2.3.1 Treap

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
282
283
284
285
286
287
288
289
290
291
292
293

/*

Maintain an ordered map, that is, an ordered collection of key-value pairs such that each possible
key appears at most once in the collection. A binary search tree (a.k.a. BST) implements this map by
inserting and deleting keys into a binary tree while upholding the BST property: for each node in
the BST, every node in its left subtree has a lesser key and every node in its right subtree has a
greater key.

A treap is a BST where each node also holds a randomly assigned priority, simultaneously maintaining
both the BST property on keys and the min-heap property on priorities (lower priority value is
closer to root). Since priorities are random, this keeps the tree balanced with high probability,
making insertions and deletions run in O(log n) on average.

The comparator `comp` defines the key ordering: `comp(a, b)` is true when `a` precedes `b`. It
defaults to `std::less<K>`; to customize the ordering, instantiate `Treap<K, V, Compare>` and pass
the comparator to the constructor.

- `Treap<K, V>()` constructs an empty map.
- `size()` returns the size of the map.
- `empty()` returns whether the map is empty.
- `insert(k, v)` adds an entry with key `k` and value `v` to the map, returning `true` if a new
  entry was added or `false` if the key already exists (in which case the map is unchanged and the
  old value associated with the key is preserved).
- `erase(k)` removes the entry with key `k` from the map, returning `true` if the removal was
  successful or `false` if the key to be removed was not found.
- `find(k)` returns a pointer to a const value associated with key `k`, or `nullptr` if the key was
  not found.
- `entries()` returns all key-value entries in comparator order.
- `min()` and `max()` return the first and last entries in comparator order, or `std::nullopt` if
  the map is empty.
- `lower_bound(k)`, `upper_bound(k)`, `prev(k)`, and `next(k)` return the matching entry according
  to the comparator, or `std::nullopt` if no such entry exists. These navigation routines depend
  only on the BST property and can be adapted to the other BST variants later in this chapter.

Time Complexity:
- O(1) per call to the constructor, `size()`, and `empty()`.
- O(log n) on average per call to `insert()`, `erase()`, `find()`, `min()`, `max()`,
  `lower_bound()`, `upper_bound()`, `prev()`, and `next()`, where $n$ is the number of entries
  currently in the map.
- O(n) per call to `entries()`.

Space Complexity:
- O(n) for storage of the map elements.
- O(log n) auxiliary stack space on average for `insert()`, `erase()`, `entries()`, and destruction,
  and O(n) in the worst case.
- O(n) for the vector returned by `entries()`.
- O(1) auxiliary for all other operations.

*/

#include <cstdint>
#include <functional>
#include <optional>
#include <random>
#include <utility>
#include <vector>

template<typename K, typename V, typename Compare = std::less<K>>
class Treap {
  struct Node {
    static uint32_t rand32() {
      static uint32_t x = std::random_device{}() | 1U;
      x ^= x << 13;
      x ^= x >> 17;
      x ^= x << 5;
      return x;
    }

    K key;
    V value;
    uint32_t priority;
    Node *left, *right;

    Node(const K &k, const V &v)
        : key(k), value(v), priority(rand32()), left(nullptr), right(nullptr) {}
  } *root;

  int num_nodes;
  Compare comp;

  static void rotate_left(Node *&n) {
    Node *tmp = n;
    n = n->right;
    tmp->right = n->left;
    n->left = tmp;
  }

  static void rotate_right(Node *&n) {
    Node *tmp = n;
    n = n->left;
    tmp->left = n->right;
    n->right = tmp;
  }

  bool insert(Node *&n, const K &k, const V &v) {
    if (n == nullptr) {
      n = new Node(k, v);
      num_nodes++;
      return true;
    }
    if (comp(k, n->key) && insert(n->left, k, v)) {
      if (n->left->priority < n->priority) {
        rotate_right(n);
      }
      return true;
    }
    if (comp(n->key, k) && insert(n->right, k, v)) {
      if (n->right->priority < n->priority) {
        rotate_left(n);
      }
      return true;
    }
    return false;
  }

  bool erase(Node *&n, const K &k) {
    if (n == nullptr) {
      return false;
    }
    if (comp(k, n->key)) {
      return erase(n->left, k);
    } else if (comp(n->key, k)) {
      return erase(n->right, k);
    }
    if (n->left != nullptr && n->right != nullptr) {
      if (n->left->priority < n->right->priority) {
        rotate_right(n);
        return erase(n->right, k);
      }
      rotate_left(n);
      return erase(n->left, k);
    }
    Node *tmp = (n->left != nullptr) ? n->left : n->right;
    delete n;
    n = tmp;
    num_nodes--;
    return true;
  }

  static void collect_entries(Node *n, std::vector<std::pair<K, V>> &res) {
    if (n != nullptr) {
      collect_entries(n->left, res);
      res.emplace_back(n->key, n->value);
      collect_entries(n->right, res);
    }
  }

  static void clean_up(Node *n) {
    if (n != nullptr) {
      clean_up(n->left);
      clean_up(n->right);
      delete n;
    }
  }

 public:
  explicit Treap(Compare comp = Compare{}) : root(nullptr), num_nodes(0), comp(std::move(comp)) {}

  ~Treap() { clean_up(root); }
  Treap(const Treap &) = delete;
  Treap &operator=(const Treap &) = delete;
  int size() const { return num_nodes; }
  bool empty() const { return root == nullptr; }
  bool insert(const K &k, const V &v) { return insert(root, k, v); }
  bool erase(const K &k) { return erase(root, k); }

  const V *find(const K &k) const {
    Node *n = root;
    while (n != nullptr) {
      if (comp(k, n->key)) {
        n = n->left;
      } else if (comp(n->key, k)) {
        n = n->right;
      } else {
        return &(n->value);
      }
    }
    return nullptr;
  }

  std::vector<std::pair<K, V>> entries() const {
    std::vector<std::pair<K, V>> res;
    res.reserve(num_nodes);
    collect_entries(root, res);
    return res;
  }

  std::optional<std::pair<K, V>> min() const {
    Node *n = root;
    if (n == nullptr) {
      return std::nullopt;
    }
    while (n->left != nullptr) {
      n = n->left;
    }
    return std::pair<K, V>{n->key, n->value};
  }

  std::optional<std::pair<K, V>> max() const {
    Node *n = root;
    if (n == nullptr) {
      return std::nullopt;
    }
    while (n->right != nullptr) {
      n = n->right;
    }
    return std::pair<K, V>{n->key, n->value};
  }

  std::optional<std::pair<K, V>> lower_bound(const K &k) const {
    Node *n = root, *best = nullptr;
    while (n != nullptr) {
      if (!comp(n->key, k)) {
        best = n;
        n = n->left;
      } else {
        n = n->right;
      }
    }
    return best ? std::make_optional(std::pair{best->key, best->value}) : std::nullopt;
  }

  std::optional<std::pair<K, V>> upper_bound(const K &k) const {
    Node *n = root, *best = nullptr;
    while (n != nullptr) {
      if (comp(k, n->key)) {
        best = n;
        n = n->left;
      } else {
        n = n->right;
      }
    }
    return best ? std::make_optional(std::pair{best->key, best->value}) : std::nullopt;
  }

  std::optional<std::pair<K, V>> prev(const K &k) const {
    Node *n = root, *best = nullptr;
    while (n != nullptr) {
      if (comp(n->key, k)) {
        best = n;
        n = n->right;
      } else {
        n = n->left;
      }
    }
    return best ? std::make_optional(std::pair{best->key, best->value}) : std::nullopt;
  }

  std::optional<std::pair<K, V>> next(const K &k) const { return upper_bound(k); }
};

/*** Example Usage ***/

#include <cassert>
using namespace std;

int main() {
  Treap<int, char> t;
  t.insert(2, 'b');
  t.insert(1, 'a');
  t.insert(3, 'c');
  t.insert(5, 'e');
  assert(t.insert(4, 'd'));
  assert(*t.find(4) == 'd');
  assert(!t.insert(4, 'd'));
  assert(t.min()->first == 1 && t.min()->second == 'a');
  assert(t.max()->first == 5 && t.max()->second == 'e');
  assert(t.lower_bound(4)->first == 4);
  assert(t.upper_bound(4)->first == 5);
  assert(t.prev(4)->first == 3);
  assert(t.next(4)->first == 5);
  assert(!t.prev(1));
  assert(!t.next(5));
  assert(
      (t.entries() == vector<pair<int, char>>{{1, 'a'}, {2, 'b'}, {3, 'c'}, {4, 'd'}, {5, 'e'}})
  );
  assert(t.erase(1));
  assert(!t.erase(1));
  assert(t.find(1) == nullptr);
  assert((t.entries() == vector<pair<int, char>>{{2, 'b'}, {3, 'c'}, {4, 'd'}, {5, 'e'}}));

  Treap<int, char, greater<int>> descending;
  for (int key : {2, 1, 3}) {
    descending.insert(key, '0' + key);
  }
  assert((descending.entries() == vector<pair<int, char>>{{3, '3'}, {2, '2'}, {1, '1'}}));
  assert(descending.min()->first == 3 && descending.max()->first == 1);
  assert(descending.lower_bound(2)->first == 2);
  assert(descending.upper_bound(2)->first == 1);
  assert(descending.prev(2)->first == 3 && descending.next(2)->first == 1);
  assert(descending.erase(2) && descending.find(2) == nullptr);
  return 0;
}