assignment 1 maximum pairwise product

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• 1.2 max pairwise product.py

Maximum Pairwise Product

Cloistered Monkey

2018-06-24 16:42

Since the test only uses python standard library I'll try and stick to that, but since the stress-test isn't part of the assignment I'll cheat and use numpy to generate the random input.

Problem Statement

Find the maximum product of two distinct numbers in a sequence of non-negative integers.

• Input: A sequence of non-negative integers.
• Output: The maximum value that can be obtained by multiplying two different elements from the sequence.

Given a sequence of non-negative numbers \(a_1,\ldots,a_n\), compute

\[ \max_{1 \le i \neq j \le n} a_i a_j \]

\(i\) and \(j\) should be different, although \(a_i\) and \(a_j\) might be equal.

Example Values

Some Constants

This is just a translation of some of the problem statement values to python so we can use them.

These are some functions to help validate the algorithms.

Brute Force Implementation

This is given as part of the problem. It traverses all the values and finds the largest product.

Because we are traversing all the numbers twice, the brute-force version has a run time of \(O(n^2)\). Since the \(n\) can be from \(2\) to \(2 \times 10^5\) that means our maximum run time will be \(4 \times 10^10\), which is too large.

Running this through the grader gives this output.

Instead of using nested for-loops, we can search the numbers twice to find the two biggest numbers, this changes the run time to \(2n\) or \(O(n)\).

This one passes the grader, doing surprisingly well, even though I was thinking it would need more optimizing.

Another Improvement

Rather than go through the second loop, I thought that since the previous maximum value is always the next highest value so far, we can just save it directly.

Stress Test

Even thought we're already passing, part of the assignment was to create a stress test to really exercise the algorithm once you have it passing.

Boosted Brute Force

To try and get this working I'm going to use numba to (hopefully) speed it enough to make the stress test runnable.

Since we need the top two values we can get a more efficient algorithm by sorting the values.

Maximum pairwise product →

Sep 2, 2017 • algorithms , C++

Let’s take a quick look at a simple problem:

Given a list of numbers, how would you determine the maximum pairwise product of numbers in the list?

For example, if we have the list [1; 3; 5; -2] the maximum pairwise product is 15 , which is the product of 3 and 5 .

The straigt-forward approach is to compute all possible pairwise products and identify the maximum among the intermediate results:

Since we have two loops that iterate over the list of numbers, the time complexity is O(n 2 ).

Can we do better? Yes, we can. The idea is to find the maximum and the second maximum elements in the list - and their product would be the maximum pairwise product.

Simple and sweet!

Nkwachi's Cave

Solving the maximum pairwise product algorithm.

What is the Maximum Pairwise Product? It means that we have to find the maximum product of two distinct numbers in a given array of non-negative numbers.

Problem Statement

Given a sequence of non-negative integers a0,…,an−1, find the maximum pairwise product, that is, the largest integer that can be obtained by multiplying two different elements from the sequence.

Input format:

The first line of the input contains an integer n. The next line contains n non-negative integers a0,…,an−1 (separated by spaces).

Constraints:

2≤n≤2⋅105; 0≤a0,…,an−1≤105.

Output format:

Output a single number — the maximum pairwise product.

One approach is finding the two largest numbers in the array and storing their value in two variables and then multiplying both numbers but this approach makes you run into a problem when you have an array where the largest number is repeated e.g [2,5,9,9]. When finding the second largest number you compare against the largest number to ensure they aren't the same and you'll end up with 9 and 5 giving a maximum pairwise product of 45 instead of 81.

A better approach is to store the indices of the largest number and the second-largest number, this approach eliminates the error stated above.

My solution

Here is my solution implemented in Kotlin.

First of I created two variables to store the indices of the largest number and second-largest number. The next step was to find the largest number which was quite straightforward. When finding the second largest number I had to check if the index of the largest number is 0 and initialize the index of the second largest number to 1 instead or else I get 0 as the index of the second-largest number which is wrong.

Conclusion:

My solution has a complexity of O(n) but it can be better as I go through the array twice and it's possible to go through the array once while keeping track of the largest number and the second-largest number simultaneously. I intend to solve the algorithm using that approach so look out for part 2 on "Solving the Maximum Pairwise Product Algorithm".

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Pairwise Max Product

This python program takes user input of the number of elements in a list in the variable b and then takes user input of the list elements in list a . It then finds out the maximum product of a pair of the list elements.

For example, if the list elements are 5,4 and 6, then the maximum product is 30.

Can we simplify it in a way such that it takes less time than this?

The error I am getting is:

Failed case #4/17: time limit exceeded (Time used: 9.99/5.00, memory used: 22740992/2147483648.)

• time-limit-exceeded

Making it faster

The maximum product of two numbers in the input will be either:

• the product of the two highest numbers
• note that this is possible when the lowest numbers are negative

So you can find the two highest and two lowest values, and decide which product is higher and return it.

This can be done in \$O(n)\$ time, which will be much faster than the current \$O(n^2)\$ solution for large inputs.

Beware of corner cases

If there are less than two numbers, the program outputs 0. For a list with a single element, say [3] , printing that the maximum product of two pairs of numbers is 0 would seem strange.

Use descriptive variable names

It's hard to remember what is what, when variables don't have descriptive names, such as a and b . nums and nums_count would help a lot.

Use functions

The code reads input from stdin , computes something, then prints to stdout . The computation part can be turned into a function that takes a list of numbers and returns the maximum product. It will have a single responsibility, to compute the maximum product, and a reader can focus on making it fulfill that one responsibility:

Note that nums_count = len(nums) .

Follow the style guide

Python has an official style guide PEP-8 , it's good to follow it. For example, it's recommended to put spaces around operators, for example instead of range(i+1,b) , write range(i + 1, b) .

• \$\begingroup\$ Thanks a lot! Turning down the complexity that way was helpful and also didn't think about the corner case! Thanks once again! \$\endgroup\$ –  Pratik Roy Commented Sep 23, 2022 at 21:45

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Coursera's Algorithmic toolbox assignment solutions( Sum of Two Digits, Maximum Pairwise Product ) :

Problem 1: sum of two digits, solution: (in c++), ( please guys before moving to the solution try it yourself at least 3-4 times , if you really wanna become a good coder).

This code is simple. There's no need for any explanation.  

Problem 2: Maximum Pairwise Product

Guys , if you have any queries related to a question or need more explanation, comment down below!  

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Coursera — Algorithmic Toolbox

Things I learnt from assignment 1 — max pairwise product problem:

• The dreaded O(N)
• Copying a list in Python is as simple as list.copy()
• What a “naive solution” means

Written by Sharon Woo

Nerd runner

Text to speech

How to find Maximum Pairwise Product in Java using Naive approach

In this Java tutorial, we will learn how to find the maximum pairwise product using Java. Maximum Pairwise Product in easy words you can say that it is the product of those two numbers present in an array from which we get the highest value possible.

What is the meaning of Maximum Pairwise Product?

It means that we have to find the maximum product of two distinct numbers in a given array of non-negative numbers.

Input : A sequence of non-negative numbers.

Output : The maximum value that can be obtained by multiplying two different numbers from the input sequence.

=>, For example, I’m giving you a sequence of non-negative numbers: a1 . . . . an

Input format:  The first line contains a single number whose value is equal to n. The next line contains n non-negative numbers a1 . . . . an.

Output format:  The maximum pairwise product.

Naive Method :

A naive approach is to solve the Maximum Pairwise Product Question is to find all the possible pairs from the sequence which is inputted by the user. A[1 . . . . n] =[a1 . . . . an] and then we have to find the largest product value.

First, we have to find the two largest values from the inputted sequence. Because we know that the product of the largest value is the maximum product we can get.

Note: All inputted numbers in the sequence is must be non-negative numbers.

Find Maximum Pairwise Product in Java using Native Approach

We have used StringTokenizer class here.

You can also read,

• Frequency of Repeated words in a string in Java
• Working with HashMap (Insertion, Deletion, Iteration) in Java

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6 videos • Total 48 minutes

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• Stress Test - Implementation • 8 minutes
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• Maximum Pairwise Product Programming Challenge • 10 minutes
• Using PyCharm to solve programming challenges • 10 minutes
• Acknowledgements • 2 minutes

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• Maximizing Loot • 8 minutes
• Implementation and Analysis • 10 minutes
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20 videos 5 readings 8 quizzes 1 programming assignment

20 videos • Total 157 minutes

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• Binary Search Runtime • 8 minutes
• Problem Overview and Naïve Solution • 6 minutes
• Naïve Divide and Conquer Algorithm • 7 minutes
• Faster Divide and Conquer Algorithm • 6 minutes
• What is the Master Theorem? • 4 minutes
• Proof of the Master Theorem • 9 minutes
• Problem Overview • 2 minutes
• Selection Sort • 8 minutes
• Merge Sort • 10 minutes
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• Algorithm • 9 minutes
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• Running Time Analysis (optional) • 15 minutes
• Equal Elements • 6 minutes
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5 readings • Total 40 minutes

• Resources • 10 minutes
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8 quizzes • Total 155 minutes

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problem calculating the Maximum Pairwise Product C++

This is my code for the maximum pairwise product. It takes an array of numbers, sorts them to find the first and second maximum numbers, then return the product of them. The code works for small arrays and small values. But it fails with some numbers, and also it epically fails with large numbers such as 10000 and so.

i thought it was a problem with the data type i used, so i defined the data type to be int_64t so it can handle large numbers, but still i get the same wrong results! Can anyone help me with that?

• 1 Can you clarify what you mean by "epically fails"? –  blackbrandt Commented Jun 23, 2019 at 23:11
• For example when i give it an array of 5 numbers consisting of 10000, 10,1,2,3..it's supposed to return 100000, instead it gives me some random large number, something like 17346274 or something. –  Fezofchekov Commented Jun 23, 2019 at 23:20
• 1 The error is here: for (int j=0; j<=n; j++) . Should be j < n . Same for another loop. –  user2956272 Commented Jun 23, 2019 at 23:22
• UPDATE: Now the code works fine with large numbers like 10000 and so, but it fails with small numbers! I tried a set of 3 numbers consisting of 1 2 3, it gave me a result of 9! while it was supposed to give 2*3=6, any ideas? –  Fezofchekov Commented Jun 23, 2019 at 23:55
• What is the meaning of this line: if(maxind1!=j && maxind2==-1 || numbers[j]>numbers[maxind2]) ? It could be a precedence problem. You may want to add parenthesis. –  user10957435 Commented Jun 24, 2019 at 0:14

2 Answers 2

(EDITED as per comments from others)

Besides what @dyukha noted - for (int j=0; j<=n; j++) - also: maxind2 could be negative in the code below, which is used to subscript the vector.

Solution is to change the precedence in logical expression inside if to:

• Actually the first if is fine, because C++ uses Short-circuit evaluation , which means that if maxind1==-1 is true, the part after || is not even evaluated. But it's true that the second if has a problem: if maxind1!=j is false, it will evaluate numbers[-1] . Solution: use parentheses to change the precedence, that is, if(maxind1!=j && (maxind2==-1 || numbers[j]>numbers[maxind2])) . Which is, by the way, what Chipster said in the comments. –  Fabio says Reinstate Monica Commented Jun 24, 2019 at 0:38
• @FabioTurati yes I didn't pay attention on the first. Also, didn't know the (what i see as standard logical ops) are called short-circuit. –  v01d Commented Jun 24, 2019 at 0:45
• In a logical OR, if the first is true, then you are sure the whole OR expression will be true, and this is due to the very nature of OR. It is like this in every language. What short-circuiting adds is that, in such cases, the second part is not even evaluated, as it would be a waste of time. This prevents some problems, as in this case. A typical example, when you have a pointer p , is: if (p && p->do_something()) . If p is null, evaluating the second part would lead to Undefined Behaviour (probably a segmentation fault), but short circuiting prevents it and makes this code safe to run. –  Fabio says Reinstate Monica Commented Jun 24, 2019 at 0:54
• @FabioTurati understand that, but I never heard term "short-circuit" is what's this called, or that that's a feature. I thought that's standard logic –  v01d Commented Jun 24, 2019 at 0:55
• 1 @Rania Please remember that if an answer solves your problem you should mark it as accepted, by clicking on the tick on its left. Thank you! –  Fabio says Reinstate Monica Commented Jun 24, 2019 at 8:59

SOLVED Thank you all for helping me out, it finally worked properly. The source of the problem was that i didn't use parenthesis in the second if conditions, and the condition of the for loop (use < instead of <=) Code after solving the errors:

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Algorithmic Toolbox | Week 1

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1-1: Sum of Two Digits

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Programming Assignment: Programming Assignment 1: Maximum Pairwise Product

1-2: Maximum Pairwise Product

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