Finding Minimum scalar product of two vectors
Detailed guide and Python implementation for the 'Finding Minimum scalar product of two vectors' problem.
Problem Statement
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.
- •1 <= len(v1) == len(v2) <= 10^4
- •-10^5 <= v1[i], v2[i] <= 10^5
Examples
v1 = [1, 3, -5], v2 = [-2, 4, 1]
-25
Sort v1 ascending: [-5,1,3]. Sort v2 descending: [4,1,-2]. Dot product: (-5)*4 + 1*1 + 3*(-2) = -20+1-6 = -25.
v1 = [1, 2, 3], v2 = [4, 5, 6]
32
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.
v1 = [1, 1], v2 = [1, 1]
2
Both vectors are [1,1]. Any arrangement gives 1*1 + 1*1 = 2.
Need a Hint?
Edge Cases to Watch
- Empty input structures
- Single element inputs
- Large numerical bounds
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