Problem 8.1

Probabilistic lower bounds on comparison sorting

In this problem, we prove a probabilistic $\Omega(n\lg{n})$ on the running time of any deterministic or randomized comparison sort on $n$ distinct input elements. We begin by examining a deterministic comparison sort $A$ with decision tree $T_A$. We assume that every permutation of $A$'s inputs is equally likely.

1. Suppose that each leaf of $T_A$ is labeled with the probability that it is reached given a random input. Prove that exactly $n!$ leaves are labeled $1/n!$ and the rest are labeled $0$.
2. Let $D(T)$ denote the external path length of a decision tree $T$; that is, $D(T)$ is the sum of the depths of all the leaves of $T$. Let $T$ be a decision tree with $k > 1$ leaves, and let $LT$ and $RT$ be the left and right subtrees of $T$. Show that $D(T) = D(LT) + D(RT) + k$
3. Let $d(k)$ be the minimum value of $D(T)$ over all decision trees $T$ with $k > 1$ leaves. Show that $d(k) = \min_{1 \le i \le k - 1} \{d(i) + d(k-i) + k\}$. (Hint: Consider a decision tree $T$ with $k$ leaves that achieves the minimum. Let $i_0$ be the number of leaves in $LT$ and $k - i_0$ the number of leaves in $RT$.)
4. Prove that for a given value of $k > 1$ and $i$ in the range $i \le i \le k - 1$, the function $i\lg{i} + (k-i)\lg(k-i)$ is minimized at $i = k/2$. Conclude that $d(k) = \Omega(k\lg{k})$.
5. Prove that $D(T_A) = \Omega(n!\lg(n!))$, and conclude that the average-case time to sort $n$ elements is $\Omega(n\lg{n})$.

Now consider a randomized comparison sort $B$. We can extend the decision-tree model to handle randomization by incorporating two kinds of nodes: ordinary comparison nodes and "randomization" nodes. A randomization node models a random choice of the form RANDOM(1,r) made by algorithm $B$; the node has $r$ children, each of which is equally likely to be chosen during an execution of the algorithm.

1. Show that for any randomized comparison sort $B$, there exists a deterministic comparison sort $A$ whose expected number of comparisons is no more than those made by $B$.

Probability labels

There are $n!$ permutations that the algorithm can perform and each corresponds to one of the $n!$ possible inputs. Each permutation will be a leaf in this tree and since the inputs are equally likely, the probability of reaching one will be $1/n!$. If the decision tree has more leaves, they will be unreachable.

This is an intuitive argument. It's easy to see that it is so if you think about it, but a formal proof seems tricky to me.

External path length

If we take a node in the tree, all paths go either through the left or the right child. Both children have the same paths as the node itself, except that they are shorter by one. Thus, $D(T) = D(LT) + D(RT) + \mathrm{LEAVES}(LT) + \mathrm{LEAVES}(RT) = D(LT) + D(RT) + k$.

Minimal external path length

If we take a tree with $k$ leaves that achieves the minimal external path, we know from the previous point that:

$$D(T) = D(LT) + D(RT) + k$$

There are $k - 1$ possible pairs of left-right children and one of them is the minimum. That is:

$$d(k) = D(T) = D(LT) + D(RT) + k = \min_{1 \le i \le k-1}\{d(i) + d(k-i) + k\}$$

Minimal value

Let:

\begin{aligned} f(i) &= i\lg{i} + (k-i)\lg(k-i) \\ f'(i) &= \lg{i} + 1 - \lg(k-i) - 1 = \lg\frac{i}{k-i} \\ f'(i) = 0 & \Leftrightarrow \lg\frac{i}{k-i} = 0 \Rightarrow i/(k-i) = 1 \Rightarrow i = \frac k 2 \end{aligned}

Since $f'(i)$ is monotonously increasing, $i = k/2$ is a local minimum.

Intuitively said, the minimum is reached when the tree is balanced (as in the way merge sort halves is decision-tree on each step.

Average-case time

In $T_A$ there are $n!$ leaves, thus $D(n) > d(k) = \Omega(n!\lg(n!))$. Each permutation has an equal probability of $1/n!$, thus the expected time to sort it is:

$$\frac{\Omega(n!\lg(n!))}{n!} = \Omega(n\lg(n!)) = \Omega(n\lg{n})$$

The randomized algorithm

A deterministic algorithm $A$ corresponding to $B$ would be one that has made its "random" choices in advance. To construct it we just remove the randomized nodes by replacing them by a child we pick. The new result is a subtree (in respect to the non-randomized nodes) and its number of choices is less than or equal to the one of the randomized algorithm. Since any subtree we pick is $\Omega(n\lg{n})$, this implies that $B$ is $\Omega(n\lg{n})$.