opBinTree.H

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/*
                          Aleph_w
 
  Data structures & Algorithms
  version 2.0.0b
  https://github.com/lrleon/Aleph-w
 
  This file is part of Aleph-w library
 
  Copyright (c) 2002-2026 Leandro Rabindranath Leon
 
  Permission is hereby granted, free of charge, to any person obtaining a copy
  of this software and associated documentation files (the "Software"), to deal
  in the Software without restriction, including without limitation the rights
  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
  copies of the Software, and to permit persons to whom the Software is
  furnished to do so, subject to the following conditions:
 
  The above copyright notice and this permission notice shall be included in all
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  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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*/
 
 
# ifndef OPBINTREE_H 
# define OPBINTREE_H
 
# include <limits>
# include <tpl_dynArray.H>
 
namespace Aleph {
 
// Internal macros for 2D array access using 1D storage
# define COST(i,j) (cost[(i)*(n+1) + (j)])
# define TREE(i,j) (tree[(i)*(n+1) + (j)])
 
static inline void compute_optimal_costs(DynArray<double> & cost, double p[], 
                                         const size_t n, DynArray<int> & tree)
{
  // Precompute prefix sums for O(1) range sum queries
  // prefix[i] = p[0] + p[1] + ... + p[i-1], prefix[0] = 0
  DynArray<double> prefix(n + 2);
  prefix[0] = 0.0;
  for (size_t i = 0; i < n; ++i)
    prefix[i + 1] = prefix[i] + p[i];
 
  // sum_p(i, j) = prefix[j] - prefix[i-1] for 1-indexed i,j
  auto sum_p = [&prefix](size_t i, size_t j) noexcept -> double {
    return prefix[j] - prefix[i - 1];
  };
 
  // Base cases: empty intervals have zero cost
  for (size_t i = 1; i <= n + 1; ++i)
    COST(i, i - 1) = 0;
 
  // Base cases: single-element intervals - the only choice is the element itself
  for (size_t i = 1; i <= n; ++i)
    {
      TREE(i, i) = static_cast<int>(i);
      COST(i, i) = p[i - 1];  // Cost is just the probability (depth = 1)
    }
 
  // Fill DP table by increasing interval length
  // Knuth's optimization: search only in [TREE(i, j-1), TREE(i+1, j)]
  for (size_t len = 2; len <= n; ++len)  // len = j - i + 1
    {
      for (size_t i = 1; i + len - 1 <= n; ++i)
        {
          const size_t j = i + len - 1;
 
          // Knuth bounds: root[i,j-1] <= root[i,j] <= root[i+1,j]
          const size_t lo = static_cast<size_t>(TREE(i, j - 1));
          const size_t hi = static_cast<size_t>(TREE(i + 1, j));
 
          double min_cost = std::numeric_limits<double>::max();
          size_t best_root = lo;
 
          for (size_t r = lo; r <= hi; ++r)
            {
              const double c = COST(i, r - 1) + COST(r + 1, j);
              if (c < min_cost)
                {
                  min_cost = c;
                  best_root = r;
                }
            }
 
          TREE(i, j) = static_cast<int>(best_root);
          COST(i, j) = min_cost + sum_p(i, j);
        }
    }
}
 
template <class Node, typename Key>
[[nodiscard]] static inline Node * compute_tree(Key keys[], DynArray<int> & tree, 
                                                const size_t n, 
                                                const size_t i, const size_t j)
{
  if (i > j) 
    return Node::NullPtr;
 
  const int root_idx = TREE(i, j);
  Node * root = new Node(keys[root_idx - 1]);
  LLINK(root) = compute_tree<Node, Key>(keys, tree, n, i, root_idx - 1);
  RLINK(root) = compute_tree<Node, Key>(keys, tree, n, root_idx + 1, j);
  return root;
}
 
template <class Node, typename Key>
[[nodiscard]] Node * build_optimal_tree(Key keys[], double p[], const size_t n)
{
  DynArray<int> tree((n + 1) * (n + 1));
  DynArray<double> cost((n + 1) * (n + 1));
  compute_optimal_costs(cost, p, n, tree);
  return compute_tree<Node, Key>(keys, tree, n, 1, n);
}
 
 
# undef COST
# undef TREE
} // end namespace Aleph
# endif // OPBINTREE_H