TAOCP 6.3 Exercise 12
Let $T_n$ be a digital search tree constructed by Algorithm D from $n$ keys $K_1,\dots,K_n$, where the keys are independent infinite binary sequences with each bit independently $0$ or $1$ with probab...
Section 6.3: Digital Searching
Exercise 12. [25] [25] After a random element is deleted from a random digital search tree constructed by Algorithm D, is the resulting tree still random? (See exercise 11 and Theorem 6.2.2H.)
Verified: yes
Solve time: 2m04s
Let $T_n$ be a digital search tree constructed by Algorithm D from $n$ keys $K_1,\dots,K_n$, where the keys are independent infinite binary sequences with each bit independently $0$ or $1$ with probability $1/2$. By definition, $T_n$ is a random digital search tree.
We delete a random element from $T_n$, meaning we first choose an index $I$ uniformly from ${1,\dots,n}$, independent of the key values, and then delete the key $K_I$. Let the remaining set of keys be $S_{n-1}$, and let $T'$ be the digital search tree constructed from $S_{n-1}$ by Algorithm D.
We must determine whether $T'$ is distributed as a random digital search tree with $n-1$ keys.
Step 1: Distribution of the remaining keys
The original sequence $(K_1,\dots,K_n)$ is an i.i.d. family of random binary strings. Because the index $I$ is chosen uniformly and independently of the keys, the remaining set $S_{n-1}$ is obtained by deleting one element from an exchangeable collection.
By symmetry (exchangeability of i.i.d. variables), every $(n-1)$-subset of ${K_1,\dots,K_n}$ is equally likely to be the remaining set. Moreover, conditional on the values of the keys, deleting a uniformly chosen index produces a multiset distributed exactly as $n-1$ i.i.d. samples from the same distribution. Hence $S_{n-1}$ has the same distribution as ${K_1,\dots,K_{n-1}}$.
This is the key probabilistic fact used in Exercise 11 and Theorem 6.2.2H: random deletion from an exchangeable sample preserves the i.i.d. structure of the remaining sample.
Step 2: Effect on the digital search tree
A digital search tree constructed by Algorithm D depends only on the set of keys and not on their labeling or order. Therefore, the distribution of the tree built from a collection of keys is completely determined by the joint distribution of those keys.
Since $S_{n-1}$ has the same distribution as $n-1$ independent random binary strings, the tree $T'$ has the same distribution as a digital search tree constructed from $n-1$ independent keys.
Step 3: Conclusion
A random digital search tree on $n$ keys, after deleting a uniformly chosen random key, produces a tree distributed exactly as a random digital search tree on $n-1$ keys.
Therefore, the resulting tree is still random in the sense of TAOCP: it satisfies the same distributional model as Algorithm D applied to $n-1$ random inputs.
Final answer
Yes. After deleting a random element from a random digital search tree, the resulting tree is still a random digital search tree (with one fewer node), because the remaining keys are distributed as a set of $n-1$ independent random binary sequences, and Algorithm D depends only on this distribution.