Problem:
Score : 400 points

Problem Statement

Given is a string S. Each character in S is either a digit (0, ..., 9) or ?.

Among the integers obtained by replacing each occurrence of ? with a digit, how many have a remainder of 5 when divided by 13? An integer may begin with 0.

Since the answer can be enormous, print the count modulo 10^9+7.

Constraints

S is a string consisting of digits (0, ..., 9) and ?.

1 \leq |S| \leq 10^5

Input

Input is given from Standard Input in the following format:

S

Output

Print the number of integers satisfying the condition, modulo 10^9+7.

Sample Input 1

??2??5

Sample Output 1

768

For example, 482305, 002865, and 972665 satisfy the condition.

Sample Input 2

?44

Sample Output 2

1

Only 044 satisfies the condition.

Sample Input 3

7?4

Sample Output 3

0

We may not be able to produce an integer satisfying the condition.

Sample Input 4

?6?42???8??2??06243????9??3???7258??5??7???????774????4?1??17???9?5?70???76???

Sample Output 4

153716888

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)


import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStream;
import java.io.InputStreamReader;
import java.util.ArrayList;
import java.util.Collections;
import java.util.Comparator;
import java.util.HashMap;
import java.util.List;
import java.util.Map.Entry;
import java.util.StringTokenizer;

public class Main implements Runnable { // クラス名はMain1

	long dp[][];
	boolean dpbool[][];
	static int MOD = 1000000000 + 7;

	public static void main(String[] args) {
		Thread.setDefaultUncaughtExceptionHandler((t, e) -> System.exit(1));
		new Thread(null, new Main(), "", 1024 * 1024 * 1024).start(); // 16MBスタックを確保して実行
	}

	public void run() {
		FastScannerMain1 sc = new FastScannerMain1(System.in);

		try {
			/**
			 * 表示系
			 *   System.out.println();
			 *   System.out.println("i:" + i + " ,j:" + j);
			 *   System.out.println(Arrays.toString(A));
			 *
			 * Sort (Arrays)
			 *  Integer[] A = new Integer[N];
			 *    Desc : Arrays.sort(A , Collections.reverseOrder());
			 *    Asc  : Arrays.sort(A);
			 *
			 * 定義系
			 *   int[] s = new int[M];
			 *   ArrayDeque<Integer> deq= new ArrayDeque<>();//先頭と末尾への要素の追加・先頭と末尾の要素の取り出しと削除が定数時間
			 *   ArrayList<Integer> arr = new ArrayList<Integer>();//定数時間
			 *   HashMap<Integer,Integer> hm = new HashMap<>(); //定数時間
			 *
			 *   //要素の追加と、最小の要素の取り出しと削除が対数時間
			 *   PriorityQueue<Integer> pq = new PriorityQueue<>();//昇順
			 *   PriorityQueue<Integer> pq = new PriorityQueue<>(Collections.reverseOrder()); //逆順。降順
			 *
			 *   TreeSet<Integer> ts = new TreeSet<>();//対数時間(重複はできない)
			 *
			 * 文字系
			 *   String S = sc.nextLine().trim();
			 *   for (char i = 'a'; i <= 'z'; i++) {}
			 *   S.substring(1,2);//あいうえお → い
			 *   S.substring(0,1); // 1文字目をとりたいとき
			 *   文字列のSort
			 *    String sorted = S.chars().sorted().collect(StringBuilder::new,StringBuilder::appendCodePoint, StringBuilder::append) .toString();
			 */
			// long l = sc.nextLong();

			String S = sc.nextLine().trim();
			int N = S.length();

			//System.out.println(Arrays.toString(A));

			dp = new long[N + 1][13];
			dpbool = new boolean[N + 1][13];

			long ans = rec(S, N, 0, 0, true);

			System.out.println(ans);

		} catch (ArithmeticException ae) {
			ae.printStackTrace();
		} catch (Exception e) {
			e.printStackTrace();
		}

	}

	private long rec(String S, int N, int k, int sum, boolean tight) {

		if (dpbool[k][sum]) {
			//System.out.println("k:" + k + " ,sum:" + sum);
			return dp[k][sum];
		}
		if (k == N) {
			//System.out.println(sum);
			if (sum % 13 == 5) {
				//System.out.println(sum);
				return 1;
			} else {
				return 0;
			}
		}

		long res = 0;

		int x;
		if (S.charAt(k) == '?') {
			x = 9;
		} else {
			x = S.charAt(k) - '0';
		}

		if (S.charAt(k) == '?') {
			for (int i = 0; i <= 9; i++) {

				if (tight && i == x) {
					res += rec(S, N, k + 1, (sum * 10 + i) % 13, true);
				} else {
					res += rec(S, N, k + 1, (sum * 10 + i) % 13, false);
				}
				res %= MOD;
				dp[k][sum] = res;
				dpbool[k][sum] = true;
			}
		} else {
			res += rec(S, N, k + 1, (sum * 10 + x) % 13, true);
			res %= MOD;
			dp[k][sum] = res;
			dpbool[k][sum] = true;
		}

		return res;
	}

	// 下記はオーバーフローでExceptionをキャッチするよう
	private long multiple(long a, long b) {
		long ans = 0;
		try {
			ans = Math.multiplyExact(a, b);
			return ans;
		} catch (ArithmeticException ae) {
			throw new ArithmeticException();
		}
	}

	private int multiple(int a, int b) {
		int ans = 0;
		try {
			ans = Math.multiplyExact(a, b);
			return ans;
		} catch (ArithmeticException ae) {
			throw new ArithmeticException();
		}
	}

	private long add(long a, long b) {
		long ans = 0;
		try {
			ans = Math.addExact(a, b);
			return ans;
		} catch (ArithmeticException ae) {
			throw new ArithmeticException();
		}
	}

	private int add(int a, int b) {
		int ans = 0;
		try {
			ans = Math.addExact(a, b);
			return ans;
		} catch (ArithmeticException ae) {
			throw new ArithmeticException();
		}
	}

	/*
	 * HashmapのSort
	 * hmはhashmap
	 * keyvalは"key" or "val"を指定して
	 * ascdescは"asc" or "desc"を指定して
	 */
	private List<Entry<Integer, Integer>> sortHashMapValue(HashMap<Integer, Integer> hm, String keyval,
			String ascdesc) {
		// 1.Map.Entryのリストを作成する
		List<Entry<Integer, Integer>> list_entries = new ArrayList<Entry<Integer, Integer>>(hm.entrySet());

		// 2.比較関数Comparatorを使用してMap.Entryの値を比較する(昇順)
		Collections.sort(list_entries, new Comparator<Entry<Integer, Integer>>() {
			public int compare(Entry<Integer, Integer> obj1, Entry<Integer, Integer> obj2) {
				if (keyval.equals("val")) {
					if (ascdesc.equals("asc")) {
						// 昇順(value)
						return obj1.getValue().compareTo(obj2.getValue());
					} else {
						// 降順(value)
						return obj2.getValue().compareTo(obj1.getValue());
					}
				} else {
					if (ascdesc.equals("asc")) {
						// 昇順(value)
						return obj1.getKey().compareTo(obj2.getKey());
					} else {
						// 降順(value)
						return obj2.getKey().compareTo(obj1.getKey());
					}
				}
			}
		});
		/*
		// 3. ループで要素順に値を取得する
		for (Entry<Integer, Integer> entry : list_entries) {
			System.out.println(entry.getKey() + " : " + entry.getValue());
		}
		*/
		return list_entries;
	}
}

class Exm implements Comparable<Exm> {

	public int a;
	public int b;

	@Override
	public int compareTo(Exm o) {

		//return this.a - o.a;//昇順
		// return o.a - this.a;//降順

		//複数項目で並び替え。aの降順、aが同じならbの降順
		if (this.a == o.a) {
			return o.b - this.b;
		} else {
			return o.a - this.a;
		}
	}

}

// 高速なScanner
class FastScannerMain1 {
	private BufferedReader reader = null;
	private StringTokenizer tokenizer = null;

	public FastScannerMain1(InputStream in) {
		reader = new BufferedReader(new InputStreamReader(in));
		tokenizer = null;
	}

	public String next() {
		if (tokenizer == null || !tokenizer.hasMoreTokens()) {
			try {
				tokenizer = new StringTokenizer(reader.readLine());
			} catch (IOException e) {
				throw new RuntimeException(e);
			}
		}
		return tokenizer.nextToken();
	}

	public String nextLine() {
		if (tokenizer == null || !tokenizer.hasMoreTokens()) {
			try {
				return reader.readLine();
			} catch (IOException e) {
				throw new RuntimeException(e);
			}
		}
		return tokenizer.nextToken("\n");
	}

	public long nextLong() {
		return Long.parseLong(next());
	}

	public int nextInt() {
		return Integer.parseInt(next());
	}

	public double nextDouble() {
		return Double.parseDouble(next());
	}

	public int[] nextIntArray(int n) {
		int[] a = new int[n];
		for (int i = 0; i < n; i++)
			a[i] = nextInt();
		return a;
	}

	public long[] nextLongArray(int n) {
		long[] a = new long[n];
		for (int i = 0; i < n; i++)
			a[i] = nextLong();
		return a;
	}
}