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The Perfect Combination of High-Dimensional String Pattern Matching and Digit Dynamic Programming

Published at: 2026-05-26 02:45 Last updated: 2026-06-06 13:04
#C++ #Tutorial

Core Logic and Mathematical Principles

In the previous chapters, we traversed through basic digit representation, specialized number theory, tree structures, and matrix acceleration. The ultimate form of Digit Dynamic Programming (Digit DP) is high-dimensional string pattern matching Digit DP combined with Automata theory.

The core feature of such problems is: the digits in the range $[L, R]$ must satisfy complex textual pattern constraints in a specific base representation (e.g., must not include certain given substrings, must include specific regular expressions, or must be interwoven with a specific string in a minor language).

When the restrictions on digits evolve from simple "adjacent digit differences" to "global string multi-pattern matching", traditional single-state records (such as pre codes) will collapse logically. To eliminate aftereffects, we must introduce the Aho-Corasick Automaton (AC Automaton) or Deterministic Finite Automaton (DFA), directly materializing the state nodes of the automaton into the high-dimensional control dimensions of Digit DP.

1. Topological Alignment of Algebraic Space: Mapping Digits to State Machine Nodes

Assume the problem requires that the decimal representation of the number must not contain "49" and "375". We first construct a Trie tree from these forbidden strings and upgrade it to an AC Automaton using failure pointers (Fail Pointer). Each node $u$ on the AC Automaton represents the longest matching state feature of the current filled digit prefix within the entire forbidden system.

When the Digit DP progresses from high digits to low digits, preparing to fill in digit $d$:

2. Solidification of the Non-Aftereffect Nature of the Automaton Digit System

Since the node transitions of the AC Automaton depend entirely on the current node $u$ and the input character $d$, and its transition network is static and deterministic. Therefore, when limit = false and lead = false, the total number of valid numbers that can be generated from the automaton node $u$ to lower digits is constant and isomorphic. This allows us to perfectly solidify the state into the memoization grid: f[pos][u], where $u$ is the current automaton node number.

State Design and Algorithm Derivation

Taking the classic top-tier problem "Avoiding 62 and 49 in large number multi-pattern matching" as an example (introducing the composite form of the AC Automaton): we seek to find how many numbers in the range $[L, R]$ do not contain any of the given $M$ forbidden substrings in their decimal string representation, where $R \le 10^{100}$ (given as a super-long string).

1. Automaton Network Construction

Insert the $M$ forbidden strings into the Trie tree and construct generalized failure edges through BFS: if the transition edge trie[u][d] exists for node $u$, its child node's failure pointer points to trie[fail[u]][d]; if it does not exist, to maintain continuity of the Digit DP transition, we directly link the empty pin trie[u][d] to trie[fail[u]][d] through Trie Graph Optimization. Additionally, if a node's failure ancestor is illegal, that node must inherit the illegal mark.

2. Memoization Framework of Digit DP

Design the DFS state space as: long long dfs(int pos, int u, bool lead, bool limit)

In the current position, enumerate the digits to fill $i \in [0, \text{up}]$:

C++ Standard Source Code (NOIP/Provincial Selection Ultimate Automaton Digit Template)

The following code demonstrates how to perfectly combine the AC Automaton with super-long number Digit DP to achieve a full score solution for industrial-grade multi-pattern substring filtering counting.

#include <iostream>
#include <vector>
#include <string>
#include <cstring>
#include <queue>
#include <algorithm>

using namespace std;

const int MOD = 1e9 + 7;

// AC Automaton/Trie Graph Component
int trie[2005][10];   // Transition matrix of Trie tree nodes, decimal character set is 0~9
int fail[2005];       // Failure pointer
bool danger[2005];    // Dangerous node mark (contains or ends with any forbidden substring)
int node_cnt = 0;

string L_str, R_str;  // Super-long interval boundaries
int digits[205];
long long f[205][2005]; // f[pos][u] core memoization grid

// Automaton operator: Insert forbidden pattern string into Trie tree
void insert_pattern(const string& s) {
    int u = 0;
    for (char ch : s) {
        int d = ch - '0';
        if (!trie[u][d]) {
            trie[u][d] = ++node_cnt;
        }
        u = trie[u][d];
    }
    danger[u] = true; // Mark the end of the pattern string
}

// Automaton operator: Construct AC Automaton failure chain and Trie graph optimization
void build_ac_automaton() {
    queue<int> q;
    for (int d = 0; d < 10; ++d) {
        if (trie[0][d]) {
            q.push(trie[0][d]);
        }
    }
    while (!q.empty()) {
        int u = q.front();
        q.pop();

        // Pass dangerous mark: if the failure ancestor is dangerous, the current node is also dangerous
        if (danger[fail[u]]) danger[u] = true;

        for (int d = 0; d < 10; ++d) {
            if (trie[u][d]) {
                fail[trie[u][d]] = trie[fail[u]][d];
                q.push(trie[u][d]);
            } else {
                // Trie graph path compression optimization: eliminate failure jumps, achieving O(1) state machine transitions
                trie[u][d] = trie[fail[u]][d];
            }
        }
    }
}

// Core control network of Digit DP
// pos: current digit; u: current AC automaton node; lead: leading zeros; limit: upper limit highest bit lock
long long dfs(int pos, int u, bool lead, bool limit) {
    if (pos < 0) return 1; // Successfully assembled a valid large number without hitting the red line

    if (!limit && !lead && f[pos][u] != -1) {
        return f[pos][u];
    }

    int up = limit ? digits[pos] : 9;
    long long res = 0;

    for (int i = 0; i <= up; ++i) {
        int next_u = u;

        if (lead && i == 0) {
            // Leading zero special case: the number has not truly started, the automaton remains at root node 0
            next_u = 0;
        } else {
            // Standard topological state machine transition
            next_u = trie[u][i];
        }

        // Forcefully intercept any search branches that hit forbidden patterns
        if (danger[next_u]) continue;

        res = (res + dfs(pos - 1, next_u, lead && (i == 0), limit && (i == up))) % MOD;
    }

    if (!limit && !lead) {
        f[pos][u] = res;
    }
    return res;
}

// Large number decomposition operator
long long solve(const string& num_str) {
    int len = num_str.length();
    for (int i = 0; i < len; ++i) {
        digits[i] = num_str[len - 1 - i] - '0'; // Align in reverse order, lower digits at lower indices
    }
    return dfs(len - 1, 0, true, true);
}

// Large number decrement operator: since the input L is a super-long string, L-1 needs to perform standard high-precision large number subtraction
string decrease_one(string s) {
    int i = s.length() - 1;
    while (i >= 0) {
        if (s[i] > '0') {
            s[i]--;
            break;
        } else {
            s[i] = '9';
            i--;
        }
    }
    // Remove any leading zeros that may have been produced
    if (s.length() > 1 && s[0] == '0') {
        s = s.substr(1);
    }
    return s;
}

int main() {
    ios_base::sync_with_stdio(false);
    cin.tie(NULL);

    int m;
    if (!(cin >> L_str >> R_str >> m)) return 0;

    for (int i = 0; i < m; ++i) {
        string pattern;
        cin >> pattern;
        insert_pattern(pattern);
    }

    build_ac_automaton();
    memset(f, -1, sizeof(f)); // Global cache processed once

    // Standard prefix sum difference formula: Ans(R) - Ans(L-1)
    string L_minus_1 = decrease_one(L_str);
    long long ans_r = solve(R_str);
    long long ans_l = solve(L_minus_1);

    long long final_ans = (ans_r - ans_l + MOD) % MOD;
    cout << final_ans << "\n";

    return 0;
}

NOIP/Provincial Selection Practical Pitfall Guide

1. Leading Zeros Causing Illegal Transitions in the Automaton (Leading Zero Mis-transition)

2. Missing Dangerous Gene Transmission When Constructing the AC Automaton (Danger Pointer Missing)

Classic Provincial Selection/NOI Real Questions

1. Luogu P4052 [JSOI2007] Text Generator

$$\text{Number of satisfying strings} = \text{Total number of strings}(26^M) - \text{Strings that do not contain any keywords}$$

"Not containing any" perfectly fits the standard automaton digit DP template provided above. We will insert all keywords into the AC Automaton, and forcibly intercept all that hit dangerous nodes. Running an unrestricted digit DP of length $M$ on this clean network (equivalent to limit = false, fully free transitions), we can find the total number of all safe schemes, and finally subtract the safe count from the total to obtain the final full score answer.

2. Luogu P3311 [SDOI2014] Counting

Original Source: Local_Import
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