What is the Lower Bound of Comparison Sort?

We have already discussed several sorting algorithms that can sort n numbers in both O(n²) and O(nlogn) time complexity in the worst case. 

These algorithms determine the sorted order based only on comparisons between input elements. We call such sorting algorithms comparison sort, where comparisons is a critical operation between elements to gain order information about an input sequence A[0], A[1], . . . , A[n-1]. In other words, given two elements A[i] and A[j], we perform one of the tests A[i] <= A[j], A[i] >= A[j], A[i] = A[j] to determine their relative order.

Decision tree model of comparison sorting

We can visualize comparison sorts in terms of decision trees. A decision tree is a full binary tree structure that represents the comparisons performed by sorting algorithms when it operates on an input of a given size. We only consider comparison operations and ignore other operations like control, data movement, etc.

Here is an example of a decision tree model for insertion sort operating on three elements. There are 3! = 6 possible permutations of input elements, so decision tree must have at least 6 leaves. Think!

Decision tree model of comparison based sorting

Each internal node in the decision tree is represented by a comparison between a pair of elements (A[i], A[j]), where 0 ≤ i, j ≤ n-1. If A[i] ≤ A[j] is true, we follow the left path. Otherwise, we follow the right path.

Similarly, each leaf node represents a permutation of the given input. In other words, each permutation on n elements (n! Permutations) must appear as one of the decision tree leaves for sorting algorithm to sort correctly.

How decision tree model of comparison sorting works?

The execution of sorting algorithm corresponds to tracing a path from the root of decision tree to a leaf. At each internal node, a comparison A[i] ≤ A[j] is made. The left subtree then dictates subsequent comparisons for A[i] ≤ A[j] and the right subtree dictates subsequent comparisons for A[i] > A[j]. When we come to a leaf node, sorting algorithm will ensure the sorted ordering of input elements.

The height of decision tree (Length of the longest path from the root to any of its leaves) represents the worst-case count of comparisons performed by the sorting algorithm. A lower bound on the heights of decision trees is, therefore, a lower bound on the running time of any comparison sort algorithm.

Idea: Any decision tree that sorts n elements has a height always greater than nlogn.

Proof: Consider a decision tree of height h that sorts n elements. Since there are n! permutations of n elements, each permutation representing a distinct sorted order, the tree must have at least n! leaves. Another side, we know that the binary tree of height h has no more than 2^h leaves.

=> 2^h ≥ n!. So, h ≥ log(n!) From Stirling’s formula, n! > (n/e)^n => h ≥ log(n/e)^n => h ≥ n(log n — log e) => h ≥ nlogn — nloge => h > O(nlogn)

Therefore, any comparison-based sorting algorithm must make at least O(nlogn) comparisons to sort the input array. 

Critical ideas to think about!

  • Heapsort and merge sort are optimal comparison sorts.
  • What is the smallest possible depth of a leaf in a decision tree?
  • Prove that 2n - 1 comparisons are necessary in the worst case to merge two sorted arrays containing n elements each.

Introduction to linear time sorting

Some sorting algorithms run faster than O(nlogn) time complexity but require special assumptions about the input sequence. These sorting algorithms use operations other than comparisons to determine the sorted order and work in O(n) time complexity.

Examples of sorting algorithms that run in linear time are counting sort, radix sort, bucket sort, etc. Counting sort and radix sort assume that the input consists of integers in a small range. At the same time, bucket sort assumes that the input is generated by a random process that distributes elements uniformly over the interval.

Reference: Algorithms by CLRS

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