topcoder srm 515 div1

problem1 link

暴力枚举即可。

problem2 link

一共有24小时，所以最多有24个顾客。设$f[x][y][z]$表示还剩下$x$把刀，现在时间是$y$,以及来过的顾客集合为$z$可以获得的最大值。

那假设第$y$小时来的顾客为$t$，来的概率为$p$,有三种情况：

（1）之前它来过，那么$f[x][y][z]=f[x][y+1][z]$

（2）之前没来过，现在也没来，$f[x][y][z]=(1-p)*f[x][y+1][z]$

（3）之前没来过，现在来了。可以选择买给他一把刀，$c_{1}=f[x-1][y+1][z|2^{t}]+v_{y}$;或者不卖给他,$c_{2}=f[x][y+1][z|2^{t}]$，所以$f[x][y][z]=max(c_{1},c_{2})*p$

另外，假设一个人第一次，第二次来的概率分别为0.2和0.4。那么上面使用的$p$是假设他之前没来过。如果第一次没来，那么第二次来的概率为$frac{0.4}{1-0.2}$

problem3 link

首先枚举字母'R'和‘F’的位置$p_{1},p_{2}$。然后计算和其他所有‘L’的答案。

首先，对于每个位置$(x,y)$,计算其函数值$g(x,y)=distance(p_{1},p_{x,y})+distance(p_{2},p_{x,y})$

然后新建一系列节点$S_{0},S_{1},S_{2},...$。

对于每个$(x,y)$，设$t=g(x,y)$。那么节点$S_{t-1}$向点$(x,y)$连有向边。

另外节点$S_{i}$到$S_{i+1}$有一条有向边。每个点$(x,y)$向四周的格子有四条有向边。

所有边的长度均为1.然后从节点$S_{0}$开始进行bfs，计算到达每个点的距离。每个‘L’字母计算的距离就是选择$p_{1},p_{2}$和该位置的答案。 

code for problem1

class RotatedClock {
   public:
    string getEarliest(int hourHand, int minuteHand) {
        for (int i = 0; i < 12; ++i) {
            for (int j = 0; j < 60; j += 2) {
                for (int k = 0; k < 12; ++k) {
                    if (Check(i, j, k, hourHand, minuteHand)) {
                        return ToString(i) + ":" + ToString(j);
                    }
                }
            }
        }
        return "";
    }

    string ToString(int x) {
        stringstream ss;
        if (x < 10) {
            ss << "0" << x;
        } else {
            ss << x;
        }
        return ss.str();
    }

    bool Check(int h, int m, int b, int hHand, int mHand) {
        return GetHourHand(h, m, b) == hHand && GetMinuteHand(m, b) == mHand;
    }

    int GetHourHand(int h, int m, int b) {
        int result = 0;
        if (h >= b) {
            result = (h - b) * 30 + m / 2;
        } else {
            result = 360 - (b - h) * 30 + m / 2;
        }
        return result;
    }
    int GetMinuteHand(int m, int b) {
        b *= 5;
        if (m >= b) {
            return (m - b) * 6;
        }
        else {
            return 360 - (b - m) * 6;
        }
    }

};

code for problem2

double cache[25][24][1 << 12];
int visited[25][24][1 << 12];

class NewItemShop {
   public:
    double getMaximum(int swords, vector<string> customers) { 
        Init(customers);
        memset(visited, 0, sizeof(visited));
        return dfs(swords, 0, 0);
    }

    double dfs(int swords, int hour, int mask) {
        if (swords == 0 || hour == 24) {
            return 0;
        }
        if (visited[swords][hour][mask] != 0) {
            return cache[swords][hour][mask];
        }
        visited[swords][hour][mask] = 1;
        if (!hour_[hour].inited_) {
            return cache[swords][hour][mask] = dfs(swords, hour + 1, mask);
        }


        int cMask = hour_[hour].mask();
        double p = hour_[hour].probility_;
        int val = hour_[hour].value_;

        if ((mask & cMask) > 0) {
            return cache[swords][hour][mask] = dfs(swords, hour + 1, mask);
        }

        cache[swords][hour][mask] = (1 - p) * dfs(swords, hour + 1, mask);

        double c0 = dfs(swords, hour + 1, mask | cMask);
        double c1 = dfs(swords - 1, hour + 1, mask | cMask) + val;

        cache[swords][hour][mask] += max(c0, c1) * p;

        return cache[swords][hour][mask];
    }

   private:


    struct HourInfo {
        bool is_multiple_times_;
        int multiple_index_;
        double probility_;
        int value_;

        bool inited_;

        HourInfo():inited_(false) {}

        int mask() const {
            if (!is_multiple_times_) {
                return 0;
            }
            return 1 << multiple_index_;
        }


    }hour_[24];

    void Init(const vector<string>& customers) {
        for (int i = 0; i < 24; ++ i) {
            hour_[i].inited_ = false;
        }
        int multiple_num = 0;
        for (int i = 0; i < (int)customers.size(); ++ i) {
            const vector<int> a = Split(customers[i]);
            const int n = (int)a.size();
            const int idx = n > 3 ? multiple_num ++ : 0;
            double cur = 1.0;
            for (int j = 0; j < n; j += 3) {
                int h = a[j];
                int v = a[j + 1];
                double p = a[j + 2] / 100.0 / cur;
                cur -= a[j + 2] / 100.0;
                if (n > 3) {
                    hour_[h].is_multiple_times_ = true;
                    hour_[h].multiple_index_ = idx;
                }
                else {
                    hour_[h].is_multiple_times_ = false;
                }
                hour_[h].probility_ = p;
                hour_[h].value_ = v;
                hour_[h].inited_ = true;
            }
        }
    }

    std::vector<int> Split(const std::string& s) {
        std::vector<int> result;
        const int len = (int)s.length();
        int idx = 0;
        while (idx < len) {
            while (idx < len && !IsDigit(s[idx])) {
                ++ idx;
            }
            if (idx >= len) {
                break;
            }
            int x = 0;
            while (idx < len && IsDigit(s[idx])) {
                x = x * 10 + s[idx++] - '0';
            }
            result.push_back(x);
        }
        return result;
    }

    static bool IsDigit(char c) {
        return '0' <= c && c <= '9';
    }
};

　　

code for problem3

class MeetInTheMaze {
   public:
    string getExpected(vector<string> maze) {
        Initialize(maze);
        std::vector<std::pair<int, int>> all_f_positions;
        std::vector<std::pair<int, int>> all_r_positions;
        int l_number = 0;
        for (int i = 0; i < height_; ++i) {
            for (int j = 0; j < width_; ++j) {
                char ch = maze_[i][j];
                if (ch == 'F') {
                    all_f_positions.push_back(std::make_pair(i, j));
                } else if (ch == 'R') {
                    all_r_positions.push_back(std::make_pair(i, j));
                } else if (ch == 'L') {
                    ++l_number;
                }
            }
        }
        int total = 0;
        for (auto f : all_f_positions) {
            for (auto r : all_r_positions) {
                int t = Calculate(f.first, f.second, r.first, r.second);
                if (t == -1) {
                    return "";
                }
                total += t;
            }
        }
        int d = (int)all_f_positions.size() * (int)all_r_positions.size() *
                l_number;
        int g = Gcd(total, d);
        total /= g;
        d /= g;
        std::stringstream ss;
        ss << total << "/" << d;
        return ss.str();
    }

    ~MeetInTheMaze() { delete[] distance_; }

   private:
    int Calculate(const int f_x, const int f_y, const int r_x, const int r_y) {
        std::vector<std::vector<std::pair<int, int>>> new_graph(
            (width_ + height_) << 1);
        for (int i = 0; i < height_; ++i) {
            for (int j = 0; j < width_; ++j) {
                if (maze_[i][j] != '#') {
                    int d1 = GetDistance(f_x, f_y, i, j);
                    int d2 = GetDistance(r_x, r_y, i, j);
                    if (d1 == -1 || d2 == -1) {
                        continue;
                    }
                    int d = d1 + d2;
                    if (d > (int)new_graph.size()) {
                        new_graph.resize(d);
                    }
                    new_graph[d - 1].push_back(std::make_pair(i, j));
                }
            }
        }
        const int nodes = (int)new_graph.size();
        std::vector<int> dist(height_ * width_ + nodes, -1);

        dist[height_ * width_] = 0;
        std::queue<int> a_queue;
        a_queue.push(height_ * width_);
        const int dx[] = {1, -1, 0, 0};
        const int dy[] = {0, 0, 1, -1};
        while (!a_queue.empty()) {
            const int start = a_queue.front();
            a_queue.pop();
            if (start < width_ * height_) {
                const int sx = start / width_;
                const int sy = start % width_;
                for (int i = 0; i < 4; ++i) {
                    int x = sx + dx[i];
                    int y = sy + dy[i];
                    if (0 <= x && x < height_ && 0 <= y && y < width_) {
                        if (maze_[x][y] == '#') {
                            continue;
                        }
                        if (dist[x * width_ + y] == -1) {
                            dist[x * width_ + y] = dist[start] + 1;
                            a_queue.push(x * width_ + y);
                        }
                    }
                }
            } else {
                const int new_graph_index = start - width_ * height_;
                for (auto son : new_graph[new_graph_index]) {
                    int x = son.first;
                    int y = son.second;
                    int key = x * width_ + y;
                    if (dist[key] == -1) {
                        dist[key] = dist[start] + 1;
                        a_queue.push(key);
                    }
                }
                if (new_graph_index + 1 < nodes) {
                    dist[start + 1] = dist[start] + 1;
                    a_queue.push(start + 1);
                }
            }
        }
        int total = 0;
        for (int i = 0; i < height_; ++i) {
            for (int j = 0; j < width_; ++j) {
                if (maze_[i][j] == 'L') {
                    if (dist[i * width_ + j] == -1) {
                        return -1;
                    }
                    total += dist[i * width_ + j];
                }
            }
        }
        return total;
    }

    int Gcd(int x, int y) { return y == 0 ? x : Gcd(y, x % y); }

    void Initialize(const std::vector<std::string> &maze) {
        maze_ = maze;
        height_ = (int)maze_.size();
        width_ = (int)maze_[0].size();
        const int total_size = width_ * width_ * height_ * height_;
        distance_ = new int[total_size];
        memset(distance_, -1, sizeof(int) * total_size);
    }

    int GetDistance(int start_x, int start_y, int end_x, int end_y) {

        if (start_x > end_x || (start_x == end_x && start_y > end_y)) {
            std::swap(start_x, end_x);
            std::swap(start_y, end_y);
        }

        int key = GetKey(start_x, start_y, end_x, end_y);
        if (distance_[key] == -1) {
            CalculateDistanceFrom(start_x, start_y);
        }
        return distance_[key];
    }

    void CalculateDistanceFrom(const int start_x, const int start_y) {
        std::queue<std::pair<int, int>> a_queue;
        a_queue.push(std::make_pair(start_x, start_y));
        distance_[GetKey(start_x, start_y, start_x, start_y)] = 0;
        const int dx[] = {1, -1, 0, 0};
        const int dy[] = {0, 0, 1, -1};
        while (!a_queue.empty()) {
            const int sx = a_queue.front().first;
            const int sy = a_queue.front().second;
            a_queue.pop();
            const int from_key = GetKey(start_x, start_y, sx, sy);
            for (int i = 0; i < 4; ++i) {
                int x = sx + dx[i];
                int y = sy + dy[i];
                if (0 <= x && x < height_ && 0 <= y && y < width_) {
                    if (maze_[x][y] == '#') {
                        continue;
                    }
                    int key = GetKey(start_x, start_y, x, y);
                    if (distance_[key] != -1) {
                        continue;
                    }
                    distance_[key] = distance_[from_key] + 1;
                    a_queue.push(std::make_pair(x, y));
                }
            }
        }
    }

    int GetKey(int x1, int y1, int x2, int y2) const {
        return x1 * width_ * height_ * width_ + y1 * height_ * width_ +
               x2 * width_ + y2;
    }

    int width_;
    int height_;
    std::vector<std::string> maze_;
    int *distance_;  
};