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
#include <cstdio>
#include <vector>
#include <set>
#include <queue>
#include <algorithm>
#include <climits>

using namespace std;

typedef long long ll_t;

struct node_t {
    int parent;
    ll_t profit;
    ll_t to_parent_cost;
    ll_t subtree_maximum_profit;
    // id of node for which profit of traversing to is equal to subtree_maximum_profit
    int subtree_optimal_node;

    set<int> children;

    // initially stores all edge information - it will be used to build the rooted tree and populate children
    set<pair<int, ll_t> > neighbors;

    int level; // level in the rooted tree

    node_t(ll_t profit, int id) {
        this->profit = profit;
        this->subtree_maximum_profit = profit;
        this->subtree_optimal_node = id;
        this->parent = -1; // -1 means the parent is currently unknown, 0 means the node is the root
        this->level = LONG_MAX;
    }
};

// G[i] contains node of id = i
vector<node_t> G;

ll_t max_profit;
int min_optimal_node_id;

bool cmp_desc_by_level(int id1, int id2) {
    return G[id1].level > G[id2].level;
}

void build_rooted_tree() {
    queue<int> Q;
    G[1].parent = 0; // make node 1 the root
    G[1].level = 0;
    Q.push(1);

    while (!Q.empty()) {
        int parent = Q.front();
        Q.pop();
        for (set<pair<int, ll_t> >::iterator it = G[parent].neighbors.begin(); it != G[parent].neighbors.end(); it++) {
            if (G[it->first].parent == -1) { // parent unknown
                int child = it->first;
                G[parent].children.insert(child);
                G[child].parent = parent;
                G[child].level = G[parent].level + 1;
                G[child].to_parent_cost = it->second;
                Q.push(child);
            }
        }
    }
}

void calculate_maximum_profit(int id) {
    G[id].subtree_maximum_profit = G[id].profit;
    G[id].subtree_optimal_node = id;

    for (set<int>::iterator it = G[id].children.begin(); it != G[id].children.end(); it++) {
        int child_id = *it;
        if (G[child_id].subtree_maximum_profit - G[child_id].to_parent_cost > G[id].subtree_maximum_profit
                || (G[child_id].subtree_maximum_profit - G[child_id].to_parent_cost == G[id].subtree_maximum_profit
                    && G[child_id].subtree_optimal_node < G[id].subtree_optimal_node)) {
            G[id].subtree_maximum_profit = G[child_id].subtree_maximum_profit - G[child_id].to_parent_cost;
            G[id].subtree_optimal_node = G[child_id].subtree_optimal_node;
        }
    }
}

void update_node_weight(int id, ll_t new_weight) {
    G[id].profit = new_weight;
    if (new_weight > G[id].subtree_maximum_profit) {
        G[id].subtree_maximum_profit = new_weight;
        G[id].subtree_optimal_node = id;
    } else {
        G[id].subtree_maximum_profit = new_weight;
        G[id].subtree_optimal_node = id;
        for (set<int>::iterator it = G[id].children.begin(); it != G[id].children.end(); it++) {
            int child_id = *it;
            if (G[child_id].subtree_maximum_profit - G[child_id].to_parent_cost > G[id].subtree_maximum_profit
                    || (G[child_id].subtree_maximum_profit - G[child_id].to_parent_cost == G[id].subtree_maximum_profit
                        && G[child_id].subtree_optimal_node < G[id].subtree_optimal_node)) {
                G[id].subtree_maximum_profit = G[child_id].subtree_maximum_profit - G[child_id].to_parent_cost;
                G[id].subtree_optimal_node = G[child_id].subtree_optimal_node;
            }
        }
    }
    bool profit_updated = true;
    int child_id = id;
    id = G[id].parent;
    while (id != 0 && profit_updated) {
        ll_t candidate_profit = G[child_id].subtree_maximum_profit - G[child_id].to_parent_cost;
        if (candidate_profit > G[id].subtree_maximum_profit ||
                (candidate_profit == G[id].subtree_maximum_profit && G[child_id].subtree_optimal_node < G[id].subtree_optimal_node)) {
            G[id].subtree_maximum_profit = G[child_id].subtree_maximum_profit - G[child_id].to_parent_cost;
            G[id].subtree_optimal_node = G[child_id].subtree_optimal_node;
        } else {
            profit_updated = false;
        }
        child_id = id;
        id = G[id].parent;
    }
}

void update_edge_weight(int id, int child_id, ll_t new_weight) {
    G[child_id].to_parent_cost = new_weight;
    while (id != 0) {
        calculate_maximum_profit(id);
        child_id = id;
        id = G[id].parent;
    }
}

void candidate(ll_t profit, int id) {
    if (profit > max_profit || (profit == max_profit && id < min_optimal_node_id)) {
        max_profit = profit;
        min_optimal_node_id = id;
    }
}

int next_optimal_node(int starting_node) {
    max_profit = LLONG_MIN;
    min_optimal_node_id = LONG_MAX;

    candidate(G[starting_node].subtree_maximum_profit, G[starting_node].subtree_optimal_node);

    int child_id = starting_node;
    int id = G[starting_node].parent;
    ll_t path_cost = 0LL;

    while (id != 0) {
        path_cost += G[child_id].to_parent_cost;
        candidate(G[id].subtree_maximum_profit - path_cost, G[id].subtree_optimal_node);
        child_id = id;
        id = G[id].parent;
    }

    if (min_optimal_node_id == starting_node) {
        // search again, this time ignoring starting_node
        max_profit = LLONG_MIN;
        min_optimal_node_id = LONG_MAX;
        for (set<int>::iterator it = G[starting_node].children.begin(); it != G[starting_node].children.end(); it++) {
            child_id = *it;
            candidate(G[child_id].subtree_maximum_profit - G[child_id].to_parent_cost, G[child_id].subtree_optimal_node);
        }
        child_id = starting_node;
        id = G[starting_node].parent;
        path_cost = 0LL;
        while (id != 0) {
            path_cost += G[child_id].to_parent_cost;
            candidate(G[id].profit - path_cost, id);
            for (set<int>::iterator it = G[id].children.begin(); it != G[id].children.end(); it++) {
                if (*it != child_id) {
                    int other_child_id = *it;
                    candidate(G[other_child_id].subtree_maximum_profit - path_cost - G[other_child_id].to_parent_cost, G[other_child_id].subtree_optimal_node);
                }
            }
            child_id = id;
            id = G[id].parent;
        }
    }
    return min_optimal_node_id;
}

int main() {

    int n, q;
    scanf("%d%d", &n, &q);
    G.push_back(node_t(0, 0)); // dummy node of id = 0
    for (int i = 1; i <= n; i++) {
        ll_t z;
        scanf("%lld", &z);
        G.push_back(node_t(z, i));
    }
    for (int i = 0; i < n - 1; i++) {
        int a, b;
        ll_t c;
        scanf("%d%d%lld", &a, &b, &c);
        G[a].neighbors.insert(make_pair(b, c));
        G[b].neighbors.insert(make_pair(a, c));
    }

    build_rooted_tree();
    vector<int> node_ids;
    for (int id = 1; id <= n; id++) {
        node_ids.push_back(id);
    }

    sort(node_ids.begin(), node_ids.end(), cmp_desc_by_level);

    for (int i = 0; i < n; i++) {
        calculate_maximum_profit(node_ids[i]);
    }

    int current_node = 1;

    for (int i = 0; i < q; i++) {
        char type;
        int a, b, v;
        ll_t d;

        scanf("%d", &type);
        if (type == 1) {
            scanf("%d%lld", &v, &d);
            update_node_weight(v, d);
        } else {
            scanf("%d%d%lld", &a, &b, &d);
            // make sure a is parent of b
            if (G[b].parent != a) {
                swap(a, b);
            }
            update_edge_weight(a, b, d);
        }

        current_node = next_optimal_node(current_node);
        printf("%d ", current_node);
    }
    putchar('\n');

    return 0;
}