Podcast
Questions and Answers
Fill in the blank for the pseudocode for an algorithm based on linear search that takes as input a valley list a1 , a2 ,..., an and returns the location of the valley. procedure LinearValley(a1 , a2 ,..., an : valley list) i.for i := 1 to n − 1 ii.if ___ iii.return i iv.return n
Fill in the blank for the pseudocode for an algorithm based on linear search that takes as input a valley list a1 , a2 ,..., an and returns the location of the valley. procedure LinearValley(a1 , a2 ,..., an : valley list) i.for i := 1 to n − 1 ii.if ___ iii.return i iv.return n
ai == 1
What is a loop invariant of LinearValley that you could use to prove the correctness of LinearValley?
What is a loop invariant of LinearValley that you could use to prove the correctness of LinearValley?
At each iteration, the index i will always point to the current position being checked in the valley list.
Fill in the blank for the algorithm based on binary search that takes as input a valley list a1 , a2 ,..., an and returns the location of the valley. procedure BinaryValley(a1 , a2 ,..., an : valley list) i.i := 1 ii.j := n iii.while (i < j) iv.m := ⌊ i+j ___ 2 ⌋ v.if ___ vi.i := m + 1 vii.else viii.j := m ix.return i
Fill in the blank for the algorithm based on binary search that takes as input a valley list a1 , a2 ,..., an and returns the location of the valley. procedure BinaryValley(a1 , a2 ,..., an : valley list) i.i := 1 ii.j := n iii.while (i < j) iv.m := ⌊ i+j ___ 2 ⌋ v.if ___ vi.i := m + 1 vii.else viii.j := m ix.return i
2, ai < am
What is a loop invariant of BinaryValley that you could use to prove the correctness of BinaryValley?
What is a loop invariant of BinaryValley that you could use to prove the correctness of BinaryValley?
In the worst case, which algorithm uses fewer comparisons on a list of length n? LinearValley or BinaryValley?
In the worst case, which algorithm uses fewer comparisons on a list of length n? LinearValley or BinaryValley?
A list of n distinct integers cannot be a valley list if all the numbers are increasing.
A list of n distinct integers cannot be a valley list if all the numbers are increasing.
In a valley list, the position of the valley is indicated by the element that equals 1.
In a valley list, the position of the valley is indicated by the element that equals 1.
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Study Notes
Homework Instructions
- Group work of 1 to 4 students, no external assistance.
- Submission via Gradescope by one representative, due by 11:59 PM on August 18, 2024.
- Grace period of 2 hours allowed.
- Review scanned work before final submission; multiple resubmissions permitted until deadline.
- Extra credit for typed solutions; hand-drawn diagrams accepted.
- Homework grading based on correctness and clarity of presentation.
- Justifications required for each answer, using mathematically sound reasoning.
Key Concepts
- Focus areas: Sorting, Searching, Loop Invariants, Runtimes, Big O notation, Algorithm runtime analysis, Master Theorem.
- Understanding valley lists: A sequence of numbers that first decreases and then increases, defining key characteristics.
Searching the Dark Valley
- Valley List Definition: A list of distinct integers that decreases to a minimum point and then increases; the "valley" is the index where the minimum occurs.
- Example: In the list [3, 2, 1, 6, 7], the valley is at index 3 (value 1).
- Special cases:
- An increasing list has a valley at index 1.
- A decreasing list has a valley at the last index.
Algorithms
-
Linear Search Algorithm (LinearValley):
- Searches each element sequentially to find the valley.
- Pseudocode structure: Loop through indices, check for valley condition, and return the index.
-
Binary Search Algorithm (BinaryValley):
- Efficiently narrows down the search space using a divide-and-conquer approach.
- Pseudocode: Initialize pointers, perform binary search until the valley is found, and return its index.
Loop Invariants
- Essential for proving the correctness of algorithms.
- LinearValley: A specific statement that maintains true throughout the execution of the algorithm, helping ensure the valley is identified correctly.
- BinaryValley: Similar to LinearValley, but focused on the division of search intervals to confirm the valley’s presence.
Comparison of Algorithms
- Analyze the maximum number of comparisons made by each algorithm in the worst-case scenario.
- LinearValley employs O(n) comparisons; BinaryValley has a logarithmic O(log n) complexity.
- Justification for the most efficient algorithm, taking into account the number of comparisons required for a list of length n.
Out of Bounds or In?
- Evaluation of true or false statements based on established mathematical rules.
- Include justifications using limit laws and function properties.
- Reference established formulas for sums, sequences, and factorial calculations.
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