Stock Price Analysis Quiz

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Questions and Answers

What is the goal in analyzing a series of stock prices in Euro?

  • To calculate the average stock price
  • To determine the number of stock price fluctuations
  • To find the maximum stock price
  • To find the tuple (i, j) such that A[j] − A[i] is maximized (correct)

The solution to finding the maximum difference in stock prices is always optimal using brute force.

True (A)

What is the formula for the run time of the brute force solution in terms of n?

n² - n

For all admissible (i, j), the calculation involves finding A[j] − A[_____].

<p>i</p> Signup and view all the answers

Match the following terms with their definitions:

<p>Brute Force = An exhaustive method of finding solutions Admissible Tuple = Pair (i, j) where 1 ≤ i &lt; j ≤ n Maximum Difference = Largest value of A[j] − A[i] for all pairs Run Time = Time complexity of an algorithm based on input size n</p> Signup and view all the answers

Which series of stock prices in Euro is denoted by A[1..n]?

<p>A collection of stock prices indexed from 1 to n (D)</p> Signup and view all the answers

The total number of operations performed in the brute force approach increases linearly with respect to n.

<p>False (B)</p> Signup and view all the answers

What does the term 'admissible tuples' refer to in the context of stock price analysis?

<p>Pairs (i, j) where 1 ≤ i &lt; j ≤ n</p> Signup and view all the answers

What is the runtime of the MaxPartSum algorithm for n greater than 1?

<p>$Θ(n ext{ log } n)$ (B)</p> Signup and view all the answers

The runtime TMPS(n) for n > 1 is equal to 2 · TMPS(n/2) + n.

<p>True (A)</p> Signup and view all the answers

What is the value of TMPS(1) in the context of the algorithm?

<p>Θ(1)</p> Signup and view all the answers

The maximum sum is calculated such that L-sum is less than ______.

<p>L-sum</p> Signup and view all the answers

Match the following components with their descriptions:

<p>L-max = Index of the maximum left sum R-max = Index of the maximum right sum sum = Current sum being calculated TMPS = Runtime of the MaxPartSum algorithm</p> Signup and view all the answers

In the loop structure, what happens when 'sum' is greater than 'L-sum'?

<p>L-sum is updated to the value of sum. (B)</p> Signup and view all the answers

The total number of additions for the MaxPartSum algorithm is virtually unbounded.

<p>False (B)</p> Signup and view all the answers

The total contributions of L-sum and R-sum are returned as ______.

<p>L-sum + R-sum</p> Signup and view all the answers

What is the run time complexity of CMS when n is a power of two?

<p>O(n log2 n) (D)</p> Signup and view all the answers

For values of a and b greater than or equal to 2, Θ(loga n) is not equal to Θ(logb n).

<p>False (B)</p> Signup and view all the answers

What is the formula for TMPS when n is greater than 1?

<p>TMPS(n) ≈ 2 · TMPS(n/2) + n</p> Signup and view all the answers

The run time of MaxPartSum for n = 1 is _____

<p>Θ(1)</p> Signup and view all the answers

What can be inferred about solving MaxPartSum in O(n) time?

<p>It is a brainteaser. (B)</p> Signup and view all the answers

MaxPartSum has a direct application in stock price analysis.

<p>True (A)</p> Signup and view all the answers

What is the base case for the run time of TMPS?

<p>TMPS(1) = Θ(1)</p> Signup and view all the answers

What is the primary design principle discussed in the content for finding the largest partial sum?

<p>Divide &amp; Conquer (C)</p> Signup and view all the answers

The left part of the largest partial sum can be computed independently of the right part.

<p>True (A)</p> Signup and view all the answers

What should be true about the left and right parts for the largest partial sum that contains the middle?

<p>Both parts must be maximum.</p> Signup and view all the answers

The process of finding the largest partial sum using the divide and conquer method involves __________ and __________.

<p>dividing, combining</p> Signup and view all the answers

Match the terms with their descriptions:

<p>Divide = Split into two roughly equal parts Conquer = Recursion on each part Combine = Check all partial sums Insight = Maximum sums on either side of the middle</p> Signup and view all the answers

What is the upper bound for the run time of the problem described?

<p>O(n²) (C)</p> Signup and view all the answers

The goal of the problem is to find the minimum sum of the series.

<p>False (B)</p> Signup and view all the answers

What is the task in the problem described?

<p>Calculate the maximum sum for all admissible pairs (i, j).</p> Signup and view all the answers

The indices (i, j) in the problem must satisfy the condition _____ ≤ i ≤ j ≤ n.

<p>1</p> Signup and view all the answers

Match the following concepts to their definitions:

<p>Brute Force = An exhaustive approach to solving problems Tuple = An ordered collection of elements Series = A sequence of numbers Maximum Sum = The largest possible total from selected elements</p> Signup and view all the answers

Given the series 7, 4, -9, 1, -3, 3, 1, 12, 0, -2, 0, 4, 2, -8, 2, which tuple gives the maximum sum?

<p>(6, 13) (A)</p> Signup and view all the answers

The value of A[10] in the series is 0.

<p>True (A)</p> Signup and view all the answers

What is the significance of calculating 'k = i A[k]' in the problem?

<p>It computes the sum of elements from index i to j.</p> Signup and view all the answers

What is the main operation being analyzed in the brute-force algorithm?

<p>Additions (A)</p> Signup and view all the answers

The number of sums with exactly s terms is represented by the formula n - s + 1.

<p>True (A)</p> Signup and view all the answers

How many additions are required to calculate k = i A[k] for all admissible tuples (i, j)?

<p>Dependent on s</p> Signup and view all the answers

The total number of additions involved in the brute-force algorithm can be calculated using the summation from s equals 1 to ___ of (n - s + 1) · (s - 1).

<p>n</p> Signup and view all the answers

Match the following variables with their descriptions:

<p>i = Index of the first element in the tuple j = Index of the second element in the tuple k = Calculated value based on A[k] s = Number of terms in the sum</p> Signup and view all the answers

In the analysis, the number of additions resulting from sums with s terms is calculated using which of the following expressions?

<p>$(n - s + 1)(s - 1)$ (B)</p> Signup and view all the answers

The run time of the brute-force algorithm is directly proportional to the number of elements in the input array.

<p>False (B)</p> Signup and view all the answers

What pattern emerges when you calculate the total number of additions related to terms s as n increases?

<p>It forms a series that decreases as s increases.</p> Signup and view all the answers

Flashcards

Maximum Difference Problem

Find indices (i, j) such that A[j] - A[i] is maximum with 1 ≤ i < j ≤ n.

Brute Force Solution

Calculate A[j] - A[i] for all admissible pairs (i, j).

Admissible Tuples

Pairs of indices (i, j) that satisfy 1 ≤ i < j ≤ n.

Run Time Calculation

The time complexity is roughly the number of admissible tuples: n(n-1)/2.

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Stock Price Series

A sequence of prices A[1..n] representing stock values in Euro.

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Calculate Difference

For each pair (i, j), compute A[j] - A[i] to find the maximum profit.

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Combinatorics Concept

Calculating pairs involves combinatorics: C(n, 2) = n(n-1)/2.

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Time Complexity

The overall complexity of brute force is O(n^2).

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Maximum Subarray Problem

Finding the tuple (i, j) that maximizes the sum of elements in the array A from index i to j.

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Sum Calculation

The process of adding elements in A from index i to j.

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Complexity Upper Bound

The maximum possible time complexity of an algorithm, often denoted as O(n^2).

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Admissible Pairs

Pairs of indices (i, j) that meet the condition 1 ≤ i ≤ j ≤ n.

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Run Time Analysis

Estimating the time it takes for an algorithm based on input size, here O(n^2) for this problem.

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Tuple Definition

An ordered pair of elements, in this case (i, j) representing indices.

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Integer Series in Programming

A sequence of integers stored in an array format, like A[1..n].

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Divide and Conquer

A strategy to break a problem into smaller subproblems.

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Partial Sum

The sum of a subset of consecutive elements in a sequence.

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Middle Element

The central element in a sequence used for analysis.

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Maximum Left Part

The maximum sum found in the left section of a split sequence.

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Maximum Right Part

The maximum sum found in the right section of a split sequence.

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Brute-force algorithm

An algorithm that checks all possible solutions to find the best one.

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Additions in brute-force

The number of addition operations required by the brute-force algorithm.

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Number of sums

Total sums calculated with a specified number of terms, s.

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Maximum of differences

The largest value obtained by subtracting one sum from another.

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Sums with s terms

The sums formed using exactly s number of terms from n terms.

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Formula for additions

The equation that calculates the total number of additions performed by the algorithm.

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Loop Invariant

A property that holds true at the start and end of iterations in a loop.

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MaxPartSum Runtime

The recursive runtime of MaxPartSum is defined by T(n) = 2*T(n/2) + n.

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TMPS(1)

The base case for MaxPartSum runtime which equals Θ(1).

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Recursive Formula

For n > 1, TMPS(n) = 2 * TMPS(n/2) + n, showing a divide-and-conquer approach.

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CMS(n)

The cumulative complexity of the MaxPartSum problem, which equals O(n log2 n) for n being a power of two.

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L-max and R-max

They represent the maximum sums obtained from the left and right parts of the array, respectively.

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L-sum and R-sum

L-sum is the sum from the left side, and R-sum is from the right side.

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TMMS(n)

The additional time complexity used in the recursive runtime for MaxPartSum calculation.

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Run time of MaxPartSum

TMPS (n) for n > 1 is 2 · TMPS (n/2) + n.

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Relationship of logs

Θ(loga n) = Θ(logb n) for a, b ≥ 2.

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MaxPartSum recurrence relation

TMPS (n) ≈ 2 · TMPS (n/2) + TMMS (n).

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Finding linear time solution

Challenge to solve MaxPartSum in O(n) time.

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TMMS (n) in MaxPartSum

TMMS (n) contributes linearly to the recursion.

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Stock price relation to MaxPartSum

MaxPartSum is used to analyze stock price trends.

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Study Notes

Algorithms and Data Structures - 4th Lecture

  • Topic: Run Time Analysis - Examples
  • Stock price analysis
  • A series of stock prices (A[1..n]) is given
  • Find the difference in stock price to maximize the profit (A[j] - A[i])
  • Brute-force solution has a time complexity of O(n²)
  • A faster solution is needed for bigger data sets

Analysis of Stock Prices

  • Data source: finanzen.net/chart/Nordex
  • Stock Prices: Example data (dates and prices shown in the slides)
  • Problem statement: Find the tuple (i, j) where 1 ≤ i < j ≤ n, such that A[j] - A[i] is the maximum profit.

A similar problem

  • Problem statement: Given a series A[1..n] of integer numbers. Find a tuple (i,j) with 1 ≤ i ≤ j≤ n, such that ∑k=i A[k] is the maximum sum.
  • Example data: 7, -4, -9, 1, -3, 3, 1, 12, 0, -2, 0, 4, 20, -8, 2
  • Solution is "brute force" for all pairs (i, j) to determine the sum and find the maximum
  • Run time: O(n²)

A faster solution

  • Idea: Divide and conquer - subproblems.
  • Steps: Divide the problem into two roughly equal parts recursively, solve each subproblem, and combine the results.
  • Further subproblems can be recursively approached
  • In total: Θ(n²) additions
  • How to calculate: Calculate Sii, Si,i+1, ..., Sin (sum of elements from i to n) for each 'i' between 1 and n.
  • Finding the maximum sum: Focus on finding the sums that need to be evaluated.

Run time analysis

  • Exact analysis of the brute force algorithm runtime
  • The number of admissible tuples is (n − 1) + (n − 2) + ... + 2 + 1 = n(n-1)/2 which is O(n²).
  • The number of sums with exactly S terms is n - s + 1
  • The number of additions is ∑s=1(n - s + 1)(s - 1) = Θ(n³).
  • Divide and conquer run time O(n log n)

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