Start with the original problem, recursively break it down, and cache results as you go.

// Fibonacci with memoization
function fib(n: number, memo: Map<number, number> = new Map()): number {
  if (n <= 1) return n;
  if (memo.has(n)) return memo.get(n)!;
  const result = fib(n - 1, memo) + fib(n - 2, memo);
  memo.set(n, result);
  return result;
}

Bottom-Up (Tabulation)

Build a table from the base cases up to the final answer.

// Fibonacci with tabulation
function fib(n: number): number {
  if (n <= 1) return n;
  const dp = new Array(n + 1);
  dp[0] = 0;
  dp[1] = 1;
  for (let i = 2; i <= n; i++) {
    dp[i] = dp[i - 1] + dp[i - 2];
  }
  return dp[n];
}

Which to use? Memoization is easier to write (just add caching to recursion). Tabulation is more efficient (no recursion stack) and preferred in interviews for complex problems.

The 5-Step DP Framework

Use this framework for every dynamic programming interview problem:

1. Define the state — What does dp[i] (or dp[i][j]) represent?
2. Find the transition — How does dp[i] relate to previous states?
3. Identify base cases — What are the trivially known answers?
4. Determine the order — In what order should you fill the table?
5. Extract the answer — Where in the table is the final answer?

Common DP Patterns

Pattern 1: Linear DP (1D)

Problems: Climbing Stairs, House Robber, Maximum Subarray, Coin Change

The state depends on previous elements in a sequence.

// House Robber: dp[i] = max money robbing houses 0..i
// Can't rob adjacent houses
function rob(nums: number[]): number {
  if (nums.length === 0) return 0;
  if (nums.length === 1) return nums[0];

  let prev2 = nums[0];
  let prev1 = Math.max(nums[0], nums[1]);

  for (let i = 2; i < nums.length; i++) {
    const curr = Math.max(prev1, prev2 + nums[i]);
    prev2 = prev1;
    prev1 = curr;
  }
  return prev1;
}

Pattern 2: Grid DP (2D)

Problems: Unique Paths, Minimum Path Sum, Dungeon Game

Navigate a grid with constraints, building solutions cell by cell.

// Unique Paths: dp[i][j] = number of paths to reach (i, j)
function uniquePaths(m: number, n: number): number {
  const dp = Array.from({ length: m }, () => new Array(n).fill(1));
  for (let i = 1; i < m; i++) {
    for (let j = 1; j < n; j++) {
      dp[i][j] = dp[i - 1][j] + dp[i][j - 1];
    }
  }
  return dp[m - 1][n - 1];
}

Pattern 3: String DP

Problems: Longest Common Subsequence (LCS), Edit Distance, Longest Palindromic Subsequence

Two-dimensional DP comparing characters from two strings (or one string with itself).

// Longest Common Subsequence
function lcs(s1: string, s2: string): number {
  const m = s1.length, n = s2.length;
  const dp = Array.from({ length: m + 1 }, () => new Array(n + 1).fill(0));

  for (let i = 1; i <= m; i++) {
    for (let j = 1; j <= n; j++) {
      if (s1[i - 1] === s2[j - 1]) {
        dp[i][j] = dp[i - 1][j - 1] + 1;
      } else {
        dp[i][j] = Math.max(dp[i - 1][j], dp[i][j - 1]);
      }
    }
  }
  return dp[m][n];
}

Pattern 4: Decision-Making DP

Problems: 0/1 Knapsack, Partition Equal Subset Sum, Target Sum

At each step, make a binary decision (include or exclude an item).

// 0/1 Knapsack
function knapsack(weights: number[], values: number[], capacity: number): number {
  const n = weights.length;
  const dp = Array.from({ length: n + 1 }, () => new Array(capacity + 1).fill(0));

  for (let i = 1; i <= n; i++) {
    for (let w = 0; w <= capacity; w++) {
      dp[i][w] = dp[i - 1][w]; // skip item
      if (weights[i - 1] <= w) {
        dp[i][w] = Math.max(dp[i][w], dp[i - 1][w - weights[i - 1]] + values[i - 1]);
      }
    }
  }
  return dp[n][capacity];
}

Pattern 5: Interval DP

Problems: Burst Balloons, Matrix Chain Multiplication, Palindrome Partitioning

Solve problems over intervals [i, j] by trying all split points.

How to Identify DP Problems in Interviews

Look for these signals:

• The problem asks for the maximum, minimum, or count of something
• You need to make a sequence of decisions
• A greedy approach doesn't work (or feels too simple)
• The problem has overlapping subproblems — the same calculation appears multiple times in a recursive solution
• Keywords: "optimal", "minimum cost", "number of ways", "is it possible"

Common Mistakes

• Jumping to code too fast — Always define your state and transition on paper first.
• Wrong state definition — If your transition doesn't work cleanly, revisit the state definition.
• Forgetting base cases — Edge cases like empty strings, zero capacity, or single elements.
• Not optimizing space — Many 2D DP problems can be reduced to 1D (rolling array). Mention this optimization to impress interviewers.

Practice Roadmap

Follow this progression for your DP preparation:

1. Week 1: 1D DP — Climbing Stairs, House Robber, Coin Change, Maximum Subarray
2. Week 2: 2D DP — Unique Paths, Minimum Path Sum, LCS, Edit Distance
3. Week 3: Decision DP — 0/1 Knapsack, Partition Equal Subset Sum, Target Sum
4. Week 4: Advanced — Burst Balloons, Longest Increasing Subsequence, Word Break

For each problem: solve it yourself first, then study optimal solutions, then re-solve without looking. Use tools like InterviewAlly to get hints when you're stuck without revealing the full solution.

Conclusion

Dynamic programming becomes manageable once you learn to recognize patterns and apply a systematic framework. Focus on the 5 core patterns, practice 5-8 problems per pattern, and use the 5-step framework for every new problem. Combined with mastery of other essential LeetCode patterns and a solid FAANG preparation plan, you'll be ready to tackle any DP question in your interview.

Stuck on a DP problem? Try InterviewAlly free and get AI-powered hints and solutions in real-time during your practice sessions.