binary_lifting.hpp

#pragma once
#include <cassert>
#include <vector>

// Binary lifting (Doubling) on functional graphs
template <class S, S (*op)(S, S)> class binary_lifting {
    int n = 0;
    std::vector<std::vector<int>> _nexts;
    std::vector<std::vector<S>> _prods;

    void build_next() {
        std::vector<int> _next(n);
        std::vector<S> _prod(n);

        for (int i = 0; i < n; ++i) {
            if (int j = _nexts.back().at(i); isin(j)) {
                _next.at(i) = _nexts.back().at(j);
                _prod.at(i) = op(_prods.back().at(i), _prods.back().at(j));
            } else {
                _next.at(i) = j;
                _prod.at(i) = _prods.back().at(i);
            }
        }

        _nexts.emplace_back(std::move(_next));
        _prods.emplace_back(std::move(_prod));
    }

    inline bool isin(int i) const noexcept { return 0 <= i and i < n; }

public:
    // (up to) 2^d steps from `s`
    // Complexity: O(d) (Already precalculated) / O(nd) (First time)
    int pow_next(int s, int d) {
        assert(isin(s));
        while (int(_nexts.size()) <= d) build_next();
        return _nexts.at(d).at(s);
    }

    // Product of (up to) 2^d elements from `s`
    const S &pow_prod(int s, int d) {
        assert(isin(s));
        while (int(_nexts.size()) <= d) build_next();
        return _prods.at(d).at(s);
    }

    binary_lifting() = default;
    binary_lifting(const std::vector<int> &g, const std::vector<S> &w)
        : n(g.size()), _nexts(1, g), _prods(1, w) {
        assert(g.size() == w.size());
    }

    // (up to) k steps from `s`
    template <class Int> int kth_next(int s, Int k) {
        for (int d = 0; k > 0 and isin(s); ++d, k >>= 1) {
            if (k & 1) s = pow_next(s, d);
        }
        return s;
    }

    // Product of (up to) `len` elements from `s`
    template <class Int> S prod(int s, Int len) {
        assert(isin(s));
        assert(len > 0);
        int d = 0;
        while (!(len & 1)) ++d, len /= 2;

        S ret = pow_prod(s, d);
        s = pow_next(s, d);
        for (++d, len /= 2; len and isin(s); ++d, len /= 2) {
            if (len & 1) {
                ret = op(ret, pow_prod(s, d));
                s = pow_next(s, d);
            }
        }
        return ret;
    }

    // `start` から出発して「`left_goal` 以下または `right_goal` 以上」に到達するまでのステップ数
    // 単調性が必要
    int distance_monotone(int start, int left_goal, int right_goal) {
        assert(isin(start));

        if (start <= left_goal or right_goal <= start) return 0;

        int d = 0;
        while (left_goal < pow_next(start, d) and pow_next(start, d) < right_goal) {
            if ((1 << d) >= n) return -1;
            ++d;
        }

        int ret = 0, cur = start;
        for (--d; d >= 0; --d) {
            if (int nxt = pow_next(cur, d); left_goal < nxt and nxt < right_goal) {
                ret += 1 << d, cur = nxt;
            }
        }

        return ret + 1;
    }

    template <class F> long long max_length(const int s, F f, const int maxd = 60) {
        assert(isin(s));
        int d = 0;
        while (d <= maxd and f(pow_prod(s, d))) {
            if (!isin(pow_next(s, d))) return 1LL << maxd;
            ++d;
        }
        if (d > maxd) return 1LL << maxd;

        --d;

        int cur = pow_next(s, d);
        long long len = 1LL << d;
        S p = pow_prod(s, d);

        for (int e = d - 1; e >= 0; --e) {
            if (S nextp = op(p, pow_prod(cur, e)); f(nextp)) {
                std::swap(p, nextp);
                cur = pow_next(cur, e);
                len += 1LL << e;
            }
        }

        return len;
    }
};