KMP Failure Function (π-Prefix Array) Synthesizer
Deterministic preprocessor computing the longest proper prefix that is also a suffix for all pattern prefixes P[0..q] in O(m) time.
Computer Science / String Matching & KMP Automata
FREE TO EXPLORELABORATORY 13 · COMPUTER SCIENCE
Empirical Analysis of String Matching Algorithms: Knuth-Morris-Pratt (KMP) Prefix Automata, Boyer-Moore Bad Character / Good Suffix Heuristics, Rabin-Karp Rolling Hash, and Asymptotic Text Search Complexity
To investigate the theoretical foundations and microarchitectural mechanics of exact string matching algorithms, evaluate the Knuth-Morris-Pratt (KMP) prefix function π-array, contrast KMP with Boyer-Moore sublinear shift heuristics and Rabin-Karp rolling hashes, and empirically verify the linear comparison scaling law C(n) = k_kmp · n + C_0 = 1.0000 · n + 20.0000 comparisons across text stream lengths n ∈ [500, 10000] characters for pattern length m = 20. Explore interactive search algorithms, compare operation counts and analyse complexity scaling.
COMPUTE · BENCHMARK · ANALYSE
Your virtual algorithm laboratory.
KMP CHARACTER COMPARISONS C(N)—
EXECUTION STATUS
Ready to execute text length benchmark.
Adjust text length 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 scalar CPU comparison registers where every character equality evaluation executes with deterministic cost of 1 operation (C(n) = n + m). In physical semiconductor microprocessors, wall-clock execution latency is modulated by CPU L1 data cache hit ratios during text buffer streaming, hardware branch predictor mispredictions during character mismatch redirects (`if (P[j] == T[i])`), Translation Lookaside Buffer (TLB) page table walks across large corpus buffers, and hardware SIMD vector unit availability. In the Knuth-Morris-Pratt (KMP) text search benchmark with pattern length m = 20, the algorithm executes in two distinct phases: (1) Preprocessing the failure function π[0 .. m-1], which consumes exactly C_0 = m = 20 comparisons; and (2) Scanning the text stream T[0 .. n-1], where the text pointer i strictly advances from 0 to n - 1. Because the text pointer never moves backward and pattern pointer fallback via π[j-1] is bounded by prior increments, the scanning phase executes with an empirical scaling rate of exactly k_kmp = 1.0000 comparison per text character on uniform corpus data. Total character comparisons scale strictly as C(n) = 1.0000 · n + 20.0000 comparisons, yielding an exact linear relationship with zero residual variance across all benchmark trials.
Computational tools & environment.
Boyer-Moore Bad Character & Good Suffix Lookup Tables
Dual-heuristic jump tables calculating right-to-left shift vectors based on mismatch character identity and matching suffix patterns.
Rabin-Karp Rolling Hash & Modular Arithmetic Register
Hardware polynomial hash accumulator evaluating rolling substring signatures modulo large prime q using Horner's method.
Aho-Corasick Multi-Pattern Trie & Dictionary Matcher
Automaton compiler constructing trie hierarchies, failure transitions, and dictionary output links for simultaneous multi-string scanning.
Hardware SIMD Vector Matching Unit (AVX-512 / ARM Neon)
Microarchitectural vector execution register executing 64-byte parallel character comparisons via byte-mask instructions.
Deterministic English Corpus & Synthetic Text Generator
Configurable text stream synthesizer generating controlled alphabets, natural language corpora, and pathological repeated-prefix sequences.
FOLLOW THE METHOD
A reason behind every step.
To investigate the theoretical foundations and microarchitectural mechanics of exact string matching algorithms, evaluate the Knuth-Morris-Pratt (KMP) prefix function π-array, contrast KMP with Boyer-Moore sublinear shift heuristics and Rabin-Karp rolling hashes, and empirically verify the linear comparison scaling law C(n) = k_kmp · n + C_0 = 1.0000 · n + 20.0000 comparisons across text stream lengths n ∈ [500, 10000] characters for pattern length m = 20. Suggested independent settings: 500, 1000, 2000, 4000, 6000, 8000, 10000 chars.
01
Initialize Pattern Register & Compute Failure Array
Load the target pattern P[0 .. 19] of length m = 20 characters. Execute the failure function preprocessor to compute π[0 .. 19], logging exactly C_0 = 20 preprocessing comparison operations into the telemetry register.
02
Select Corpus Text Stream Length
Adjust the independent variable control slider to configure the input text stream length n between 200 and 10,000 characters across the designated benchmark sample points.
03
Execute KMP Non-Backtracking Text Scan
Trigger the KMP matching engine. Traverse text pointer i from 0 to n-1, comparing text character T[i] with pattern character P[j]. On match, advance j; on mismatch, redirect j to π[j-1] without decrementing text pointer i.
04
Capture Cumulative Character Comparisons
Read the total character comparison count C(n) from the algorithmic comparator register upon stream completion, verifying that C(n) = n + m comparisons.
05
Record Benchmark Observation in Notebook
Log the independent text length n and dependent comparison count C(n) into the laboratory notebook table and plot the corresponding data coordinate.
06
Perform Asymptotic Linear Regression & Paradigm Analysis
Perform ordinary least-squares regression across all seven benchmark trials to extract the text scan slope k_kmp and intercept C_0, evaluating residual variance, cache throughput, and comparative algorithm trade-offs.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into asymptotic insight.
Points are your recorded benchmark observations. The regression line validates String Matching & KMP Automata complexity scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
C(n) = k_kmp · n + C_0 = 1.0000 · n + 20.0000 comparisons
| # | Text length n characters (pattern length m = 20) (chars) | KMP character comparisons C(n) (comparisons) | Text stream length n (characters) | KMP character comparisons C(n) = n + m (comparisons) | Remove |
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