DP Framework - Learning Module

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Choosing Top-Down vs Bottom-Up

Two Paths to the Same Destination

Dynamic Programming can be implemented in two fundamentally different ways: top-down with memoization and bottom-up with tabulation. Both approaches solve the same problems and have the same asymptotic complexity—yet they differ significantly in implementation style, practical performance, and applicability to different problem types.

The choice between top-down and bottom-up is not purely aesthetic. Understanding when each approach excels allows you to write cleaner, faster, and more maintainable DP solutions. This page provides a comprehensive framework for making this choice.

What You Will Learn

By completing this page, you will be able to:

• Explain the fundamental differences between top-down and bottom-up DP • Identify which approach is more suitable for a given problem • Convert between top-down and bottom-up implementations • Understand the practical performance implications of each approach • Make informed decisions that balance clarity, efficiency, and maintainability

The Two Approaches Defined

Let's establish precise definitions before comparing the approaches.

Top-Down (Memoization)

•Structure: Recursive function with caching
•Direction: Start from the original problem, recurse to subproblems
•Order: Problems solved on-demand, in dependency order
•Storage: Cache (hash map or array) stores computed results
•Philosophy: 'Solve lazily—only compute what's needed'

Bottom-Up (Tabulation)

•Structure: Iterative loops filling a table
•Direction: Start from base cases, build up to original problem
•Order: All subproblems solved in predetermined order
•Storage: Table (usually array) filled systematically
•Philosophy: 'Solve eagerly—compute everything in order'

Same Problem, Both Approaches
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// PROBLEM: Fibonacci(n)
// STATE: fib(n) = n-th Fibonacci number
// TRANSITION: fib(n) = fib(n-1) + fib(n-2)
// BASE: fib(0) = 0, fib(1) = 1
 
// TOP-DOWN (MEMOIZATION)
function fibTopDown(n: number, memo: Map<number, number> = new Map()): number {
    // Base cases
    if (n <= 1) return n;
    
    // Check cache
    if (memo.has(n)) return memo.get(n)!;
    
    // Compute recursively
    const result = fibTopDown(n - 1, memo) + fibTopDown(n - 2, memo);
    
    // Store in cache
    memo.set(n, result);
    return result;
}
 
// BOTTOM-UP (TABULATION)
function fibBottomUp(n: number): number {
    // Handle base cases
    if (n <= 1) return n;
    
    // Create table
    const dp: number[] = new Array(n + 1);
    
    // Initialize base cases
    dp[0] = 0;
    dp[1] = 1;
    
    // Fill table iteratively
    for (let i = 2; i <= n; i++) {
        dp[i] = dp[i - 1] + dp[i - 2];
    }
    
    return dp[n];
}

Both Are DP

Both approaches implement Dynamic Programming. The 'dynamic' refers to storing solutions to subproblems to avoid recomputation. Whether you store them in a recursive cache or an iterative table, the fundamental insight—trading space for time through memoization—is the same.

Execution Flow Comparison

Understanding how each approach executes reveals their fundamental differences. Let's trace through Fibonacci(5):

Top-Down Execution Trace
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// TOP-DOWN: fibTopDown(5)
// 
// Call stack progression:
// 
// fibTopDown(5) 
//   → needs fib(4) and fib(3)
//   → calls fibTopDown(4)
//       → needs fib(3) and fib(2)
//       → calls fibTopDown(3)
//           → needs fib(2) and fib(1)
//           → calls fibTopDown(2)
//               → needs fib(1) and fib(0)
//               → fib(1) = 1 (base case)
//               → fib(0) = 0 (base case)
//               → fib(2) = 1, CACHE fib(2)=1
//           → calls fibTopDown(1)
//               → fib(1) = 1 (base case)
//           → fib(3) = 1 + 1 = 2, CACHE fib(3)=2
//       → calls fibTopDown(2)
//           → CACHE HIT! Returns 1
//       → fib(4) = 2 + 1 = 3, CACHE fib(4)=3
//   → calls fibTopDown(3)
//       → CACHE HIT! Returns 2
//   → fib(5) = 3 + 2 = 5
// 
// Notice: Evaluates in TOP-DOWN order: 5 → 4 → 3 → 2 → 1 → 0
// Cache prevents recomputation of fib(3) and fib(2)

Bottom-Up Execution Trace
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// BOTTOM-UP: fibBottomUp(5)
// 
// Iteration progression:
// 
// Initialize: dp[0] = 0, dp[1] = 1
// 
// i = 2: dp[2] = dp[1] + dp[0] = 1 + 0 = 1
//        Table: [0, 1, 1, _, _, _]
// 
// i = 3: dp[3] = dp[2] + dp[1] = 1 + 1 = 2
//        Table: [0, 1, 1, 2, _, _]
// 
// i = 4: dp[4] = dp[3] + dp[2] = 2 + 1 = 3
//        Table: [0, 1, 1, 2, 3, _]
// 
// i = 5: dp[5] = dp[4] + dp[3] = 3 + 2 = 5
//        Table: [0, 1, 1, 2, 3, 5]
// 
// Return dp[5] = 5
// 
// Notice: Evaluates in BOTTOM-UP order: 0 → 1 → 2 → 3 → 4 → 5
// No recursion, no cache lookups—just array access

Key Observations:

Order of evaluation: Top-down follows the dependency graph naturally through recursion. Bottom-up requires you to determine a valid order that respects dependencies.
Subproblems solved: Top-down only solves subproblems that are actually needed. Bottom-up typically solves all subproblems in the table, even if some aren't used.
Overhead: Top-down has function call overhead and cache lookup overhead. Bottom-up has only array access overhead.
Memory access pattern: Top-down has unpredictable access patterns (bad for CPU cache). Bottom-up usually has sequential access (good for CPU cache).

Advantages of Top-Down (Memoization)

Top-down memoization has several significant advantages that make it the preferred choice in many situations:

Top-Down Advantages

•Natural recursive thinking: The code structure mirrors the mathematical recurrence directly. If you can write the recurrence, you can write top-down DP.
•Lazy evaluation: Only subproblems that are actually needed get computed. If the recursion tree is sparse (many branches never explored), this saves significant work.
•No need to determine order: The recursion automatically handles dependencies. You don't need to figure out which subproblems to solve first.
•Easier to debug: The recursive structure allows stepping through with a debugger, following the natural problem decomposition.
•Handles complex state spaces: When the state space is irregular or hard to enumerate, recursive calls naturally explore only valid states.
•Easy to convert from brute force: Often, converting brute force recursion to DP is as simple as adding memoization.

When Top-Down Excels: Sparse State Space
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// PROBLEM: Count paths in a grid with blocked cells
// Some cells are obstacles; we can only move right or down.
 
// If obstacles are irregular, the valid path space is sparse.
// Top-down naturally skips unreachable states.
 
function countPaths(
    grid: number[][], 
    r: number, 
    c: number,
    memo: Map<string, number> = new Map()
): number {
    const rows = grid.length, cols = grid[0].length;
    
    // Out of bounds or obstacle
    if (r >= rows || c >= cols || grid[r][c] === 1) return 0;
    
    // Reached destination
    if (r === rows - 1 && c === cols - 1) return 1;
    
    const key = `${r},${c}`;
    if (memo.has(key)) return memo.get(key)!;
    
    // Only explore reachable cells
    const result = countPaths(grid, r + 1, c, memo) + 
                   countPaths(grid, r, c + 1, memo);
    
    memo.set(key, result);
    return result;
}
 
// If 90% of cells are obstacles, top-down only visits 10% of states
// Bottom-up would iterate over all grid cells, checking each for validity

The 'Just Add Memoization' Pattern

Many developers find top-down easier because of this workflow:

Write the naive recursive solution
Identify repeated subproblems
Add memoization (cache)
Done!

This incremental approach is less error-prone than designing bottom-up from scratch.

Advantages of Bottom-Up (Tabulation)

Bottom-up tabulation has its own compelling advantages, particularly for performance-critical applications:

Bottom-Up Advantages

•No recursion overhead: No function call stack, no stack frame allocation, no return address tracking. Pure iteration is faster.
•No stack overflow risk: Deep recursion can exceed stack limits. Bottom-up uses only heap memory for the table.
•Better cache locality: Sequential array access (dp[i-1], dp[i]) is cache-friendly. Random memo lookups are not.
•Space optimization possible: When dependencies are limited, you can reduce the table to just the needed rows/values (rolling array technique).
•Predictable performance: No variance from cache hits/misses. Execution time is consistent and analyzable.
•Easier to optimize further: Loop-based code is easier to parallelize, vectorize, or hand-optimize.

Space Optimization with Bottom-Up
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// PROBLEM: Fibonacci with O(1) space
// Possible with bottom-up because we only need last two values
 
// NAIVE BOTTOM-UP: O(n) space
function fibBU(n: number): number {
    if (n <= 1) return n;
    const dp: number[] = 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];
}
 
// SPACE-OPTIMIZED BOTTOM-UP: O(1) space
function fibOptimized(n: number): number {
    if (n <= 1) return n;
    
    let prev2 = 0; // fib(i-2)
    let prev1 = 1; // fib(i-1)
    
    for (let i = 2; i <= n; i++) {
        const current = prev1 + prev2;
        prev2 = prev1;
        prev1 = current;
    }
    
    return prev1;
}
 
// This optimization is MUCH harder with top-down memoization!
// Top-down naturally keeps the entire cache, not just recent values.

Stack Overflow Is Real

In languages like Python (default recursion limit ~1000) or JavaScript (varies by engine), deep recursion causes stack overflow. For problems with n = 10⁵ or more, top-down memoization may crash.

Bottom-up avoids this entirely. For large inputs, bottom-up is often the only viable option.

Side-by-Side Comparison

Let's compare the two approaches across multiple dimensions:

Top-Down vs Bottom-Up Comparison
Dimension	Top-Down (Memoization)	Bottom-Up (Tabulation)
Code structure	Recursive, matches recurrence	Iterative, requires order analysis
Time complexity	Same asymptotically	Same asymptotically
Constant factors	Higher (recursion + cache overhead)	Lower (pure iteration)
Space complexity	O(state space) + O(recursion depth)	O(state space), can often reduce
Subproblems computed	Only needed ones (lazy)	All of them (eager)
Stack overflow risk	Yes, for deep recursion	No
Debugging	Easier (follows problem structure)	Harder (need to verify order)
Space optimization	Difficult	Often straightforward
Cache performance	Poor (random access)	Good (sequential access)
Implementation ease	Easier (add memo to recursion)	Harder (design order, handle indices)

Summary Heuristics:

• Choose top-down for interviews, prototyping, complex state spaces, or when only some subproblems are needed.

• Choose bottom-up for production code, large inputs, when space optimization is needed, or when maximum performance matters.

The Decision Framework

Here's a systematic framework for choosing between top-down and bottom-up:

Decision Algorithm
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// STEP 1: Check for mandatory constraints
if (inputSize >= 10^4 && maxRecursionDepth >= 10^4) {
    // Risk of stack overflow
    return "BOTTOM-UP (mandatory)";
}
 
if (stateSpaceIsVeryIrregular || hardToEnumerate) {
    // Top-down naturally handles irregular spaces
    return "TOP-DOWN (strongly preferred)";
}
 
// STEP 2: Check for strong preferences
if (needsSpaceOptimization && dependenciesAreLimited) {
    return "BOTTOM-UP (for rolling array optimization)";
}
 
if (onlySmallFractionOfSubproblemsNeeded) {
    return "TOP-DOWN (exploits sparsity)";
}
 
if (maximumPerformanceRequired) {
    return "BOTTOM-UP (lower constant factors)";
}
 
// STEP 3: Default to comfort and simplicity
if (writingForInterview || prototyping) {
    return "TOP-DOWN (easier to write correctly)";
}
 
if (writingProductionCode) {
    return "BOTTOM-UP (better performance, no stack issues)";
}
 
// STEP 4: Personal preference
return "Either works—choose based on clarity";

Quick Decision Questions

•Is n potentially very large (≥10⁴)? → Prefer bottom-up to avoid stack overflow
•Is the state space irregular or sparse? → Prefer top-down for lazy evaluation
•Do you need space optimization? → Prefer bottom-up for rolling arrays
•Are you solving under time pressure? → Prefer top-down for faster writing
•Is this performance-critical production code? → Prefer bottom-up
•Is the problem structure confusing? → Start top-down, convert if needed

The Interview Strategy

In interviews, start with top-down memoization—it's faster to write and less prone to off-by-one errors. If the interviewer asks about optimization, mention bottom-up conversion and space optimization as follow-up improvements.

Converting Between Approaches

Once you understand both approaches, converting between them becomes mechanical. Here's the systematic process:

Top-Down → Bottom-Up Conversion

•Identify the state dimensions. What are the parameters of your memoized function?
•Create a table. Array dimensions match state dimensions. Size = domain of each parameter.
•Initialize base cases. Fill in the table cells corresponding to base cases.
•Determine iteration order. Ensure dependencies are computed before they're needed. Usually iterate 'towards' the final answer.
•Convert recursion to iteration. Replace recursive calls with table lookups. Replace base case returns with table initializations.
•Return the answer. Look up the cell corresponding to the original problem.

Conversion Example: Longest Common Subsequence
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// TOP-DOWN VERSION
function lcsTopDown(
    s1: string, 
    s2: string, 
    i: number = 0, 
    j: number = 0,
    memo: Map<string, number> = new Map()
): number {
    // Base case
    if (i === s1.length || j === s2.length) return 0;
    
    const key = `${i},${j}`;
    if (memo.has(key)) return memo.get(key)!;
    
    let result: number;
    if (s1[i] === s2[j]) {
        result = 1 + lcsTopDown(s1, s2, i + 1, j + 1, memo);
    } else {
        result = Math.max(
            lcsTopDown(s1, s2, i + 1, j, memo),
            lcsTopDown(s1, s2, i, j + 1, memo)
        );
    }
    
    memo.set(key, result);
    return result;
}
 
// CONVERSION PROCESS:
// 1. State dimensions: i (0 to s1.length), j (0 to s2.length)
// 2. Table: dp[i][j] for all i, j
// 3. Base cases: dp[s1.length][*] = 0, dp[*][s2.length] = 0
// 4. Order: i goes down (from end to start), j goes down
// 5. Convert recursive calls to table lookups
 
// BOTTOM-UP VERSION
function lcsBottomUp(s1: string, s2: string): number {
    const m = s1.length, n = s2.length;
    
    // Step 2: Create table (with extra row/col for base cases)
    const dp: number[][] = Array.from({ length: m + 1 }, 
        () => new Array(n + 1).fill(0));
    
    // Step 3: Base cases (already 0 from fill)
    // dp[m][*] = 0, dp[*][n] = 0
    
    // Step 4-5: Iterate in reverse order
    for (let i = m - 1; i >= 0; i--) {
        for (let j = n - 1; j >= 0; j--) {
            if (s1[i] === s2[j]) {
                dp[i][j] = 1 + dp[i + 1][j + 1];
            } else {
                dp[i][j] = Math.max(dp[i + 1][j], dp[i][j + 1]);
            }
        }
    }
    
    // Step 6: Return answer
    return dp[0][0];
}

The Iteration Order Is Critical

The trickiest part of bottom-up is determining the correct iteration order. The rule: when computing dp[i][j], all cells it depends on must already be filled.

For LCS, dp[i][j] depends on dp[i+1][j], dp[i][j+1], dp[i+1][j+1]—all 'later' positions. So we iterate backwards from the end.

Space Optimization with Bottom-Up

One of bottom-up's major advantages is the ability to optimize space when dependencies are limited. This is especially powerful for 2D DP problems.

Space Optimization Patterns

•Two Variables: When dp[i] depends only on dp[i-1] and dp[i-2] (e.g., Fibonacci) → O(1) space
•One Row: When dp[i][j] depends only on current and previous row → O(n) instead of O(m×n)
•Single Row with Care: When dp[i][j] depends on dp[i-1][j-1], dp[i-1][j], need to save old values → O(n) space
•Rolling Array: Keep alternating between two rows → O(2n) = O(n) space

Space Optimization: LCS from O(m×n) to O(n)
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// ORIGINAL: O(m × n) space
function lcsOriginal(s1: string, s2: string): number {
    const m = s1.length, n = s2.length;
    const dp: number[][] = Array.from({ length: m + 1 }, 
        () => new Array(n + 1).fill(0));
    
    for (let i = m - 1; i >= 0; i--) {
        for (let j = n - 1; j >= 0; j--) {
            if (s1[i] === s2[j]) {
                dp[i][j] = 1 + dp[i + 1][j + 1];
            } else {
                dp[i][j] = Math.max(dp[i + 1][j], dp[i][j + 1]);
            }
        }
    }
    return dp[0][0];
}
 
// OPTIMIZED: O(n) space using two rows
function lcsOptimized(s1: string, s2: string): number {
    const m = s1.length, n = s2.length;
    
    // Only keep current row and next row
    let next: number[] = new Array(n + 1).fill(0);
    let curr: number[] = new Array(n + 1).fill(0);
    
    for (let i = m - 1; i >= 0; i--) {
        for (let j = n - 1; j >= 0; j--) {
            if (s1[i] === s2[j]) {
                curr[j] = 1 + next[j + 1];
            } else {
                curr[j] = Math.max(next[j], curr[j + 1]);
            }
        }
        // Swap rows for next iteration
        [curr, next] = [next, curr];
    }
    
    // After loop, answer is in 'next' (due to swap)
    return next[0];
}
 
// FURTHER OPTIMIZED: O(n) space using single row + one variable
function lcsUltraOptimized(s1: string, s2: string): number {
    const m = s1.length, n = s2.length;
    const dp: number[] = new Array(n + 1).fill(0);
    
    for (let i = m - 1; i >= 0; i--) {
        let prev = 0; // This holds dp[i+1][j+1] from previous iteration
        for (let j = n - 1; j >= 0; j--) {
            const temp = dp[j]; // Save dp[i+1][j] before overwriting
            if (s1[i] === s2[j]) {
                dp[j] = 1 + prev;
            } else {
                dp[j] = Math.max(dp[j], dp[j + 1]);
            }
            prev = temp;
        }
    }
    
    return dp[0];
}

Tradeoff: Space vs Solution Reconstruction

Space-optimized solutions often can't reconstruct the actual solution (like the LCS string itself), only compute the optimal value. If you need the solution, not just its value, you typically need the full table.

Alternatively, use a hybrid: compute value with O(n) space, then use divide-and-conquer reconstruction (Hirschberg's algorithm for LCS).

Hybrid and Advanced Approaches

Sometimes the best solution combines elements of both approaches or uses advanced techniques:

Advanced Techniques

•Iterative with explicit stack: Simulate recursion iteratively to avoid stack overflow while maintaining top-down's flexibility
•Lazy tabulation: Fill table on-demand using a queue/stack of pending computations
•Divide-and-conquer with DP: Use D&C for reconstruction while DP handles value computation (Hirschberg's algorithm)
•Parallel bottom-up: Independence in some dimensions allows parallel filling (e.g., filling each diagonal in parallel)
•Incremental DP: For online problems, maintain and update DP table as new elements arrive

Iterative Memoization (Avoiding Stack Overflow)
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// TECHNIQUE: Simulate top-down recursion with explicit stack
// Useful when you need top-down's flexibility but can't risk stack overflow
 
interface StackFrame {
    i: number;
    j: number;
    phase: 'compute' | 'combine';
    childResults?: { skipResult?: number; takeResult?: number };
}
 
function knapsackIterativeMemo(
    weights: number[], 
    values: number[], 
    capacity: number
): number {
    const n = weights.length;
    const memo: Map<string, number> = new Map();
    const stack: StackFrame[] = [{ i: 0, j: capacity, phase: 'compute' }];
    
    while (stack.length > 0) {
        const frame = stack[stack.length - 1];
        const key = `${frame.i},${frame.j}`;
        
        // Base case
        if (frame.i >= n || frame.j <= 0) {
            memo.set(key, 0);
            stack.pop();
            continue;
        }
        
        // Already computed
        if (memo.has(key)) {
            stack.pop();
            continue;
        }
        
        if (frame.phase === 'compute') {
            // Need to compute children first
            const skipKey = `${frame.i + 1},${frame.j}`;
            const takeKey = `${frame.i + 1},${frame.j - weights[frame.i]}`;
            
            if (!memo.has(skipKey)) {
                stack.push({ i: frame.i + 1, j: frame.j, phase: 'compute' });
            }
            if (weights[frame.i] <= frame.j && !memo.has(takeKey)) {
                stack.push({ 
                    i: frame.i + 1, 
                    j: frame.j - weights[frame.i], 
                    phase: 'compute' 
                });
            }
            
            frame.phase = 'combine';
        } else {
            // Children are computed, combine results
            const skipResult = memo.get(`${frame.i + 1},${frame.j}`) || 0;
            let takeResult = 0;
            if (weights[frame.i] <= frame.j) {
                takeResult = values[frame.i] + 
                    (memo.get(`${frame.i + 1},${frame.j - weights[frame.i]}`) || 0);
            }
            
            memo.set(key, Math.max(skipResult, takeResult));
            stack.pop();
        }
    }
    
    return memo.get(`0,${capacity}`) || 0;
}

Summary: Mastering the Top-Down vs Bottom-Up Choice

Choosing between top-down and bottom-up is a skill that develops with experience. Let's consolidate the key insights:

Key Takeaways

•Both are valid DP. They solve the same problems with the same asymptotic complexity—the difference is implementation style.
•Top-down is recursive with caching. It's natural to write, handles sparse state spaces well, but has overhead and stack risks.
•Bottom-up is iterative with tables. It's faster in practice, avoids stack issues, and enables space optimization.
•Choose top-down for: interviews, prototyping, complex/sparse state spaces, when only some subproblems are needed.
•Choose bottom-up for: production code, large inputs, performance-critical systems, when space optimization matters.
•Know how to convert. The ability to transform between approaches is a valuable skill for optimization.
•Space optimization is powerful. Bottom-up enables reducing O(n²) to O(n) or O(1) when dependencies are limited.

What's Next:

Even with perfect state design and the right approach, DP solutions can have subtle bugs. The next page covers Debugging DP Solutions—systematic techniques for finding and fixing errors in your Dynamic Programming code.

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You now have a comprehensive framework for choosing between top-down and bottom-up implementations. This decision is no longer arbitrary—you can make informed choices based on problem characteristics and requirements. Next, we'll master the art of debugging DP solutions.