1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
#include <bits/stdc++.h>

using namespace std;

int L, v0;
int v[3];

double EPSILON = 1e-10;
vector<pair<double, double>> lanes[3];

void readLane(vector<pair<double, double>> &lane, bool removeFirst= false){
    string laneDescription;
    cin >> laneDescription;
    laneDescription += '.';
    int begin = -1;
    int end = -1;
    if(removeFirst)
        laneDescription[0] = '.';
    for(int i = 0; i < laneDescription.size(); i++){
        if(laneDescription[i] == '.') {
            if(begin == -1){
                continue;
            }
            lane.emplace_back(begin, end);
            begin = -1;
            end = -1;
        }
        else {
            if(begin == -1) {
                begin = i;
                end = i+1;
            }
            else {
                end++;
            }
        }
    }
}

struct positionDetails{
    double time;
    int line;
    int behindUs;
    double position;
};

int getBeforeOurPosition(int line, double time, double position) {
    pair<double, double> low_val = {position - time*v[line]+1, position - time*v[line]};
    auto it = lower_bound(lanes[line].begin(), lanes[line].end(), low_val,
                          [](pair<double, double> p1, pair<double, double> p2) -> bool {return p1.first - p2.first < EPSILON; });

    if(it == lanes[line].begin())
        return -1;
    it = prev(it, 1);

    return it - lanes[line].begin();
}

int getEndBeforeOurPosition(int line, double time, double position) {
    pair<double, double> low_val = {position, position - time*v[line]};
    auto it = upper_bound(lanes[line].begin(), lanes[line].end(), low_val,
                          [](pair<double, double> p1, pair<double, double> p2) -> bool {return p1.second - p2.second < -EPSILON; });

    if(it == lanes[line].begin())
        return -1;
    it = prev(it, 1);

    return it - lanes[line].begin();
}

struct hash_pair {
    template <class T1, class T2>
    size_t operator()(const pair<T1, T2>& p) const
    {
        auto hash1 = hash<T1>{}(p.first);
        auto hash2 = hash<T2>{}(p.second);
        return hash1 ^ hash2;
    }
};

double cutTo12Number(double number){
    auto tmp = (long long)number;
    number -= (double)tmp;
    auto tmp2 = (double)((long long)(number*1e12))/(1e12);

    return (double)tmp + tmp2;
}
unordered_map<pair<double, double>, bool, hash_pair> um[3];

bool inHashMap(double time, double position, int line){

    if(um[line].count({cutTo12Number(position), cutTo12Number(time)}) > 0)
        return true;
    um[line][{cutTo12Number(position), cutTo12Number(time)}] = true;
    return false;
}

double findFastestWay() {
    auto cmp = [] (positionDetails &p1, positionDetails &p2) {
        return p1.time > p2.time;
    };
    priority_queue<positionDetails, vector<positionDetails>, decltype(cmp)> q(cmp);
    q.push({0, 2, 0, 0});

    bool firstIteration = true;

    while (!q.empty()) {
        positionDetails pos = q.top();
        q.pop();

        if((lanes[0].empty() or ((pos.time*v[0] + lanes[0].back().second) - pos.position)<EPSILON)
           and (lanes[1].empty() or ((pos.time*v[1] + lanes[1].back().second) - pos.position)<EPSILON)
           and (lanes[2].empty() or ((pos.time*v[2] + lanes[2].back().second) - pos.position)<EPSILON)) {
            return pos.time;
        }
        if(lanes[pos.line].size() <= pos.behindUs) {
            double timeMax = 0;

            if(!lanes[0].empty()) {
                double distance = (pos.time*v[0] + lanes[0].back().second) - pos.position;
                if(distance <= 0)
                    timeMax = max(timeMax, pos.time);
                else {
                    double tmpTime = distance/(v0 - v[0]);
                    timeMax = max(timeMax, pos.time + tmpTime);
                }
            }
            if(!lanes[1].empty()) {
                double distance = (pos.time*v[1] + lanes[1].back().second) - pos.position;
                if(distance <= 0)
                    timeMax = max(timeMax, pos.time);
                else {
                    double tmpTime = distance/(v0 - v[1]);
                    timeMax = max(timeMax, pos.time + tmpTime);
                }
            }
            if(!lanes[2].empty()) {
                double distance = (pos.time*v[2] + lanes[2].back().second) - pos.position;
                if(distance <= 0)
                    timeMax = max(timeMax, pos.time);
                else {
                    double tmpTime = distance/(v0 - v[2]);
                    timeMax = max(timeMax, pos.time + tmpTime);
                }
            }
            if(!inHashMap(timeMax, pos.position + (timeMax-pos.time)*v0, pos.line))
                q.push({timeMax, pos.line, pos.behindUs, pos.position + (timeMax-pos.time)*v0});
            continue;
        }
        double nextCar = lanes[pos.line][pos.behindUs].first;

        if(pos.line > 0){ // idziemy w gore - szybsza
            double distance = nextCar + pos.time*v[pos.line] - 1;
            double time = (distance-pos.position)/(v0 - v[pos.line]);
            double nextPosition = pos.position + time*v0;
            double nextTime = pos.time + time;

            int idEndBeforeOurEnd = getEndBeforeOurPosition(pos.line - 1, nextTime, nextPosition);

            for(int i = idEndBeforeOurEnd; i <= idEndBeforeOurEnd; i++) {
                if(i == -1){
                    if(pos.line == 1 or firstIteration) {
                        firstIteration = false;
                        if(!inHashMap(pos.time, pos.position, pos.line-1))
                            q.push({pos.time, pos.line-1, 0, pos.position});
                    }

                    continue;
                }
                double carPosition = lanes[pos.line-1][i].second + pos.time*v[pos.line-1];
                if(carPosition - pos.position < EPSILON) { //CHANGED
                    if(pos.line == 1 or firstIteration) {
                        if(!inHashMap(pos.time, pos.position, pos.line-1))
                            q.push({pos.time, pos.line-1, i+1, pos.position});
                        firstIteration = false;
                    }
                } else {
                    double tmpDistance = carPosition - pos.position - 1;
                    double tmpTime = tmpDistance/(v0 - v[pos.line-1]);
                    double tmpNextPosition = pos.position + tmpTime*v0;
                    double tmpNextTime = pos.time + tmpTime;
                    if(!inHashMap(tmpNextTime, tmpNextPosition, pos.line-1))
                        q.push({tmpNextTime, pos.line-1, i+1, tmpNextPosition});
                }
            }
        }
        if(pos.line < 2) { // idziemy w dol - wolniejsza
            double distance = nextCar + pos.time*v[pos.line] - 1;
            double time = (distance-pos.position)/(v0 - v[pos.line]);
            double nextPosition = pos.position + time*v0;
            double nextTime = pos.time + time;
//            int idEndBeforeOurEnd = getEndBeforeOurPosition(pos.line + 1, nextTime, nextPosition);

            int idBeginBeforeOurPosition = getBeforeOurPosition(pos.line + 1, pos.time, pos.position);
            for(int i = (int)lanes[pos.line + 1].size() - 1;i >= idBeginBeforeOurPosition; i--) {
//            for(int i = idEndBeforeOurEnd;i >= idBeginBeforeOurPosition; i--) {
                if(i >= lanes[pos.line+1].size())
                    continue;
                if(i == -1){
                    if(pos.line == 1){
                        if(!inHashMap(pos.time, pos.position, pos.line+1))
                            q.push({pos.time, pos.line+1, 0, pos.position});
                    }
                    continue;
                }
                double carPosition = lanes[pos.line+1][i].second + pos.time*v[pos.line+1];
                if(carPosition <= pos.position) {
                    if(pos.line == 1){
                        if(!inHashMap(pos.time, pos.position, pos.line+1))
                            q.push({pos.time, pos.line+1, i+1, pos.position});
                        break;
                    }
                } else {
                    double tmpDistance = carPosition - pos.position;
                    double tmpTime = tmpDistance/(v0 - v[pos.line+1]);

                    if(nextCar + (pos.time + tmpTime)*v[pos.line] >=
                            lanes[pos.line+1][i].second + (pos.time + tmpTime)*v[pos.line+1] + 1) {
                        double tmpNextPosition = pos.position + tmpTime*v0;
                        double tmpNextTime = pos.time + tmpTime;
                        if(!inHashMap(tmpNextTime, tmpNextPosition, pos.line+1))
                            q.push({tmpNextTime, pos.line+1, i+1, tmpNextPosition});
                    } else {
                        double tmpTime1 = (lanes[pos.line+1][i].second - nextCar+1)/(v[pos.line] - v[pos.line+1]);
                        double tmpNextPosition = nextCar + v[pos.line]*tmpTime1 - 1;
                        if(!inHashMap(tmpTime1, tmpNextPosition, pos.line+1))
                            q.push({tmpTime1, pos.line+1, i+1, tmpNextPosition});
                    }
                }
//                break;
            }
        }
    }

    return 0.0;
}


int main() {
    std::ios_base::sync_with_stdio(false);
    cin >> L >> v0 >> v[0] >> v[1] >> v[2];

    readLane(lanes[0]);
    readLane(lanes[1]);
    readLane(lanes[2], true);

//    printf("%.*f\n", 10, findFastestWay());
    printf("%.13g\n", findFastestWay());
    return 0;
}