Download CS 361 Lecture 15: Midterm Preparation - Quicksort and Heaps and more Study notes Computer Science in PDF only on Docsity!
CS 361, Lecture 15
Jared Saia University of New Mexico
Outline
Midterm
- 5 questions, 20 points each
- Hard but fair
- There will be some time pressure, so make sure you can e.g. solve recurrences both quickly and correctly.
- I expect a class mean of between 50 :( and 65 :) points
Question 1
Collection of true/false questions and short answer on:
- sorting algorithms (mergesort, heapsort, bubblesort)
- heaps (heights, number of nodes, heap algorithms, where is the max?, where is the min?)
- theta notation (i give you a bunch of functions and ask you to give me the simplest possible theta notation for each)
Question 2
- A question on annihilators and recurrence trees (like problems 1-3 of hw)
- You’ll need to know the formula for sum of an infinite con- vergent series
Question 3
- A question on using annihilators to solve a recurrence with both homogeneous and non-homogeneous parts
Question 4
- A question on writing recurrences for both the result of a function and the time cost of the function, and solving both of these recurrences using annihilators
Question 5
- A question asking you to prove the correctness of an algo- rithm using loop invariants
- I’ll give you the loop invariant and ask you to prove initial- ization, maintenance and termination
- Will be for an algorithm on heaps
Questions
Review Session
There will be a review session Today at 1pm
Other Review Session Options:
- Today at 5pm
- Today at 7pm
- Tomorrow at 3pm
- Tomorrow at 5pm
In-Class Exercise
- Imagine you have a min-heap with the following operations defined and taking O(log n): - (key,data) Heap-Extract-Min (A) - Heap-Insert (A,key,data)
- Now assume you’re given k sorted lists, each of length n/k
- Use this min-heap to give a O(n log k) algorithm for merging these k lists into one sorted list of size n.
In-Class Exercise
- Q1: What is the high level idea for solving this problem?
- Q2: What is the pseudocode for solving the problem?
- Q3: What is the runtime analysis?
- Q4: What would be an appropriate loop invariant for proving correctness of the algorithm?
Correctness
Basic idea: The array is partitioned into four regions, x is the pivot
- Region 1: Region that is less than or equal to x
- Region 2: Region that is greater than x
- Region 3: Unprocessed region
- Region 4: Region that contains x only
Region 1 and 2 are growing and Region 3 is shrinking
Correctness
Basic idea: The array is partitioned into four regions, x is the pivot
- Region 1: Region that is less than or equal to x (between p and i)
- Region 2: Region that is greater than x (between i + 1 and j − 1)
- Region 3: Unprocessed region (between j and r − 1)
- Region 4: Region that contains x only (r)
Region 1 and 2 are growing and Region 3 is shrinking
Example
- Consider the array (2 6 4 1 5 3)
Loop Invariant
At the beginning of each iteration of the for loop, for any index k:
- If p ≤ k ≤ i then A[k] ≤ x
- If i + 1 ≤ k ≤ j − 1 then A[k] > x
- If k = r then A[k] = x
In Class Exercise
- Show Initialization for this loop invariant
- Show Termination for this loop invariant
- Show Maintenance for this loop invariant:
- Show Maintenance when A[j] > x
- Show Maintenance when A[j] ≤ x
Scratch Space
Scratch Space
Analysis
- The function Partition takes O(n) time. Why?
- Q: What is the runtime of Quicksort?
- A: It depends on the size of the two lists in the recursive calls
Best Case
- In the best case, the partition always splits the original list into two lists of half the size
- Then we have the recurrence T (n) = 2T (n/2) + Θ(n)
- This is the same recurrence as for mergesort and its solution is T (n) = O(n log n)
Worst Case
- In the worst case, the partition always splits the original list into a singleton element and the remaining list
- Then we have the recurrence T (n) = T (n − 1) + T (1) + Θ(n), which is the same as T (n) = T (n − 1) + Θ(n)
- The solution to this recurrence is T (n) = O(n^2 ). Why?
Todo
- Read Chapter 7
- Finish HW
- Study for Midterm!
