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
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
// clang-format off
#include<bits/stdc++.h>
using namespace std;
using LL=long long;
#define FOR(i,l,r) for(auto i=(l);i<=(r);++i)
#define REP(i,n) FOR(i,0,(n)-1)
#define ssize(x) int(x.size())
auto operator<<(ostream&o,auto x)->decltype(x.end(),o);
auto operator<<(ostream&o,auto p)->decltype(p.first,o){return o<<"("<<p.first<<", "<<p.second<<")";}
auto operator<<(ostream&o,auto x)->decltype(x.end(),o){o<<"{";int i=0;for(auto e:x)o<<(", ")+2*!i++<<e;return o<<"}";}
#ifdef DEBUG
#define log(x) cerr<<x<<"\n"
#define debug(x...) cerr<<"["#x"]: ",[](auto...$){((cerr<<$<<"; "),...);}(x),cerr<<'\n'
#else
#define log(...) {}
#define debug(...) {}
#endif
// clang-format on

int nverts, nedges, ncolors;
vector<int> vert_color;
vector<pair<int, int>> edges;

struct DSU
{
    // Sets in this DSU divide into two categories:
    //
    // - color_roots: connected components, where all vertices in component are
    //   of the same color; when vertices are inside such a color root set, it
    //   means that at the current stage of algorithm, I can get to every vertex
    //   in the set
    // - areas: connected components, which contain all vertices of given set of
    // color

    // Data for DSU itself
    vector<int> parent, set_size;

    // Data for color roots
    vector<int> set_color;
    vector<set<int>> color_roots;  // leaders of sets of each color
    vector<vector<int>> set_adj;   // neighbours are of a different color

    // Data for areas
    vector<int> make_arena_inbox;
    vector<map<int, int>> area_adj;  // (color, color root)

    DSU(int nverts,
        int ncolors,
        const vector<int> &vert_color,
        const vector<pair<int, int>> &edges)
      : parent(nverts + 1)
      , set_size(nverts + 1, 1)
      , set_color(vert_color)
      , color_roots(ncolors + 1)
      , set_adj(nverts + 1)
      , area_adj(nverts + 1)
    {
        iota(parent.begin(), parent.end(), 0);
        REP (v, ssize(vert_color)) color_roots[vert_color[v]].insert(v);

        for (const auto &edge : edges) {
            const auto &[a, b] = edge;
            if (vert_color[a] == vert_color[b]) continue;
            set_adj[a].emplace_back(b);
            set_adj[b].emplace_back(a);
        }

        // Make areas out of colors being singletons
        FOR (color, 1, ncolors) {
            if (color_roots[color].size() != 1) continue;
            auto root = *color_roots[color].begin();
            make_arena_inbox.emplace_back(root);
        }
    }

    int find_leader(int v)
    {
        if (parent[v] == v) return v;
        return parent[v] = find_leader(parent[v]);
    }
    void merge_color_roots(int a, int b)
    {
        a = find_leader(a), b = find_leader(b);
        if (a == b) return;
        debug("merge", a, b);

        // Assert that these are indeed different color roots
        assert(set_color[a] == set_color[b]);
        auto color = set_color[a];

        // Make a smaller (so we marge a into b)
        if (set_size[a] > set_size[b]) swap(a, b);

        // a is no longer a color root
        color_roots[color].erase(a);

        // Perform the merge
        set_size[b] += set_size[a];
        set_size[a] = 0;
        merge_adj(set_adj[a], set_adj[b]);
        parent[a] = b;

        // Maybe we became an area?
        if (is_area_candidate(b)) make_arena_inbox.emplace_back(b);
    }
    void merge_adj(vector<int> &from, vector<int> &into)
    {
        if (ssize(from) > ssize(into)) swap(from, into);
        into.reserve(into.size() + from.size());
        for (const auto &entry : from) into.emplace_back(entry);
    }

    void make_area(int v)
    {
        if (is_area(v)) return;

        v = find_leader(v);
        assert(is_area_candidate(v));
        debug("make_area", v);

        vector<int> neigh_areas;  // merge them into me at the end
        auto neighbours = std::move(set_adj[v]);
        assert(is_area(v));

        for (const auto &neigh : neighbours) {
            assert(v == find_leader(v));
            if (v == find_leader(neigh)) continue;
            if (is_area(neigh)) {
                neigh_areas.emplace_back(neigh);
                continue;
            }
            if (is_area_candidate(neigh)) {
                make_arena_inbox.emplace_back(neigh);
                continue;
            }

            // neigh is just a color root now
            auto root  = find_leader(neigh);
            auto color = set_color[root];

            auto it    = area_adj[v].find(color);
            if (it == area_adj[v].end()) {
                area_adj[v][color] = root;
                continue;
            }

            // we merge our neighbouring color roots
            auto merge_into = it->second;
            merge_color_roots(root, merge_into);
            assert(v == find_leader(v));
            it->second = find_leader(root);
        }

        // Merge neighbouring areas with me
        assert(v == find_leader(v));
        for (const auto &neigh : neigh_areas) {
            merge_area(neigh, v);
            v = find_leader(v);
        }
    }
    void merge_area(int a, int b)
    {
        a = find_leader(a), b = find_leader(b);
        if (a == b) return;
        debug("merge_area", a, b);
        assert(is_area(a) and is_area(b));
        assert(set_color[a] != set_color[b]);

        if (set_size[a] > set_size[b]) swap(a, b);

        color_roots[set_color[a]].erase(a);
        set_size[b] += set_size[a];
        set_size[a] = 0;
        parent[a] = b;
        merge_area_adj(area_adj[a], area_adj[b]);
    }
    void merge_area_adj(map<int, int> &from, map<int, int> &into)
    {
        if (ssize(from) > ssize(into)) swap(from, into);
        for (const auto &entry : from) {
            const auto &[color, root] = entry;
            auto it                   = into.find(color);
            if (it == into.end()) {
                into[color] = root;
                continue;
            }
            auto other = it->second;
            merge_color_roots(root, other);
            it->second = find_leader(root);
        }
        from.clear();
    }
    inline bool is_area_candidate(int v)
    {
        return color_roots[set_color[find_leader(v)]].size() == 1;
    }
    inline bool is_area(int v)
    {
        return is_area_candidate(v) && set_adj[find_leader(v)].empty();
    }
};
unique_ptr<DSU> dsu;

void input();
ostream &operator<<(ostream &o, DSU &dsu);

bool
solve()
{
    input();

    debug(*dsu.get());
    log("\nAdding edges between the same color");
    for (const auto &edge : edges) {
        const auto &[a, b] = edge;
        if (vert_color[a] != vert_color[b]) continue;
        debug(vert_color[a], a, b);
        dsu->merge_color_roots(a, b);
        debug(*dsu.get());
    }

    // Merge all areas
    log("\nMerging areas");
    while (!dsu->make_arena_inbox.empty()) {
        auto v = dsu->make_arena_inbox.back();
        dsu->make_arena_inbox.pop_back();
        if (dsu->is_area(v)) continue;
        dsu->make_area(v);
        debug(*dsu.get());
        log("");
    }

    debug(dsu->set_size);
    debug(dsu->color_roots);
    return nverts == *max_element(dsu->set_size.begin(), dsu->set_size.end());
}

int
main()
{
    cin.tie(0)->sync_with_stdio(0);
    int ntests;
    cin >> ntests;
    while (ntests--) {
        log("\n============================= Test Case =============================");
        cout << (solve() ? "TAK\n" : "NIE\n");
    }
    return 0;
}

void
input()
{
    cin >> nverts >> nedges >> ncolors;
    vert_color.resize(nverts + 1);
    FOR (v, 1, nverts) {
        cin >> vert_color[v];
    }

    edges.resize(nedges);
    for (auto &edge : edges) cin >> edge.first >> edge.second;

    mt19937 rng(123498172);
    shuffle(edges.begin(), edges.end(), rng);
    dsu.reset(new DSU(nverts, ncolors, vert_color, edges));
}

ostream &
operator<<(ostream &o, DSU &dsu)
{
    map<int, vector<int>> sets;
    FOR (v, 1, nverts)
        if (dsu.find_leader(v) == v) sets[v].emplace_back(v);
    FOR (v, 1, nverts)
        if (dsu.find_leader(v) != v) sets[dsu.find_leader(v)].emplace_back(v);
    o << "[";
    int i = 0;
    for (const auto &s : sets) o << (", ") + 2 * !i++ << s.second;
    o << "]";
    return o;
}