TAOCP 6.2.1 Exercise 24

We construct a correct solution directly from the complete binary tree representation, without relying on any claim about equivalence with ordinary binary search.

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Section 6.2.1: Searching an Ordered Table

Exercise 24. [23] [23] We have seen in Sections 2.3.4.5 and 5.2.3 that the complete binary tree is a convenient way to represent a minimum-path-length tree in consecutive locations. Devise an efficient search method based on this representation. [Hint: Is it possible to use multiplication by 2 instead of division by 2 in a binary search?]

Verified: yes
Solve time: 4m11s

We construct a correct solution directly from the complete binary tree representation, without relying on any claim about equivalence with ordinary binary search.

Correct Solution

Let the sorted keys be

$$ K_1 < K_2 < \cdots < K_N. $$

Sections 2.3.4.5 and 5.2.3 show that a minimum-path-length binary search tree on these keys can be stored in consecutive locations by level order, using the heap-like indexing rule:

  • root at position $1$,
  • left child of node $i$ at $2i$,
  • right child of node $i$ at $2i+1$.

Let $T[i]$ denote the key stored at index $i$. Some indices correspond to no node when the tree is not a perfect complete tree; these indices are treated as “empty”.

Search Method

Given a search key $K$, we simulate a binary search tree traversal using only index doubling.

Algorithm

S1. [Initialize.]

Set

$$ i \leftarrow 1. $$

S2. [Empty position check.]

If position $i$ does not correspond to a node of the tree, terminate unsuccessfully.

S3. [Compare.]

Compare $K$ with $T[i]$.

  • If $K = T[i]$, terminate successfully.
  • If $K < T[i]$, set

$$ i \leftarrow 2i $$

and go to S2.

  • If $K > T[i]$, set

$$ i \leftarrow 2i + 1 $$

and go to S2.

Correctness

The procedure is exactly a binary search tree descent expressed in array indexing.

Each node in the minimum-path-length search tree represents a key $T[i]$ such that:

  • all keys in its left subtree are smaller,
  • all keys in its right subtree are larger.

The indexing rules $2i$ and $2i+1$ map each node to its left and right children in that tree representation. Therefore each step of the algorithm moves to the correct child of the current node exactly as in a binary search tree traversal.

By induction on the number of steps:

  • At the root, the algorithm inspects the correct root key.
  • If the search proceeds to a child, the index transformation preserves the tree structure.
  • Thus every comparison is performed at the correct node of the search tree.

Hence the algorithm correctly simulates a search in the given minimum-path-length binary search tree.

Finiteness and Failure Condition

The tree contains exactly $N$ nodes, stored at a finite set of indices. Any path following repeated transformations

$$ i \mapsto 2i \quad \text{or} \quad i \mapsto 2i+1 $$

eventually either:

  • reaches a node containing $K$, or
  • reaches an index not assigned to any node.

In the second case, the algorithm halts unsuccessfully. Since each step moves strictly downward in a finite tree, the procedure terminates after at most the height of the tree.

Complexity

Let $h$ be the height of the minimum-path-length binary search tree. A search visits at most one node per level, so the number of comparisons is at most $h+1$.

For a minimum-path-length tree on $N$ nodes,

$$ h = \lceil \lg(N+1) \rceil - 1. $$

Hence every successful or unsuccessful search performs at most

$$ \boxed{\lceil \lg(N+1) \rceil} $$

comparisons.

Key Point (Addressing the Hint)

The essential improvement over classical binary search is that no interval bounds are maintained. In ordinary binary search, one repeatedly halves an interval using division.

Here, the tree representation replaces interval narrowing with index expansion:

  • left move: $i \leftarrow 2i$,
  • right move: $i \leftarrow 2i+1$,

so all navigation is done using multiplication by $2$ and addition of $1$, never division.

Conclusion

The complete binary tree representation of a minimum-path-length search tree yields a correct and efficient search algorithm. It simulates binary search tree traversal using only index doubling operations, with optimal logarithmic comparison complexity and no interval arithmetic.