This problem requires verifying whether integers in nums can be distributed to match each customer's quantity exactly. Using state transition dynamic programming with bitmasking allows efficient tracking of assignment possibilities. The key is counting frequencies of each unique number and recursively attempting distributions while pruning impossible paths.
Problem Statement
You are given an array nums of n integers, where at most 50 unique values exist, and an array quantity of m customer orders. Each quantity[i] specifies how many identical integers the ith customer requests. Determine if it's possible to assign integers from nums so that each customer receives exactly quantity[i] of identical numbers without sharing integers across customers.
Return true if such a distribution exists. For example, nums = [1,2,3,4], quantity = [2] returns false since no customer can get two identical numbers, while nums = [1,1,2,2], quantity = [2,2] returns true as each customer can receive a matching pair.
Examples
Example 1
Input: nums = [1,2,3,4], quantity = [2]
Output: false
The 0th customer cannot be given two different integers.
Example 2
Input: nums = [1,2,3,3], quantity = [2]
Output: true
The 0th customer is given [3,3]. The integers [1,2] are not used.
Example 3
Input: nums = [1,1,2,2], quantity = [2,2]
Output: true
The 0th customer is given [1,1], and the 1st customer is given [2,2].
Constraints
- n == nums.length
- 1 <= n <= 105
- 1 <= nums[i] <= 1000
- m == quantity.length
- 1 <= m <= 10
- 1 <= quantity[i] <= 105
- There are at most 50 unique values in nums.
Solution Approach
Count Frequencies
Calculate the frequency of each unique number in nums. For example, nums = [4,4,5,5,5] gives frequencies [2,3]. This allows mapping available integers to customer requests efficiently.
Use State Transition DP with Bitmask
Apply dynamic programming with bitmasking to represent which customers have been satisfied. Each DP state corresponds to a subset of customers, and transitions allocate available numbers to unfulfilled orders while pruning impossible combinations.
Recursive Backtracking with Pruning
Combine DP with backtracking to attempt distributions. Recursively assign frequencies to remaining customers, skipping paths where a frequency is insufficient. This leverages both the limited unique numbers and small customer count to reduce computation.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
Time complexity depends on the number of unique numbers and customer orders, roughly O(2^m * n) in the worst case due to all DP states. Space complexity is O(2^m) for storing DP states.
What Interviewers Usually Probe
- Counting frequencies before attempting assignment indicates understanding of data reduction for DP.
- Implementing bitmask DP shows awareness of handling subsets efficiently in state transition problems.
- Pruning impossible paths during backtracking signals optimization for hard cases with repeated numbers.
Common Pitfalls or Variants
Common pitfalls
- Failing to account for identical numbers and giving a customer distinct integers instead.
- Overlooking pruning opportunities, leading to TLE for larger frequency distributions.
- Incorrectly initializing DP states, causing invalid assignments to propagate.
Follow-up variants
- Distribute integers allowing each customer to receive either one or two identical numbers.
- Maximize the number of customers satisfied given limited repeated integers in nums.
- Distribute integers with constraints on specific integer types per customer.
How GhostInterview Helps
- Automatically counts frequencies and maps them to customer orders for efficient assignment.
- Implements bitmask-based state transition DP to explore valid distributions systematically.
- Detects impossible paths early, avoiding wasted computation and highlighting distribution failures.
Topic Pages
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FAQ
What is the main pattern used in Distribute Repeating Integers?
The problem relies on state transition dynamic programming, tracking subsets of satisfied customers via bitmasking.
How do frequencies of nums help in solving this problem?
Counting frequencies reduces the search space by letting you know exactly how many of each number are available for allocation.
Why is backtracking needed along with DP?
Backtracking allows trying different allocation orders of numbers to customers while DP avoids recomputing identical states.
What causes a false return in example inputs?
If a customer’s requested quantity cannot be fulfilled with identical numbers from nums, the function returns false.
Can this approach scale for large n?
Yes, because DP operates on at most 50 unique numbers and up to 10 customers, keeping computation feasible even if n is large.
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