Subproblem Cache Array (Memoization Table)
Allocates a 2D contiguous buffer of (n + 1) × (W + 1) 64-bit numeric cells pre-initialized to sentinel -1 to record computed subproblem values and eliminate redundant recursion branches.
Computer Science / Dynamic Programming & 0/1 Knapsack
FREE TO EXPLORELABORATORY 11 · COMPUTER SCIENCE
Empirical Analysis of Dynamic Programming: Memoization vs. Tabulation, Optimal Substructure, Overlapping Subproblems, and 0/1 Knapsack State Space Complexity
To investigate the fundamental mechanisms of Dynamic Programming (Optimal Substructure and Overlapping Subproblems), quantitatively contrast Top-Down Memoization against Bottom-Up Tabulation, analyze memory hierarchy cache locality across 2D matrix vs. 1D rolling array representations, and verify the pseudo-polynomial complexity scaling law T(n) = n · W = 100.0000 · n operations across knapsack capacity W = 100. Explore interactive search algorithms, compare operation counts and analyse complexity scaling.
COMPUTE · BENCHMARK · ANALYSE
Your virtual algorithm laboratory.
TABULATION STATE EVALUATIONS T(N)—
EXECUTION STATUS
Ready to execute item count benchmark.
Adjust item count n, then execute benchmark.
Operation counts are generated deterministically by the exact algorithmic execution model. Modern hardware benchmarks additionally reflect cache line prefetching and branch misprediction penalties.
Algorithmic assumptions and limits
The simulated execution environment models idealized state transitions where every table cell evaluation executes with deterministic cost of 1 operation (T(n) = n · W). In physical semiconductor microprocessors, wall-clock execution latency is modulated by CPU L1/L2 data cache hit ratios during row-major array streaming, branch predictor stalls during capacity feasibility checks (w ≥ w_i), hardware memory bus saturation during large matrix allocations, and operating system page table faults. In Bottom-Up Tabulation of the 0/1 Knapsack problem with fixed capacity W = 100 weight units, every subproblem cell DP[i][w] across the (n + 1) × (W + 1) state matrix is computed exactly once in O(1) time using the Bellman recurrence relation. Across n items, the total number of elementary state transition evaluations scales as T(n) = W · n = 100.0000 · n operations with exact theoretical intercept T_0 = 0.0000 states.
Computational tools & environment.
Knapsack Item Generator & Weights Matrix
Synthesizes discrete test datasets of n items with integer weights w_i ∈ [1, 25] and valuations v_i ∈ [10, 100], bounded by maximum knapsack capacity W = 100.
Recursive Activation Call Stack Tracer
Monitors recursion depth, stack frame allocations (AMD64 48 bytes/frame), and function call frequencies to quantify overhead differences between Top-Down and Bottom-Up models.
Cache-Locality CPU Performance Profiler
Measures hardware L1/L2 data cache hit ratios, memory bus transfer volumes, and instruction cycles across row-major vs. column-major 2D table traversal loops.
Space-Optimized Rolling Buffer (1D DP Array)
Simulates single-array 1D dynamic programming with reverse capacity scanning (W down to w_i), compressing spatial memory footprint from Θ(n·W) down to Θ(W).
DAG Topological Dependency Visualizer
Visualizes the Directed Acyclic Graph (DAG) state transition structure, tracking subproblem dependencies DP[i-1][w] and DP[i-1][w - w_i] to verify Bellman's principle of optimality.
FOLLOW THE METHOD
A reason behind every step.
To investigate the fundamental mechanisms of Dynamic Programming (Optimal Substructure and Overlapping Subproblems), quantitatively contrast Top-Down Memoization against Bottom-Up Tabulation, analyze memory hierarchy cache locality across 2D matrix vs. 1D rolling array representations, and verify the pseudo-polynomial complexity scaling law T(n) = n · W = 100.0000 · n operations across knapsack capacity W = 100. Suggested independent settings: 10, 20, 30, 50, 75, 100, 150 items.
01
State Space Allocation & Sentinel Table Initialization
Allocate a contiguous 2D tabulation matrix of dimensions (n + 1) × (W + 1) for n items and capacity budget W = 100. Initialize all boundary base-case entries DP[0][w] = 0 and DP[i][0] = 0, and pre-fill memoization buffers with sentinel -1.
02
Top-Down Memoized Recursion Benchmark
Invoke the recursive solver for target state DP(n, W). At each call, inspect the memo table: return the cached value immediately upon a cache hit (O(1)), or evaluate subproblem branches (exclude item i vs. include item i) and write the result into the table.
03
Bottom-Up Iterative Tabulation Execution
Execute the dual nested loops (i from 1 to n, w from 0 to W) in row-major memory order. Evaluate the Bellman recurrence relation for each cell and increment the deterministic state evaluation counter.
04
Optimal Item Subset Backtracking Reconstruction
Trace backward from cell DP[n][W] to determine the exact optimal binary selection vector x ∈ {0, 1}^n. If DP[i][w] ≠ DP[i-1][w], mark item i as included (x_i = 1) and decrement remaining capacity w ← w - w_i; otherwise mark x_i = 0.
05
Space Optimization via 1D Reverse Scanning
Allocate a compact 1D array of length W + 1. For each item i, iterate capacity w backward from W down to w_i, updating DP[w] = max(DP[w], DP[w - w_i] + v_i) to demonstrate Θ(W) memory compression.
06
Asymptotic Linear Regression & Complexity Verification
Record total state evaluations across item scales n ∈ [10, 20, 30, 50, 75, 100, 150]. Regress total evaluations T(n) against item count n to compute slope k_dp, intercept T_0, and coefficient of determination R².
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into asymptotic insight.
Points are your recorded benchmark observations. The regression line validates Dynamic Programming & 0/1 Knapsack complexity scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
T(n) = k_dp · n + T_0 = 100.0000 · n + 0.0000 operations
| # | Item count n (knapsack capacity W = 100 weight units) (items) | Tabulation state evaluations T(n) (operations) | Item count n (items) | Total tabulation state evaluations T(n) = n · W (operations) | Remove |
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