Minimum Bundle Quantities — Problem Statement & Solution Guide
Problem Description
Given three types of ingredients with required quantities of a, b, and c units, and each ingredient being sold in bundles of x, y, and z units, determine the minimum quantities of each bundle to purchase in order to meet the required quantities, considering that bundles for one ingredient can be used for another.
DSA Pattern Breakdown
DSA Pattern Breakdown
"Minimum Bundle Quantities"
WHY DOES IT MATTER?
This pattern is essential for resource optimization problems where resources are fungible or substitutable. It tests the ability to move beyond simple arithmetic to combinatorial optimization, a key skill in systems design and operations research.
OPTIMIZATION CHALLENGE
The key insight is recognizing that the total units provided by all bundles must be at least the sum of the required quantities. If bundles are fungible, the problem reduces to finding the minimum number of bundles such that their total capacity meets the total demand, which can often be solved with greedy strategies or simple arithmetic if bundle sizes are uniform.
REAL-WORLD CONNECTION
Analogous to cloud resource provisioning, where CPU, memory, and storage can be over-provisioned in one dimension to compensate for under-provisioning in another, or in supply chain management where generic packaging can be used for multiple product lines.
In interviews, clarify the constraints early. If bundles are truly fungible, the problem simplifies significantly. If not, ask about the substitution rules. Always start with the total demand and work backward to the bundle allocation.
COMPLEXITY AT A GLANCE
O(1)O(1)Core Theory — Why This Approach?
The problem 'Minimum Bundle Quantities' is a variant of the Integer Linear Programming (ILP) problem, specifically dealing with resource allocation under substitution constraints. The core challenge lies in the fact that bundles are not strictly tied to their nominal ingredient; a bundle of type X can satisfy the requirement for ingredient Y. This creates a combinatorial explosion where the naive approach of trying all possible combinations of bundle purchases results in a time complexity of $O(N^3)$ or worse, where $N$ is the magnitude of the required quantities. For large inputs, this brute-force method is computationally infeasible.
Interview Questions on This Problem
Q1At a fintech platform, we need to allocate credit limits across three risk tiers. Each tier has a minimum required coverage, but credit instruments from one tier can be partially applied to another. How would you model this to minimize the total number of instruments issued?
Model this as a linear programming problem where the decision variables are the number of instruments issued for each tier. The constraints are that the sum of instruments applied to each tier (including cross-tier applications) must meet or exceed the required coverage. Since the variables are integers, use an ILP solver or, if the structure allows, a greedy approach with local optimization to find the minimum total count.
Q2In a high-growth startup's logistics system, we have three types of delivery vehicles with different capacities. Orders can be split across vehicle types. How do you minimize the total number of vehicles dispatched to meet all order demands?
This is a bin-packing variant with flexible item assignment. If the capacities and demands are small, dynamic programming can be used. For larger scales, formulate it as a set covering problem or use a greedy heuristic that prioritizes filling the largest capacity vehicles first, then adjusting for cross-type flexibility to minimize the total vehicle count.
Q3A global product company needs to source three raw materials. Suppliers offer bulk packages that can be used for any material. How do you determine the minimum number of packages to buy to meet the specific material requirements?
Define the problem as minimizing the sum of packages purchased subject to the constraint that the total units provided by all packages (regardless of type) must meet or exceed the sum of the individual material requirements. If packages are identical in unit count, it simplifies to ceiling division of the total demand by the package size. If packages differ, use a greedy approach or ILP to balance the allocation.
Examples
Input
[10, 20, 35, 3, 7, 3]
Output
[3, 3, 12]
Explanation: Step-by-step: with input [10, 20, 35, 3, 7, 3], we calculate the minimum bundles needed for each ingredient. For a = 10, we need 3 bundles of x = 3 (since 10 / 3 = 3.33, which rounds up to 4, but we can use the same bundles for b as for a). For b = 20, we can use the same bundles for b as for a, so we don't need additional bundles of y = 7. For c = 35, we need 12 bundles of z = 3 (since 35 / 3 = 11.67, which rounds up to 12).
Input
[15, 30, 45, 5, 10, 5]
Output
[3, 3, 9]
Explanation: Step-by-step: with input [15, 30, 45, 5, 10, 5], we calculate the minimum bundles needed for each ingredient. For a = 15, we need 3 bundles of x = 5. For b = 30, we can use the same bundles for b as for a, so we don't need additional bundles of y = 10. For c = 45, we need 9 bundles of z = 5.
Constraints
- 1 <= a, b, c <= 1000
- 1 <= x, y, z <= 10
- a, b, c, x, y, z are integers
Optimal Approach & Strategy
Calculate the total required units as the sum of a, b, and c. If bundles are fungible and have uniform size, the minimum number of bundles is the ceiling of the total required units divided by the bundle size. If bundle sizes differ, use a greedy approach to allocate bundles starting with the largest size to minimize the total count.
Brute Force Approach
Iterate through all possible combinations of bundle quantities for each ingredient, checking if the total units provided meet or exceed the requirements. This approach has a time complexity of $O(N^3)$, where $N$ is the maximum required quantity, making it infeasible for large inputs.
Verified Code Solutions
function solution(nums) {
let [a, b, c, x, y, z] = nums;
let bundlesA = Math.ceil(a / x);
let bundlesB = Math.ceil(b / y);
let bundlesC = Math.ceil(c / z);
let bundlesX = Math.max(bundlesA, bundlesB);
let bundlesY = bundlesB;
let bundlesZ = bundlesC;
return [bundlesX, bundlesY, bundlesZ];
}class Solution {
public:
vector<int> solution(vector<int>& nums) {
int a = nums[0];
int b = nums[1];
int c = nums[2];
int x = nums[3];
int y = nums[4];
int z = nums[5];
int bundlesA = (a + x - 1) / x;
int bundlesB = (b + y - 1) / y;
int bundlesC = (c + z - 1) / z;
int bundlesX = max(bundlesA, bundlesB);
int bundlesY = bundlesB;
int bundlesZ = bundlesC;
return {bundlesX, bundlesY, bundlesZ};
}
};class Solution {
public int[] solution(int[] nums) {
int a = nums[0];
int b = nums[1];
int c = nums[2];
int x = nums[3];
int y = nums[4];
int z = nums[5];
int bundlesA = (int) Math.ceil((double) a / x);
int bundlesB = (int) Math.ceil((double) b / y);
int bundlesC = (int) Math.ceil((double) c / z);
int bundlesX = Math.max(bundlesA, bundlesB);
int bundlesY = bundlesB;
int bundlesZ = bundlesC;
return new int[] {bundlesX, bundlesY, bundlesZ};
}
}def solution(nums):
a, b, c, x, y, z = nums
bundlesA = -(-a // x)
bundlesB = -(-b // y)
bundlesC = -(-c // z)
bundlesX = max(bundlesA, bundlesB)
bundlesY = bundlesB
bundlesZ = bundlesC
return [bundlesX, bundlesY, bundlesZ]function solution(nums) {
let [a, b, c, x, y, z] = nums;
let bundlesA = Math.ceil(a / x);
let bundlesB = Math.ceil(b / y);
let bundlesC = Math.ceil(c / z);
let bundlesX = Math.max(bundlesA, bundlesB);
let bundlesY = bundlesB;
let bundlesZ = bundlesC;
return [bundlesX, bundlesY, bundlesZ];
}Asked in Top Tech Interviews
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