1675. Minimize Deviation in Array - Explanation

Prerequisites

Before attempting this problem, you should be comfortable with:

Heap / Priority Queue - Using min-heap or max-heap to efficiently track extreme values
Sliding Window - Maintaining a valid window while tracking which elements are represented
Sorting - Sorting pairs of values and using sorted order to find optimal ranges
Bit Manipulation - Understanding odd/even properties and operations like doubling and halving

1. Sorting + Sliding Window

Intuition

Each element in the array can only take certain values: odd numbers can become themselves or double themselves, while even numbers can be halved repeatedly until they become odd. The key insight is that we can precompute all possible values each element can take, then find the smallest range that contains at least one value from each original element.

This transforms the problem into finding the smallest window in a sorted list of values where each original array element is represented at least once. We use a sliding window approach on the sorted list of all possible values, tracking which original elements are covered.

Algorithm

For each number in the input array, generate all possible values it can take:
- If odd: it can be itself or doubled (only once).
- If even: it can be halved repeatedly down to an odd number, plus all intermediate values.
Store each value paired with its original index.
Sort all these pairs by value.
Use a sliding window with two pointers i and j:
- Expand j to include more values and track how many original elements are covered.
- When all n elements are covered, shrink from i to minimize the window range.
- Update the result with the minimum difference between the largest and smallest values in valid windows.
Return the minimum deviation found.

class Solution:
    def minimumDeviation(self, nums: List[int]) -> int:
        n = len(nums)
        arr = []
        for i, num in enumerate(nums):
            if num & 1:
                arr.append((num, i))
                arr.append((num * 2, i))
            else:
                while num % 2 == 0:
                    arr.append((num, i))
                    num //= 2
                arr.append((num, i))

        arr.sort()
        res = float("inf")

        seen = [0] * n
        count = i = 0
        for j in range(len(arr)):
            seen[arr[j][1]] += 1
            if seen[arr[j][1]] == 1:
                count += 1
                while count == n:
                    res = min(res, arr[j][0] - arr[i][0])
                    seen[arr[i][1]] -= 1
                    if seen[arr[i][1]] == 0:
                        count -= 1
                    i += 1

        return res

public class Solution {
    public int minimumDeviation(int[] nums) {
        int n = nums.length;
        List<int[]> arr = new ArrayList<>();

        for (int i = 0; i < n; i++) {
            int num = nums[i];
            if (num % 2 == 1) {
                arr.add(new int[]{num, i});
                arr.add(new int[]{num * 2, i});
            } else {
                while (num % 2 == 0) {
                    arr.add(new int[]{num, i});
                    num /= 2;
                }
                arr.add(new int[]{num, i});
            }
        }

        arr.sort(Comparator.comparingInt(a -> a[0]));
        int res = Integer.MAX_VALUE;

        int[] seen = new int[n];
        int count = 0, i = 0;

        for (int j = 0; j < arr.size(); j++) {
            seen[arr.get(j)[1]]++;
            if (seen[arr.get(j)[1]] == 1) {
                count++;
                while (count == n) {
                    res = Math.min(res, arr.get(j)[0] - arr.get(i)[0]);
                    seen[arr.get(i)[1]]--;
                    if (seen[arr.get(i)[1]] == 0) {
                        count--;
                    }
                    i++;
                }
            }
        }
        return res;
    }
}

class Solution {
public:
    int minimumDeviation(vector<int>& nums) {
        int n = nums.size();
        vector<pair<int, int>> arr;

        for (int i = 0; i < n; i++) {
            int num = nums[i];
            if (num % 2 == 1) {
                arr.emplace_back(num, i);
                arr.emplace_back(num * 2, i);
            } else {
                while (num % 2 == 0) {
                    arr.emplace_back(num, i);
                    num /= 2;
                }
                arr.emplace_back(num, i);
            }
        }

        sort(arr.begin(), arr.end());
        int res = INT_MAX;

        vector<int> seen(n, 0);
        int count = 0, i = 0;

        for (int j = 0; j < arr.size(); j++) {
            seen[arr[j].second]++;
            if (seen[arr[j].second] == 1) {
                count++;
                while (count == n) {
                    res = min(res, arr[j].first - arr[i].first);
                    seen[arr[i].second]--;
                    if (seen[arr[i].second] == 0) {
                        count--;
                    }
                    i++;
                }
            }
        }
        return res;
    }
};

class Solution {
    /**
     * @param {number[]} nums
     * @return {number}
     */
    minimumDeviation(nums) {
        let n = nums.length;
        let arr = [];

        for (let i = 0; i < n; i++) {
            let num = nums[i];
            if (num % 2 === 1) {
                arr.push([num, i]);
                arr.push([num * 2, i]);
            } else {
                while (num % 2 === 0) {
                    arr.push([num, i]);
                    num /= 2;
                }
                arr.push([num, i]);
            }
        }

        arr.sort((a, b) => a[0] - b[0]);
        let res = Infinity;

        let seen = new Array(n).fill(0);
        let count = 0,
            i = 0;

        for (let j = 0; j < arr.length; j++) {
            seen[arr[j][1]]++;
            if (seen[arr[j][1]] === 1) {
                count++;
                while (count === n) {
                    res = Math.min(res, arr[j][0] - arr[i][0]);
                    seen[arr[i][1]]--;
                    if (seen[arr[i][1]] === 0) {
                        count--;
                    }
                    i++;
                }
            }
        }
        return res;
    }
}

public class Solution {
    public int MinimumDeviation(int[] nums) {
        int n = nums.Length;
        List<int[]> arr = new List<int[]>();

        for (int i = 0; i < n; i++) {
            int num = nums[i];
            if (num % 2 == 1) {
                arr.Add(new int[] { num, i });
                arr.Add(new int[] { num * 2, i });
            } else {
                while (num % 2 == 0) {
                    arr.Add(new int[] { num, i });
                    num /= 2;
                }
                arr.Add(new int[] { num, i });
            }
        }

        arr.Sort((a, b) => a[0].CompareTo(b[0]));
        int res = int.MaxValue;

        int[] seen = new int[n];
        int count = 0, idx = 0;

        for (int j = 0; j < arr.Count; j++) {
            seen[arr[j][1]]++;
            if (seen[arr[j][1]] == 1) {
                count++;
                while (count == n) {
                    res = Math.Min(res, arr[j][0] - arr[idx][0]);
                    seen[arr[idx][1]]--;
                    if (seen[arr[idx][1]] == 0) {
                        count--;
                    }
                    idx++;
                }
            }
        }
        return res;
    }
}

func minimumDeviation(nums []int) int {
    n := len(nums)
    arr := [][]int{}

    for i := 0; i < n; i++ {
        num := nums[i]
        if num%2 == 1 {
            arr = append(arr, []int{num, i})
            arr = append(arr, []int{num * 2, i})
        } else {
            for num%2 == 0 {
                arr = append(arr, []int{num, i})
                num /= 2
            }
            arr = append(arr, []int{num, i})
        }
    }

    sort.Slice(arr, func(i, j int) bool {
        return arr[i][0] < arr[j][0]
    })
    res := math.MaxInt32

    seen := make([]int, n)
    count, idx := 0, 0

    for j := 0; j < len(arr); j++ {
        seen[arr[j][1]]++
        if seen[arr[j][1]] == 1 {
            count++
            for count == n {
                if arr[j][0]-arr[idx][0] < res {
                    res = arr[j][0] - arr[idx][0]
                }
                seen[arr[idx][1]]--
                if seen[arr[idx][1]] == 0 {
                    count--
                }
                idx++
            }
        }
    }
    return res
}

class Solution {
    fun minimumDeviation(nums: IntArray): Int {
        val n = nums.size
        val arr = mutableListOf<IntArray>()

        for (i in 0 until n) {
            var num = nums[i]
            if (num % 2 == 1) {
                arr.add(intArrayOf(num, i))
                arr.add(intArrayOf(num * 2, i))
            } else {
                while (num % 2 == 0) {
                    arr.add(intArrayOf(num, i))
                    num /= 2
                }
                arr.add(intArrayOf(num, i))
            }
        }

        arr.sortBy { it[0] }
        var res = Int.MAX_VALUE

        val seen = IntArray(n)
        var count = 0
        var idx = 0

        for (j in arr.indices) {
            seen[arr[j][1]]++
            if (seen[arr[j][1]] == 1) {
                count++
                while (count == n) {
                    res = minOf(res, arr[j][0] - arr[idx][0])
                    seen[arr[idx][1]]--
                    if (seen[arr[idx][1]] == 0) {
                        count--
                    }
                    idx++
                }
            }
        }
        return res
    }
}

class Solution {
    func minimumDeviation(_ nums: [Int]) -> Int {
        let n = nums.count
        var arr = [[Int]]()

        for i in 0..<n {
            var num = nums[i]
            if num % 2 == 1 {
                arr.append([num, i])
                arr.append([num * 2, i])
            } else {
                while num % 2 == 0 {
                    arr.append([num, i])
                    num /= 2
                }
                arr.append([num, i])
            }
        }

        arr.sort { $0[0] < $1[0] }
        var res = Int.max

        var seen = [Int](repeating: 0, count: n)
        var count = 0
        var idx = 0

        for j in 0..<arr.count {
            seen[arr[j][1]] += 1
            if seen[arr[j][1]] == 1 {
                count += 1
                while count == n {
                    res = min(res, arr[j][0] - arr[idx][0])
                    seen[arr[idx][1]] -= 1
                    if seen[arr[idx][1]] == 0 {
                        count -= 1
                    }
                    idx += 1
                }
            }
        }
        return res
    }
}

Time & Space Complexity

Time complexity: $O((n \log m) * \log (n \log m))$
Space complexity: $O(n \log m)$

Where $n$ is the size of the array $nums$ and $m$ is the maximum element in $nums$ .

2. Min-Heap

Intuition

Instead of generating all possible values upfront, we can work incrementally. First, reduce every number to its minimum possible value (divide even numbers until they become odd). Then, we repeatedly try to increase the smallest element by doubling it (if possible), since increasing smaller values is the only way to reduce the deviation.

The min-heap lets us efficiently access the smallest current value. We track the maximum value in the heap separately. Each iteration, we pop the minimum, update our best deviation, and if that minimum can still be doubled, we push the doubled value back.

Algorithm

Reduce each number to its minimum form by dividing even numbers by 2 until odd. Track the maximum allowed value each element can reach.
Push each (current value, max allowed value) pair into a min-heap. Track the current maximum across all heap elements.
While the heap contains all elements:
- Pop the minimum value and compute the current deviation (max minus min).
- If this minimum can still increase (less than its max allowed), double it and push back.
- Update the tracked maximum if needed.
- If the minimum cannot increase, stop (we cannot reduce deviation further).
Return the minimum deviation found.

class Solution:
    def minimumDeviation(self, nums: List[int]) -> int:
        minHeap, heapMax = [], 0

        for n in nums:
            tmp = n
            while n % 2 == 0:
                n //= 2
            minHeap.append((n, max(tmp, 2 * n)))
            heapMax = max(heapMax, n)

        res = float("inf")
        heapq.heapify(minHeap)

        while len(minHeap) == len(nums):
            n, nMax = heapq.heappop(minHeap)
            res = min(res, heapMax - n)

            if n < nMax:
                heapq.heappush(minHeap, (n * 2, nMax))
                heapMax = max(heapMax, n * 2)

        return res

public class Solution {
    public int minimumDeviation(int[] nums) {
        PriorityQueue<int[]> minHeap = new PriorityQueue<>((a, b) -> a[0] - b[0]);
        int heapMax = 0;

        for (int num : nums) {
            int tmp = num;
            while (num % 2 == 0) {
                num /= 2;
            }
            minHeap.offer(new int[]{num, Math.max(tmp, 2 * num)});
            heapMax = Math.max(heapMax, num);
        }

        int res = Integer.MAX_VALUE;

        while (minHeap.size() == nums.length) {
            int[] minElement = minHeap.poll();
            int n = minElement[0], nMax = minElement[1];
            res = Math.min(res, heapMax - n);

            if (n < nMax) {
                minHeap.offer(new int[]{n * 2, nMax});
                heapMax = Math.max(heapMax, n * 2);
            }
        }

        return res;
    }
}

class Solution {
public:
    int minimumDeviation(vector<int>& nums) {
        priority_queue<pair<int, int>, vector<pair<int, int>>, greater<>> minHeap;
        int heapMax = 0;

        for (int num : nums) {
            int tmp = num;
            while (num % 2 == 0) {
                num /= 2;
            }
            minHeap.push({num, max(tmp, 2 * num)});
            heapMax = max(heapMax, num);
        }

        int res = INT_MAX;

        while (minHeap.size() == nums.size()) {
            auto [n, nMax] = minHeap.top();
            minHeap.pop();
            res = min(res, heapMax - n);

            if (n < nMax) {
                minHeap.push({n * 2, nMax});
                heapMax = max(heapMax, n * 2);
            }
        }

        return res;
    }
};

class Solution {
    /**
     * @param {number[]} nums
     * @return {number}
     */
    minimumDeviation(nums) {
        const minHeap = new MinPriorityQueue((x) => x[0]);
        let heapMax = 0;

        for (let num of nums) {
            let tmp = num;
            while (num % 2 === 0) {
                num /= 2;
            }
            minHeap.enqueue([num, Math.max(tmp, num * 2)]);
            heapMax = Math.max(heapMax, num);
        }

        let res = Infinity;

        while (minHeap.size() === nums.length) {
            let [n, nMax] = minHeap.dequeue();
            res = Math.min(res, heapMax - n);

            if (n < nMax) {
                minHeap.enqueue([n * 2, nMax]);
                heapMax = Math.max(heapMax, n * 2);
            }
        }

        return res;
    }
}

public class Solution {
    public int MinimumDeviation(int[] nums) {
        var minHeap = new PriorityQueue<int[], int>();
        int heapMax = 0;

        foreach (int num in nums) {
            int tmp = num;
            int n = num;
            while (n % 2 == 0) {
                n /= 2;
            }
            minHeap.Enqueue(new int[] { n, Math.Max(tmp, 2 * n) }, n);
            heapMax = Math.Max(heapMax, n);
        }

        int res = int.MaxValue;

        while (minHeap.Count == nums.Length) {
            int[] minElement = minHeap.Dequeue();
            int val = minElement[0], nMax = minElement[1];
            res = Math.Min(res, heapMax - val);

            if (val < nMax) {
                minHeap.Enqueue(new int[] { val * 2, nMax }, val * 2);
                heapMax = Math.Max(heapMax, val * 2);
            }
        }

        return res;
    }
}

class Solution {
    fun minimumDeviation(nums: IntArray): Int {
        val minHeap = PriorityQueue<IntArray>(compareBy { it[0] })
        var heapMax = 0

        for (num in nums) {
            var n = num
            val tmp = num
            while (n % 2 == 0) {
                n /= 2
            }
            minHeap.offer(intArrayOf(n, maxOf(tmp, 2 * n)))
            heapMax = maxOf(heapMax, n)
        }

        var res = Int.MAX_VALUE

        while (minHeap.size == nums.size) {
            val minElement = minHeap.poll()
            val n = minElement[0]
            val nMax = minElement[1]
            res = minOf(res, heapMax - n)

            if (n < nMax) {
                minHeap.offer(intArrayOf(n * 2, nMax))
                heapMax = maxOf(heapMax, n * 2)
            }
        }

        return res
    }
}

class Solution {
    func minimumDeviation(_ nums: [Int]) -> Int {
        var minHeap = Heap<(Int, Int)>(comparator: { $0.0 < $1.0 })
        var heapMax = 0

        for num in nums {
            var n = num
            let tmp = num
            while n % 2 == 0 {
                n /= 2
            }
            minHeap.insert((n, max(tmp, 2 * n)))
            heapMax = max(heapMax, n)
        }

        var res = Int.max

        while minHeap.count == nums.count {
            let minElement = minHeap.remove()!
            let n = minElement.0
            let nMax = minElement.1
            res = min(res, heapMax - n)

            if n < nMax {
                minHeap.insert((n * 2, nMax))
                heapMax = max(heapMax, n * 2)
            }
        }

        return res
    }
}

struct Heap<T> {
    var elements: [T] = []
    let comparator: (T, T) -> Bool

    var count: Int { elements.count }
    var isEmpty: Bool { elements.isEmpty }

    init(comparator: @escaping (T, T) -> Bool) {
        self.comparator = comparator
    }

    mutating func insert(_ value: T) {
        elements.append(value)
        siftUp(from: elements.count - 1)
    }

    mutating func remove() -> T? {
        guard !elements.isEmpty else { return nil }
        elements.swapAt(0, elements.count - 1)
        let removed = elements.removeLast()
        if !elements.isEmpty { siftDown(from: 0) }
        return removed
    }

    private mutating func siftUp(from index: Int) {
        var child = index
        var parent = (child - 1) / 2
        while child > 0 && comparator(elements[child], elements[parent]) {
            elements.swapAt(child, parent)
            child = parent
            parent = (child - 1) / 2
        }
    }

    private mutating func siftDown(from index: Int) {
        var parent = index
        while true {
            let left = 2 * parent + 1
            let right = 2 * parent + 2
            var candidate = parent
            if left < elements.count && comparator(elements[left], elements[candidate]) {
                candidate = left
            }
            if right < elements.count && comparator(elements[right], elements[candidate]) {
                candidate = right
            }
            if candidate == parent { return }
            elements.swapAt(parent, candidate)
            parent = candidate
        }
    }
}

Time & Space Complexity

Time complexity: $O(n *\log n * \log m)$
Space complexity: $O(n)$

Where $n$ is the size of the array $nums$ and $m$ is the maximum element in $nums$ .

3. Max-Heap

Intuition

We can approach this from the opposite direction: start with all numbers at their maximum possible value, then repeatedly decrease the largest element. Odd numbers are first doubled to reach their maximum. Even numbers stay as they are initially.

Using a max-heap, we always have quick access to the current largest value. We track the minimum value separately. Each iteration, we halve the maximum (if even) and update our best deviation. The process stops when the maximum is odd, since odd numbers cannot be reduced.

Algorithm

For each number, if it is odd, double it. Push all values into a max-heap. Track the current minimum value.
While the heap is not empty:
- Pop the maximum value and compute the current deviation (max minus min).
- Update the result if this deviation is smaller.
- If the maximum is odd, stop (cannot reduce it further).
- Otherwise, halve the maximum, push it back, and update the minimum if needed.
Return the minimum deviation found.

class Solution:
    def minimumDeviation(self, nums: List[int]) -> int:
        maxHeap = []
        minVal = float("inf")

        for num in nums:
            if num % 2 == 1:
                num *= 2
            heapq.heappush(maxHeap, -num)
            minVal = min(minVal, num)

        res = float("inf")

        while maxHeap:
            maxVal = -heapq.heappop(maxHeap)
            res = min(res, maxVal - minVal)
            if maxVal % 2 == 1:
                break

            nextVal = maxVal // 2
            heapq.heappush(maxHeap, -nextVal)
            minVal = min(minVal, nextVal)

        return res

public class Solution {
    public int minimumDeviation(int[] nums) {
        PriorityQueue<Integer> maxHeap = new PriorityQueue<>((a, b) -> b - a);
        int minVal = Integer.MAX_VALUE;

        for (int num : nums) {
            if (num % 2 == 1) num *= 2;
            maxHeap.offer(num);
            minVal = Math.min(minVal, num);
        }

        int res = Integer.MAX_VALUE;

        while (!maxHeap.isEmpty()) {
            int maxVal = maxHeap.poll();
            res = Math.min(res, maxVal - minVal);

            if (maxVal % 2 == 1) break;

            int nextVal = maxVal / 2;
            maxHeap.offer(nextVal);
            minVal = Math.min(minVal, nextVal);
        }

        return res;
    }
}

class Solution {
public:
    int minimumDeviation(vector<int>& nums) {
        priority_queue<int> maxHeap;
        int minVal = INT_MAX;

        for (int num : nums) {
            if (num % 2 == 1) num *= 2;
            maxHeap.push(num);
            minVal = min(minVal, num);
        }

        int res = INT_MAX;

        while (!maxHeap.empty()) {
            int maxVal = maxHeap.top();
            maxHeap.pop();
            res = min(res, maxVal - minVal);

            if (maxVal % 2 == 1) break;

            int nextVal = maxVal / 2;
            maxHeap.push(nextVal);
            minVal = min(minVal, nextVal);
        }

        return res;
    }
};

class Solution {
    /**
     * @param {number[]} nums
     * @return {number}
     */
    minimumDeviation(nums) {
        const maxHeap = new MaxPriorityQueue();
        let minVal = Infinity;

        for (let num of nums) {
            if (num % 2 === 1) num *= 2;
            maxHeap.enqueue(num);
            minVal = Math.min(minVal, num);
        }

        let res = Infinity;

        while (!maxHeap.isEmpty()) {
            let maxVal = maxHeap.dequeue();
            res = Math.min(res, maxVal - minVal);

            if (maxVal % 2 === 1) break;

            let nextVal = maxVal / 2;
            maxHeap.enqueue(nextVal);
            minVal = Math.min(minVal, nextVal);
        }

        return res;
    }
}

public class Solution {
    public int MinimumDeviation(int[] nums) {
        var maxHeap = new PriorityQueue<int, int>(Comparer<int>.Create((a, b) => b.CompareTo(a)));
        int minVal = int.MaxValue;

        foreach (int num in nums) {
            int n = num;
            if (n % 2 == 1) n *= 2;
            maxHeap.Enqueue(n, n);
            minVal = Math.Min(minVal, n);
        }

        int res = int.MaxValue;

        while (maxHeap.Count > 0) {
            int maxVal = maxHeap.Dequeue();
            res = Math.Min(res, maxVal - minVal);

            if (maxVal % 2 == 1) break;

            int nextVal = maxVal / 2;
            maxHeap.Enqueue(nextVal, nextVal);
            minVal = Math.Min(minVal, nextVal);
        }

        return res;
    }
}

class Solution {
    fun minimumDeviation(nums: IntArray): Int {
        val maxHeap = PriorityQueue<Int>(compareByDescending { it })
        var minVal = Int.MAX_VALUE

        for (num in nums) {
            var n = num
            if (n % 2 == 1) n *= 2
            maxHeap.offer(n)
            minVal = minOf(minVal, n)
        }

        var res = Int.MAX_VALUE

        while (maxHeap.isNotEmpty()) {
            val maxVal = maxHeap.poll()
            res = minOf(res, maxVal - minVal)

            if (maxVal % 2 == 1) break

            val nextVal = maxVal / 2
            maxHeap.offer(nextVal)
            minVal = minOf(minVal, nextVal)
        }

        return res
    }
}

class Solution {
    func minimumDeviation(_ nums: [Int]) -> Int {
        var maxHeap = Heap<Int>(comparator: { $0 > $1 })
        var minVal = Int.max

        for num in nums {
            var n = num
            if n % 2 == 1 { n *= 2 }
            maxHeap.insert(n)
            minVal = min(minVal, n)
        }

        var res = Int.max

        while !maxHeap.isEmpty {
            let maxVal = maxHeap.remove()!
            res = min(res, maxVal - minVal)

            if maxVal % 2 == 1 { break }

            let nextVal = maxVal / 2
            maxHeap.insert(nextVal)
            minVal = min(minVal, nextVal)
        }

        return res
    }
}

struct Heap<T> {
    var elements: [T] = []
    let comparator: (T, T) -> Bool

    var count: Int { elements.count }
    var isEmpty: Bool { elements.isEmpty }

    init(comparator: @escaping (T, T) -> Bool) {
        self.comparator = comparator
    }

    mutating func insert(_ value: T) {
        elements.append(value)
        siftUp(from: elements.count - 1)
    }

    mutating func remove() -> T? {
        guard !elements.isEmpty else { return nil }
        elements.swapAt(0, elements.count - 1)
        let removed = elements.removeLast()
        if !elements.isEmpty { siftDown(from: 0) }
        return removed
    }

    private mutating func siftUp(from index: Int) {
        var child = index
        var parent = (child - 1) / 2
        while child > 0 && comparator(elements[child], elements[parent]) {
            elements.swapAt(child, parent)
            child = parent
            parent = (child - 1) / 2
        }
    }

    private mutating func siftDown(from index: Int) {
        var parent = index
        while true {
            let left = 2 * parent + 1
            let right = 2 * parent + 2
            var candidate = parent
            if left < elements.count && comparator(elements[left], elements[candidate]) {
                candidate = left
            }
            if right < elements.count && comparator(elements[right], elements[candidate]) {
                candidate = right
            }
            if candidate == parent { return }
            elements.swapAt(parent, candidate)
            parent = candidate
        }
    }
}

Time & Space Complexity

Time complexity: $O(n *\log n * \log m)$
Space complexity: $O(n)$

Where $n$ is the size of the array $nums$ and $m$ is the maximum element in $nums$ .

Common Pitfalls

Confusing the Operations for Odd and Even Numbers

Odd numbers can only be doubled (multiplied by 2), while even numbers can only be halved (divided by 2). A frequent mistake is applying the wrong operation or allowing both operations on any number. Remember: odd numbers increase, even numbers decrease.

Not Recognizing the Stopping Condition

When using the max-heap approach, the algorithm must stop when the maximum element is odd because odd numbers cannot be reduced further. Failing to check this condition leads to infinite loops or attempting invalid operations.

Forgetting to Track the Minimum Value Separately

When using a max-heap, you only have direct access to the maximum element. The deviation requires knowing both the max and min values. A common error is trying to find the minimum by scanning the heap, which defeats the purpose of the data structure. Track the minimum separately and update it when pushing new values.

Not Generating All Possible Values for Each Element

In the sliding window approach, each number can take multiple values. Even numbers can be halved repeatedly until odd, and odd numbers can be doubled once. Missing any possible value means the algorithm might not find the true minimum deviation.

Off-by-One Errors in the Sliding Window Count

When using the sliding window with a frequency array, ensure you correctly track when all n original elements are represented. Incrementing or decrementing the count at the wrong time can cause the window to be considered valid prematurely or remain invalid when it should be valid.