Math Problem Statement

numbers = [24, 13, 29, 10, 17, 51, 62, 52, 54, 53]

Partition(numbers, 5, 9) is called. So, only elements in indices 5 to 9 are partitioned. Elements in indices 0 to 4 are not affected.

Assume quicksort always chooses the element at the midpoint as the pivot.What is the pivot?What is the low partition?What is the high partition?What is numbers after Partition(numbers, 5, 9) completes?

Solution

Let's break down the process of partitioning and sorting the list based on the rules you've provided.

The numbers list is:

[24, 13, 29, 10, 17, 51, 62, 52, 54, 53]

You are partitioning only the sublist from indices 5 to 9 (i.e., elements: 51, 62, 52, 54, 53).

Step 1: Identify the Pivot

In quicksort, the pivot is the element chosen for comparison, and it splits the list into two partitions: those less than the pivot and those greater than it.

Since the pivot is always chosen as the midpoint of the sublist (in this case, the sublist is numbers[5:10] which contains [51, 62, 52, 54, 53]), the midpoint index for this range is:

Midpoint index=5+92=7\text{Midpoint index} = \frac{5 + 9}{2} = 7

So, the pivot element is:

Pivot=numbers[7]=52\text{Pivot} = numbers[7] = 52

Step 2: Partitioning the Sublist

We now partition the sublist [51, 62, 52, 54, 53] around the pivot element 52.

  • Low partition: Elements less than the pivot (52).
  • High partition: Elements greater than the pivot (52).

After comparing each element to the pivot, we get:

  • Elements less than 52: [51]
  • Elements greater than 52: [62, 54, 53]

Step 3: Rearrange the Elements

After partitioning, we rearrange the elements such that:

  • The elements less than the pivot (51) come first.
  • The pivot (52) comes next.
  • The elements greater than the pivot (62, 54, 53) come last.

The resulting list after partitioning will be:

[51, 52, 62, 54, 53]

Step 4: Final Result for numbers

After the partition operation on the sublist from indices 5 to 9, the list numbers will be updated as follows:

[24, 13, 29, 10, 17, 51, 52, 62, 54, 53]

Summary of the Results:

  • Pivot: 52
  • Low partition: [51]
  • High partition: [62, 54, 53]
  • Final numbers list after partitioning:
    [24, 13, 29, 10, 17, 51, 52, 62, 54, 53]
    

Do you need more details on how quicksort works or the partitioning steps? Let me know if you have further questions!


Here are some related questions:

  1. How does quicksort's partitioning step work when choosing the first or last element as the pivot?
  2. What would happen if we used a different partitioning method, like the "Lomuto partition scheme"?
  3. How would you apply quicksort recursively to the entire list?
  4. Can quicksort perform better on already sorted data? Why or why not?
  5. How does the choice of pivot affect the efficiency of quicksort in practice?

Tip: When analyzing quicksort performance, always consider the nature of the input list. If the input list is already sorted or nearly sorted, quicksort can degrade to O(n^2) performance.

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Math Problem Analysis

Mathematical Concepts

Sorting Algorithms
Quicksort
Partitioning

Formulas

Midpoint index formula: (low + high) / 2

Theorems

Quicksort Partitioning Theorem

Suitable Grade Level

Grades 9-12