LeetCode Problem Workspace
Minimum Numbers of Function Calls to Make Target Array
Calculate the minimum number of modify calls needed to convert an all-zero array into a given target array using greedy and bit manipulation techniques.
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Practice Focus
Medium · Greedy choice plus invariant validation
Answer-first summary
Calculate the minimum number of modify calls needed to convert an all-zero array into a given target array using greedy and bit manipulation techniques.
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This problem requires computing the fewest operations to make a zero-initialized array equal to a given target array. Using a greedy approach combined with bit manipulation, we can work backwards from the target to reduce operations. By repeatedly halving even numbers and subtracting one from odd numbers, we track the total modify calls efficiently.
Problem Statement
You are given an integer array nums of length n. Initially, you have an array arr of the same length filled with zeros, and a modify function that can either increment any single element by 1 or double all elements of arr in one call. Your task is to determine the minimum number of function calls needed to transform arr into nums.
Return the minimum number of function calls required to make arr identical to nums. The array length n can be up to 10^5 and each nums[i] can range from 0 to 10^9, requiring careful tracking of incremental and doubling operations while applying a greedy choice plus invariant validation pattern.
Examples
Example 1
Input: nums = [1,5]
Output: 5
Increment by 1 (second element): [0, 0] to get [0, 1] (1 operation). Double all the elements: [0, 1] -> [0, 2] -> [0, 4] (2 operations). Increment by 1 (both elements) [0, 4] -> [1, 4] -> [1, 5] (2 operations). Total of operations: 1 + 2 + 2 = 5.
Example 2
Input: nums = [2,2]
Output: 3
Increment by 1 (both elements) [0, 0] -> [0, 1] -> [1, 1] (2 operations). Double all the elements: [1, 1] -> [2, 2] (1 operation). Total of operations: 2 + 1 = 3.
Example 3
Input: nums = [4,2,5]
Output: 6
(initial)[0,0,0] -> [1,0,0] -> [1,0,1] -> [2,0,2] -> [2,1,2] -> [4,2,4] -> .
Constraints
- 1 <= nums.length <= 105
- 0 <= nums[i] <= 109
Solution Approach
Work Backwards from Target
Instead of incrementing from zero, analyze the problem in reverse: for each element, if it is even, divide by two; if odd, subtract one. Each subtraction represents an increment operation, each division represents a doubling operation, allowing greedy accumulation of minimum calls.
Track Operations Efficiently
Maintain a counter for total increments (for odd elements) and track the maximum number of doubling operations needed across all elements. This ensures no redundant doubling is counted and leverages the greedy pattern where doubling applies simultaneously to all elements.
Combine Counts for Final Answer
After processing all elements, sum the total increment operations and the maximum doubling operations. This sum represents the minimum number of function calls required to construct the target array from zeros following the greedy plus invariant validation pattern.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
Time complexity is O(n * log(max(nums[i]))) because each element may be halved up to log(max value) times. Space complexity is O(1) beyond input storage since we only track counters and the maximum doubling depth.
What Interviewers Usually Probe
- Mentions working backwards or reverse operations from target array.
- Hints at bit manipulation or handling even/odd numbers separately.
- Focuses on greedy accumulation of increments and maximum doubling.
Common Pitfalls or Variants
Common pitfalls
- Incrementing forwards without considering doubling causes overcounting operations.
- Treating doubling per element rather than globally leads to suboptimal solutions.
- Ignoring edge cases where elements are zero or powers of two can miscount operations.
Follow-up variants
- Allow decrement operations instead of increments and recalculate minimal steps.
- Change the doubling operation to triple all elements, requiring updated greedy logic.
- Target array contains negative numbers, requiring separate handling for subtractive steps.
FAQ
What is the core pattern in Minimum Numbers of Function Calls to Make Target Array?
The core pattern is a greedy choice plus invariant validation, where you work backwards from the target array, subtracting one for odd numbers and halving even numbers.
How do I calculate doubling operations efficiently?
Track the maximum number of consecutive divisions by two across all elements; each division corresponds to one doubling operation in the forward construction.
Why work backwards instead of forwards from zeros?
Working backwards avoids overcounting increments and ensures doubling operations are applied only when necessary, aligning with the greedy minimal operation pattern.
Can this method handle large arrays up to 10^5 elements?
Yes, because each element is processed in O(log(max value)) steps, ensuring overall efficiency even for maximum constraints.
How are odd and even numbers treated differently?
Odd numbers require an increment operation (subtract one when reversing), while even numbers are halved (reverse of doubling), reflecting the pattern used to minimize total function calls.
Solution
Solution 1
#### Python3
class Solution:
def minOperations(self, nums: List[int]) -> int:
return sum(v.bit_count() for v in nums) + max(0, max(nums).bit_length() - 1)Continue Topic
array
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