Digit Dynamic Programming: In-Depth Analysis of Efficient Counting and State Management

Core Logic and Mathematical Principles

The essence of Digit Dynamic Programming (Digit DP) is to reduce the macro comparison of numerical sizes into a dimensional mapping represented as a "string-like" topological state transition of digits from high to low in a specific base.

Digit DP is specifically designed to solve counting problems of the type "how many integers in the interval $[L, R]$ satisfy a certain specific digit characteristic (e.g., not containing 49, the difference between adjacent digits $ ge 2$, the sum of digits being prime, etc.)". Its core idea lies in utilizing prefix sums, memoized search, and upper limit constraints controlling high digits over low digits.

1. Geometric Splitting of Space: Prefix Sum Differentiation

Directly finding the numbers that meet the conditions in the interval $[L, R]$ often faces complex bilateral boundary constraints. Digit DP strictly adheres to the prefix sum differentiation principle, transforming any interval query into an independent unidirectional query starting from 0:

$$ \text{Ans}(L, R) = \text{Ans}(0, R) - \text{Ans}(0, L - 1) $$

This allows us to focus solely on solving the core proposition of "how many numbers in $[0, N]$ satisfy the conditions".

2. Topological Decoupling of Numerical Upper Limits: limit Marker Principle

Assuming $N = 3125$. When we start filling digits from the highest place (thousands place) to the lower places:

• If the thousands place is filled with 1 or 2, then any digit from 0 to 9 can be freely filled in the hundreds place without exceeding the limit of $N$. At this point, the filling of lower places completely loses the upper limit constraint from higher places, and its local state is entirely isomorphic.
• If the thousands place is filled with 3, then the hundreds place can at most be filled with 1. If the hundreds place is filled with 0, the tens place can be freely filled with 0 to 9; if the hundreds place is filled with 1, the tens place can at most be filled with 2.

This dynamic constraint caused by the choice of high digits over the range of options for low digits is referred to in Digit DP as the limit constraint:

• When limit = true, the upper limit of the current digit that can be filled is strictly constrained by the current digit of $N$;
• When limit = false, the upper limit of the current digit that can be filled is automatically liberated to the maximum digit of that base (which is 9 in decimal).

State Design and Memoized Search Network

The most standard and error-free implementation scheme of Digit DP in competitive programming is memoized search based on DFS.

1. Standard Control Skeleton of DFS State Space

A standard Digit DP search function usually includes the following core control parameters: int dfs(int pos, int state, bool lead, bool limit)

• pos: The current digit being decided. Typically starts from the highest place and decreases to 0 (the lowest place), returning 1 when it reaches the bottom.
• state: Records the constraint state imposed on the current digit by the preceding digit (e.g., what digit was filled in the previous place, the sum of all previous digits, etc.).
• lead: Leading Zero Marker. If lead = true, it indicates that all digits before the current place are 0 (e.g., in the number 0025). This is crucial in some problems, as leading zeros are not part of the valid composition of a number. For example, if the problem requires that "adjacent digits cannot be the same", if lead = true, filling 0 in the current place is valid (i.e., the number 0 becomes a valid number), and it should not be mistakenly judged as the same as the previous place's "leading zero".
• limit: Upper limit lock on the highest place, controlling the physical upper bound of digit enumeration in the current place.

2. Core Conditions for State Memoization

The prerequisite for the memoization array f[pos][state] in Digit DP to be reusable is: the current search branch must be in the state of limit = false and lead = false.

Physical Reason: If limit = true, it indicates that the current count of numbers obtained is subject to the local constraint of $N$, and it is not a complete set, thus cannot be written into the global memoization array. Only when the upper limit is fully released and leading zeros are completely detached can the solutions found in lower places be universal enough to be solidified into the f array.

C++ Standard Source Code (NOIP/Comprehensive Template for Improvement Group)

The following code solves the classic introductory problem of Digit DP: "count the number of integers in the interval $[L, R]$ that do not contain the digit 4 and do not contain consecutive digits 62" (Luogu P2602 / Classic Wind Flow Digit DP Model), strictly compatible with the Linux g++ -O2 environment.

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

using namespace std;

int digits[20]; // Decomposed storage of the digits of number N
int f[20][10];  // f[pos][pre_digit] memoized state grid

// pos: current place; pre: the digit filled in the previous place; lead: whether there are leading zeros; limit: whether there is a highest place limit
int dfs(int pos, int pre, bool lead, bool limit) {
    // Termination condition: all digit decisions are completed, successfully forming a valid number, return 1
    if (pos < 0) return 1;

    // Memoization trigger: only in a "free state" without any restrictions and without leading zeros can the old cache be referenced
    if (!limit && !lead && f[pos][pre] != -1) {
        return f[pos][pre];
    }

    // Dynamically calculate the upper limit of the current digit enumeration
    int up = limit ? digits[pos] : 9;
    int res = 0;

    for (int i = 0; i <= up; ++i) {
        // Pruning interception: cannot contain 4
        if (i == 4) continue;
        // Pruning interception: cannot contain consecutive 62
        if (pre == 6 && i == 2) continue;

        // Recursive control transmission:
        // 1. New leading zero state: if current is a leading zero and the current place is filled with 0, the next place is still a leading zero
        // 2. New highest place limit: if current is limited and the current place has filled the theoretical upper limit digit, the next place will have the highest place limit
        res += dfs(pos - 1, i, lead && (i == 0), limit && (i == up));
    }

    // Solidifying state cache: only the free state can be recorded
    if (!limit && !lead) {
        f[pos][pre] = res;
    }

    return res;
}

// Solve the total number of valid numbers in [0, n]
int solve(int n) {
    if (n < 0) return 0;
    if (n == 0) return 1; // 0 itself is a valid number (does not contain 4, does not contain 62)

    int len = 0;
    // Decompose the decimal number from low to high into the array
    while (n > 0) {
        digits[len++] = n % 10;
        n /= 10;
    }

    // Start recursion: begin from the highest valid place (len - 1), assign the previous place a safe sentinel value of 0, initially in the highest place limit and leading zero state
    return dfs(len - 1, 0, true, true);
}

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

    memset(f, -1, sizeof(f)); // The global state grid only needs to be initialized once

    int l, r;
    while (cin >> l >> r && (l || r)) {
        // Utilize prefix sum differentiation to capture interval answers without loss
        cout << solve(r) - solve(l - 1) << "\n";
    }

    return 0;
}

NOIP Practical Avoidance Guide

1. Misinitialization of Memoization Array in solve Function

• Phenomenon: Many contestants tend to execute memset(f, -1, sizeof(f)) within the solve(n) function every time it is called. This is fine for single queries in an interval but completely undermines the core advantage of Digit DP (cross-value isomorphic state reuse) when querying multiple datasets or large intervals. Each clear means that each dataset needs to rerun a complete state DFS, directly leading to a constant explosion causing TLE (Time Limit Exceeded).
• Avoidance Strategy: Since Digit DP separates limit and lead, the local state entirely depends on the current position pos and digit statistics state state, and has nothing to do with the specific $N$. Therefore, the memoization array f should be initialized once at the beginning of the main function and reused for a lifetime.

2. Ignoring Boundary Underflow of solve(L - 1) When $L=0$

• Phenomenon: When the given interval starting point $L = 0$, the program will execute solve(-1). If there is no safe interception for negative numbers at the beginning of the solve function, it can lead to array out-of-bounds or logical collapse.
• Avoidance Strategy: A strict boundary sentinel should be added at the entrance of solve: if (n < 0) return 0;.

Classic NOIP/Luogu Real Problems

1. Luogu P2602 [ZJOI2010] Digit Counting

• Problem Description: Given two positive integers $a$ and $b$, count how many times each digit ($0 \sim 9$) appears among all integers in the interval $[a, b]$. $a, b \le 10^{12}$.
• Essence of the Problem and Solution Idea: Digit DP with independent counters.
• Core Idea: To count the specific occurrences of each digit, we can run Digit DP separately for the 10 digits from $0 \sim 9$, or add an additional dimension count in the DFS state to record how many times the target digit has appeared so far. In this problem, leading zero determination (i.e., lead) is crucial: when counting the occurrence of the digit 0, if it is a leading zero (like in 00023 where the first three zeros), it must not be included in the answer. Only zeros that come after lead = false are valid components of the number.

2. Luogu P4999 Annoying Math Homework

• Problem Description: Calculate the total sum of the sum of digits of all integers in the interval $[L, R]$. The answer should be taken modulo $10^9+7$. $L, R \le 10^{18}$.
• Essence of the Problem and Solution Idea: Digit DP for summing digits with weights.
• Core Idea: This problem requires not just counting the number of "valid numbers", but summing the indicators of all numbers.
• State Design: dfs(pos, sum, lead, limit). Here, sum records the sum of digits already selected from high places.
• Return Value Correction: When pos < 0 reaches the bottom, it should return the accumulated sum directly instead of returning 1. This way, the DFS can naturally accumulate the digit sums of all paths during backtracking. Due to the data scale reaching $10^{18}$, all intermediate results and memoization arrays must be maintained using long long and take modulus immediately after each addition.