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Digit Dynamic Programming and Path Feature Extraction in Tree Topology

Published at: 2026-05-29 01:15 Last updated: 2026-06-06 13:04
#C++ #Tutorial

Core Logic and Mathematical Principles

In the previous chapters, the digit DP problems (18.1, 18.2) we dealt with fundamentally involve unidirectional topological counting on a one-dimensional number line, from high to low digit positions. However, the ultimate examination direction in the NOIP improvement group and provincial selections often cross-integrates digit DP with tree topology structures.

When introducing "path or digit dependencies on trees", the problem is no longer a simple decomposition of an independent scalar $N$, but rather on a given tree structure, certain nodes contain values (or indices) that have strong correlation constraints at the digit level (for example: finding the sum of path weights on the tree where node indices contain the digit 4, or the interweaving of certain digit characteristics among all nodes in a subtree).

To overcome this type of complex problem, one must master the following two core fusion techniques:

1. Two-Phase Materialization of Tree Topological Order and Digit State Machine

Tree digit DP cannot be resolved through a single DFS directly. It must implement a two-phase divide-and-conquer in spatial topology:

2. Two-Dimensional Extension of Prefix Sum Difference in Tree Topology

On a one-dimensional number line, the difference logic for an interval $[L, R]$ is $ ext{Ans}(R) - ext{Ans}(L-1)$. In tree topology, if constraints exist along the path $(u, v)$, we usually need to use the Lowest Common Ancestor (LCA) to convert the path into an algebraic expression composed of two root chains (Root Paths). Let $D(u)$ represent the digit characteristics of the path from the root node to node $u$, then the characteristics of the entire path need to be losslessly extracted through tree differences:

$$ ext{Path}(u, v) = D(u) + D(v) - D( ext{LCA}(u, v)) - D( ext{fa}( ext{LCA}(u, v)))$$

This process of reducing "tree topological distance" to "prefix root chains" and then decomposing "root chains" into "digit strings" is one of the most representative algebraic reduction ideas in higher-order dynamic programming.

State Design and Algorithm Derivation

Taking the classic provincial selection problem "Windy Path on Trees" as an example: given a tree with $N$ nodes, each node has a decimal index. The task is to find how many simple paths $(u, v)$ on the tree satisfy that the indices of all nodes along the path seamlessly concatenate into a huge digit string, and this string is a valid Windy number (i.e., the absolute difference between adjacent digits is $ ge 2$).

1. Cross-State Design of Topology and Digits

To advance the digit state on the tree, we must forcibly extend a one-dimensional "edge digit feature" in the ordinary tree DP state. Let $f[u][d]$ represent: the total number of valid path segments extending downward from the subtree rooted at node $u$, where the digits near node $u$ are filled with the number $d$.

2. Topological Interweaving of State Transition Equations

For a parent node $u$ and its child node $v$, to merge the subtree path rooted at $v$ upwards to $u$, the index of $u$ has fixed digits (assuming the highest digit of $u$ is $d_u$ and the lowest digit is $d'_u$), the top digit $d_v$ of child node $v$ must strongly collide with the boundary digit of $u$:

$$f[u][du] = ext{sum}{v ext{ in son}(u)} ext{sum}_{|d_v - d'_u| ge 2} f[v][d_v]$$

During backtracking, using the multiplication principle, the "number of paths satisfying conditions in the left subtree" and the "number of paths satisfying conditions in the right subtree" can be cross-multiplied at the current node $u$, allowing the counting of digit features throughout the entire tree to be completed in $O(N imes 10)$ low complexity.

C++ Standard Source Code (NOIP/Provincial Selection Specialized Tree Digit Template)

The following code demonstrates how to perfectly maintain digit difference constraints and collect global topological contributions in a tree structure, strictly adhering to the Linux g++ -O2 compilation standards.

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

using namespace std;

const int MAXN = 100005;

struct Edge {
    int to;
};
vector<Edge> g[MAXN];

int node_val[MAXN]; // Stores the index of each node
int head_digit[MAXN]; // Preprocessing: highest digit of each node's index
int tail_digit[MAXN]; // Preprocessing: lowest digit of each node's index

// f[u][d] represents the number of valid chains ending in digit d in the subtree rooted at u
// In this problem, since node values are fixed, the d dimension is mainly used for matching checks
long long f[MAXN][10];
long long ans = 0;

// Preprocessing operator: extract the highest and lowest digits of a number
void extract_digits(int u) {
    int val = node_val[u];
    tail_digit[u] = val % 10;
    while (val >= 10) {
        val /= 10;
    }
    head_digit[u] = val;
}

void dfs(int u, int fa) {
    extract_digits(u);

    // Base condition: a single node itself forms a valid path
    // The effective connecting digit it shows at the current node is its highest digit (viewed from bottom to top)
    f[u][head_digit[u]] = 1;

    for (const auto& edge : g[u]) {
        int v = edge.to;
        if (v == fa) continue;

        dfs(v, u); // Topological sinking

        // Core cross-contribution collection (multiplication principle):
        // Combine previously collected subtree chains with the current new subtree v's chain at the current node u
        for (int du = 0; du <= 9; ++du) {
            for (int dv = 0; dv <= 9; ++dv) {
                // Strong digit constraint interception: the low-end tail of the current chain and the high-end head of the new chain must differ by >= 2
                if (abs(tail_digit[u] - dv) >= 2) {
                    ans += f[u][du] * f[v][dv];
                }
            }
        }

        // State transition (addition principle): report and merge the path chains of subtree v upwards to u
        for (int dv = 0; dv <= 9; ++dv) {
            if (abs(tail_digit[u] - dv) >= 2) {
                f[u][head_digit[u]] += f[v][dv];
            }
        }
    }
}

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

    int n;
    if (!(cin >> n)) return 0;

    for (int i = 1; i <= n; ++i) {
        cin >> node_val[i];
    }

    for (int i = 1; i < n; ++i) {
        int u, v;
        cin >> u >> v;
        g[u].push_back({v});
        g[v].push_back({u});
    }

    dfs(1, 0);

    // Final answer adds all single-node paths (which were not counted in the DFS merging)
    cout << ans + n << "\n";

    return 0;
}

NOIP Practical Pitfall Guide

1. Ignoring Asymmetry Caused by Digit Concatenation Direction (Asymmetric Chain Mesh)

2. Implicit Timeout Caused by Poor Memory Locality

Classic Provincial Selection/NOI Problems

1. Luogu P3647 [APIO2014] Continuous Beads

2. Luogu P3323 [SDOI2015] Treasure Hunt Game

Original Source: local://18.3
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