What is the optimal time complexity of solving Finding Minimum scalar product of two vectors?

The optimized solution achieves a time complexity of O(n) by using sorting, mathematical shortcuts, or hash table lookups.

What is the auxiliary space complexity for Finding Minimum scalar product of two vectors?

The space complexity is O(1) since we only use a minimal/constant amount of extra memory for tracking variables (or auxiliary space if dynamic structures are allocated).

What are the typical edge cases we must handle in Finding Minimum scalar product of two vectors?

Key edge cases include empty collections, negative or large bounds causing integer overflow, arrays with only one element, or inputs where target criteria are not met.

How can we optimize the brute force approach for Finding Minimum scalar product of two vectors?

Brute force methods often inspect all possible combinations. We optimize this by sorting the input list to enable binary search, or tracking processed items in a set to avoid redundant nested loop lookups.

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Python BasicsEasy

Finding Minimum scalar product of two vectors

Learn how to solve the 'Finding Minimum scalar product of two vectors' problem. This detailed resource details brute force and optimized approaches.

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Problem Statement

Easy

Write a function min_scalar_product(v1, v2) that takes two lists of integers of equal length and returns the minimum possible scalar (dot) product. You can rearrange elements in both vectors in any order before computing the dot product. The dot product is sum(v1[i] * v2[i]) for all i. To minimize, pair the largest of one with the smallest of the other.

Constraints

•1 <= len(v1) == len(v2) <= 10^4
•-10^5 <= v1[i], v2[i] <= 10^5

Examples

Example 1

Input

v1 = [1, 3, -5], v2 = [-2, 4, 1]

Output

-25

Explanation

Sort v1 ascending: [-5,1,3]. Sort v2 descending: [4,1,-2]. Dot product: (-5)*4 + 1*1 + 3*(-2) = -20+1-6 = -25.

Example 2

Input

v1 = [1, 2, 3], v2 = [4, 5, 6]

Output

Explanation

Sort v1 ascending: [1,2,3]. Sort v2 descending: [6,5,4]. Dot: 1*6+2*5+3*4 = 6+10+12 = 32? Wait: minimum is 1*6+2*5+3*4=32. Alternative: 1*4+2*5+3*6=32. Actually min is 1*6+2*5+3*4=32.

Example 3

Input

v1 = [1, 1], v2 = [1, 1]

Output

Explanation

Both vectors are [1,1]. Any arrangement gives 1*1 + 1*1 = 2.

Need a Hint?

Use simple arithmetic operators (like modulo `%`, division `//`), conditional checks, or loops to inspect number properties.

Edge Cases to Watch

Empty list or null input variables
Single item lists/arrays
Extremely large input bounds causing integer or stack overflow

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