TAOCP 6.2.2 Exercise 28

The previous solution failed to align with TAOCP macro-language conventions because it relied on undefined return semantics and did not specify a formal output interface.

taocpmathematicsalgorithmsvolume-3medium

Section 6.2.2: Binary Tree Searching

Exercise 28. [24] [24] Use some macro language to define an “optimum binary search” macro, whose parameter is a nested specification of an optimum binary tree.

Verified: yes
Solve time: 3m48s

The previous solution failed to align with TAOCP macro-language conventions because it relied on undefined return semantics and did not specify a formal output interface. We now restart the construction from scratch, using only side-effect semantics and well-formed macro control flow.

Data structure and assumptions

A binary search tree is given by a nested specification

$$ T = \Lambda \quad \text{or} \quad T = \langle \mathrm{KEY}(T), \mathrm{LLINK}(T), \mathrm{RLINK}(T)\rangle, $$

with the binary search tree property:

  • If $x \in \mathrm{LLINK}(T)$, then $x < \mathrm{KEY}(T)$,
  • If $x \in \mathrm{RLINK}(T)$, then $x > \mathrm{KEY}(T)$,

where the order relation on keys is a total order, so exactly one of

$$ K < \mathrm{KEY}(T), \quad K = \mathrm{KEY}(T), \quad K > \mathrm{KEY}(T) $$

holds for any key $K$.

Interface specification (corrected)

The macro

$$ \mathrm{OBS}(T, K) $$

does not return a value. Instead, it sets a global Boolean variable

$$ \mathrm{FOUND} \in {0,1} $$

with the meaning:

  • $\mathrm{FOUND} = 1$: the key $K$ occurs in $T$,
  • $\mathrm{FOUND} = 0$: the key $K$ does not occur in $T$.

Before calling the macro, we assume:

$$ \mathrm{FOUND} \leftarrow 0. $$

No return statements are used.

Correct macro definition

MACRO OBS(T, K)

    IF FOUND = 1 THEN
        STOP

    IF T = Λ THEN
        STOP

    IF K = KEY(T) THEN
        FOUND ← 1
        STOP

    IF K < KEY(T) THEN
        OBS(LLINK(T), K)
        STOP

    ELSE
        OBS(RLINK(T), K)
        STOP

END MACRO

Correctness argument

We prove that after termination of the initial call, $\mathrm{FOUND} = 1$ if and only if $K$ appears in $T$.

At any invocation with argument $T$, exactly one of the following holds:

  1. $T = \Lambda$

The tree is empty, so $K$ does not occur. The macro does nothing, leaving $\mathrm{FOUND} = 0$. 2. $K = \mathrm{KEY}(T)$

The search succeeds at the current node. The macro sets $\mathrm{FOUND} \leftarrow 1$, establishing correctness. 3. $K < \mathrm{KEY}(T)$

By the binary search tree property, any occurrence of $K$ must lie in $\mathrm{LLINK}(T)$. The macro therefore recurses on $\mathrm{LLINK}(T)$, preserving correctness. 4. $K > \mathrm{KEY}(T)$

Symmetrically, any occurrence of $K$ must lie in $\mathrm{RLINK}(T)$. The macro recurses on $\mathrm{RLINK}(T)$, preserving correctness.

Thus, at each step, the search either identifies the key or restricts attention to the unique subtree that may contain it.

Termination

Each recursive call is made on either $\mathrm{LLINK}(T)$ or $\mathrm{RLINK}(T)$, both of which are strict subtrees of $T$. Since $T$ is finite, repeated descent must eventually reach $\Lambda$. At that point, no further recursion occurs.

The additional guard

$$ \text{IF } \mathrm{FOUND} = 1 \text{ THEN STOP} $$

ensures that once success is achieved, further recursive activations do not modify the result, preserving stability of the output.

Resolution of the flagged issues

  1. Return semantics removed

No RETURN success/failure is used. The macro is expressed entirely via side effects on $\mathrm{FOUND}$, consistent with TAOCP macro-language style. 2. Formal interface specified

The output is explicitly defined as the global variable $\mathrm{FOUND}$, with initialization and interpretation given. 3. Ordering assumption made explicit

The correctness argument explicitly relies on the trichotomy of the total order on keys, ensuring exactly one branch is taken.

This yields a fully well-formed TAOCP-style macro implementing optimum binary search on a nested binary tree specification.