2.2.3 Skew Heap

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/*

Maintains a mergeable priority queue, returning the minimum element by default. The comparator
`comp` defines priority: `comp(a, b)` is true when `a` has higher priority than `b`, so
`std::greater<T>` instead creates a max-priority queue. A skew heap attempts to maintain balance by
unconditionally swapping all nodes in the merge path when merging. Every other operation reduces to
this merge: insertion merges a one-node heap, and extraction merges the root's two subtrees. To
customize the ordering, instantiate `SkewHeap<T, Compare>` and pass the comparator to either
constructor.

- `SkewHeap<T>()` constructs an empty priority queue.
- `SkewHeap<T>(lo, hi)` constructs a priority queue from the elements in the half-open iterator
  range $[`lo`, `hi`)$.
- `size()` returns the size of the priority queue.
- `empty()` returns whether the priority queue is empty.
- `push(v)` inserts the value `v` into the priority queue.
- `pop()` removes the highest-priority element from the priority queue.
- `top()` returns the highest-priority element in the priority queue.
- `absorb(h)` inserts every value from the distinct heap `h` and sets `h` to the empty priority
  queue. Both heaps must use equivalent comparators.

Time Complexity:
- O(1) per call to the first constructor, `size()`, `empty()`, and `top()`.
- O(log n) amortized per call to `push()`, `pop()`, and `absorb()`, where $n$ is the number of
  elements in the priority queue.
- O(n log n) amortized per call to the second constructor, where $n$ is the distance between `lo`
  and `hi`.

Space Complexity:
- O(n) for storage of the priority queue elements.
- O(n) auxiliary stack space in the worst case for `push()`, `pop()`, `absorb()`, and destruction.
- O(1) auxiliary for all other operations.

*/

#include <cassert>
#include <functional>
#include <utility>

template<typename T, typename Compare = std::less<T>>
class SkewHeap {
  struct Node {
    T value;
    Node *left, *right;

    explicit Node(const T &v) : value(v), left(nullptr), right(nullptr) {}
  } *root;

  int num_nodes;
  Compare comp;

  Node *merge(Node *a, Node *b) {
    if (a == nullptr) {
      return b;
    }
    if (b == nullptr) {
      return a;
    }
    if (comp(b->value, a->value)) {
      std::swap(a, b);
    }
    std::swap(a->left, a->right);
    a->left = merge(b, a->left);
    return a;
  }

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

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

  template<typename It>
  SkewHeap(It lo, It hi, Compare comp = Compare{})
      : root(nullptr), num_nodes(0), comp(std::move(comp)) {
    while (lo != hi) {
      push(*(lo++));
    }
  }

  ~SkewHeap() { clean_up(root); }
  SkewHeap(const SkewHeap &) = delete;
  SkewHeap &operator=(const SkewHeap &) = delete;
  int size() const { return num_nodes; }
  bool empty() const { return root == nullptr; }

  void push(const T &v) {
    root = merge(root, new Node(v));
    num_nodes++;
  }

  void pop() {
    assert(!empty());
    Node *tmp = root;
    root = merge(root->left, root->right);
    delete tmp;
    num_nodes--;
  }

  const T &top() const {
    assert(!empty());
    return root->value;
  }

  void absorb(SkewHeap &h) {
    assert(this != &h);
    root = merge(root, h.root);
    num_nodes += h.num_nodes;
    h.root = nullptr;
    h.num_nodes = 0;
  }
};

/*** Example Usage ***/

#include <vector>
using namespace std;

int main() {
  SkewHeap<int> h, h2;
  assert(h.empty());
  h.push(12);
  h.push(10);
  h2.push(5);
  h2.push(-1);
  h2.push(0);
  h.absorb(h2);
  assert(h.size() == 5);
  assert(h2.empty());
  vector<int> popped;
  while (!h.empty()) {
    popped.push_back(h.top());
    h.pop();
  }
  assert((popped == vector<int>{-1, 0, 5, 10, 12}));

  SkewHeap<int, greater<int>> max_heap;
  max_heap.push(1);
  max_heap.push(3);
  assert(max_heap.top() == 3);
  return 0;
}