3701. Compute Alternating Sum
Problem Description
You are given an integer array nums.
The alternating sum of nums is calculated by adding the elements located at even indices and subtracting the elements located at odd indices. In other words, the result is:
nums[0] - nums[1] + nums[2] - nums[3] + ...
Your task is to return an integer representing this alternating sum.
For example, if nums = [4, 2, 5, 3], the alternating sum is 4 - 2 + 5 - 3 = 4. Index 0 holds 4 (added), index 1 holds 2 (subtracted), index 2 holds 5 (added), and index 3 holds 3 (subtracted).
Note that indices start from 0, so the first element is always added. If the array contains only one element, the answer is simply that element itself.
How We Pick the Algorithm
Why Simulation / Basic DSA?
This problem maps to Simulation / Basic DSA through a short path in the full flowchart.
Adding even-indexed elements and subtracting odd-indexed ones is a simple linear accumulation with no search or optimization.
Open in FlowchartIntuition
The problem directly tells us the rule: elements at even indices contribute positively to the result, while elements at odd indices contribute negatively. There is no hidden trick or transformation needed — we just need to follow the definition.
This naturally leads to two equivalent ways of thinking:
-
Simulation: Walk through the array one element at a time. For each index
i, check its parity — ifiis even, addnums[i]to a running total; ifiis odd, subtract it. After one pass, the running total is the answer. -
Grouping: Instead of deciding add-or-subtract per element, we can split the array into two groups — all elements at even indices and all elements at odd indices. The answer is simply:
sum(even-indexed elements) - sum(odd-indexed elements)In Python, slicing makes this elegant:
nums[0::2]picks every element starting at index0with step2(the even positions), andnums[1::2]picks the odd positions. Subtracting the two sums gives the result in one line.
Both views perform the same arithmetic and visit every element exactly once, so the time complexity is O(n). The simulation uses O(1) extra space, while the slicing version uses O(n) space for the temporary sublists — a small trade-off for conciseness.
Solution Approach
Solution 1: Simulation
Following the reference approach, we traverse the array nums once and maintain a running answer:
- Initialize an accumulator, say
ans = 0. - Iterate over each index
ifrom0ton - 1:- If
iis even (i % 2 == 0), updateans += nums[i]. - If
iis odd, updateans -= nums[i].
- If
- Return
ansafter the loop finishes.
A compact way to express the per-element sign is (-1) ** i, since it evaluates to 1 for even i and -1 for odd i.
The provided solution implements the same logic using Python's slice syntax instead of an explicit loop:
class Solution:
def alternatingSum(self, nums: List[int]) -> int:
return sum(nums[0::2]) - sum(nums[1::2])
How it works:
nums[0::2]— starts at index0and takes every second element, collecting all even-indexed values.nums[1::2]— starts at index1and takes every second element, collecting all odd-indexed values.- The final result is
sum(even part) - sum(odd part), which matches the definitionnums[0] - nums[1] + nums[2] - nums[3] + ...exactly.
Data structures and patterns: no auxiliary data structure beyond the array itself is required — this is a straightforward linear scan / accumulation pattern. The slicing version trades a bit of memory for readability by materializing two sublists.
Complexity analysis:
- Time complexity:
O(n), wherenis the length ofnums, since every element is visited exactly once. - Space complexity:
O(1)for the explicit loop version;O(n)for the slicing version because the two slices create temporary lists.
An equivalent loop-based implementation in Python looks like this:
class Solution:
def alternatingSum(self, nums: List[int]) -> int:
ans = 0
for i, x in enumerate(nums):
if i % 2 == 0:
ans += x
else:
ans -= x
return ans
Both versions produce identical results; choose the slicing one for brevity or the loop for constant extra space.
Example Walkthrough
Let's trace the solution with nums = [4, 2, 5, 3]. The expected answer is 4 - 2 + 5 - 3 = 4.
Approach 1: Simulation (linear scan)
We initialize ans = 0 and walk through the array, checking the parity of each index:
| Step | Index i | nums[i] | Parity of i | Operation | ans after step |
|---|---|---|---|---|---|
| 1 | 0 | 4 | even | ans += 4 | 4 |
| 2 | 1 | 2 | odd | ans -= 2 | 2 |
| 3 | 2 | 5 | even | ans += 5 | 7 |
| 4 | 3 | 3 | odd | ans -= 3 | 4 |
After the loop finishes, we return ans = 4. ✓
Approach 2: Slicing (grouping by parity)
Instead of deciding the sign element by element, we split the array into two groups:
- Even-indexed elements:
nums[0::2]→[4, 5], sosum(nums[0::2]) = 9 - Odd-indexed elements:
nums[1::2]→[2, 3], sosum(nums[1::2]) = 5
The result is 9 - 5 = 4. ✓
Both approaches perform exactly the same arithmetic — the simulation interleaves additions and subtractions (4 - 2 + 5 - 3), while the slicing version regroups them ((4 + 5) - (2 + 3)). Since addition is associative and commutative, the two computations are equivalent and visit each of the n = 4 elements exactly once.
Edge case check: for a single-element array like nums = [7], the even slice is [7] and the odd slice is empty (sum = 0), giving 7 - 0 = 7 — the element itself, as the problem specifies.
Solution Implementation
1from typing import List
2
3
4class Solution:
5 def alternatingSum(self, nums: List[int]) -> int:
6 # Sum of elements at even indices (0, 2, 4, ...)
7 even_index_sum = sum(nums[0::2])
8 # Sum of elements at odd indices (1, 3, 5, ...)
9 odd_index_sum = sum(nums[1::2])
10 # The alternating sum is the even-index sum minus the odd-index sum
11 return even_index_sum - odd_index_sum
12```
13
14**Explanation:**
15
161. **Slicing with step:** `nums[0::2]` takes every element starting from index 0 with a step of 2, collecting all even-indexed elements. Similarly, `nums[1::2]` collects all odd-indexed elements.
172. **Computing the result:** The alternating sum is defined as `nums[0] - nums[1] + nums[2] - nums[3] + ...`, which simplifies to (sum of even-indexed elements) − (sum of odd-indexed elements).
183. **Complexity:** Time complexity is O(n) and space complexity is O(n) due to slicing creating new lists.
19
20**Alternative approach (O(1) extra space):**
21
22```python3
23from typing import List
24
25
26class Solution:
27 def alternatingSum(self, nums: List[int]) -> int:
28 # Accumulate the result by adding even-indexed values
29 # and subtracting odd-indexed values
30 result = 0
31 for index, value in enumerate(nums):
32 # Add when index is even, subtract when index is odd
33 result += value if index % 2 == 0 else -value
34 return result
351class Solution {
2 /**
3 * Calculates the alternating sum of the given array.
4 * Elements at even indices are added, while elements at odd indices are subtracted.
5 *
6 * @param nums the input array of integers
7 * @return the alternating sum of the array
8 */
9 public int alternatingSum(int[] nums) {
10 // Initialize the result to store the alternating sum
11 int result = 0;
12
13 // Iterate through each element of the array
14 for (int index = 0; index < nums.length; index++) {
15 if (index % 2 == 0) {
16 // Add the element if its index is even
17 result += nums[index];
18 } else {
19 // Subtract the element if its index is odd
20 result -= nums[index];
21 }
22 }
23
24 // Return the final alternating sum
25 return result;
26 }
27}
281class Solution {
2public:
3 int alternatingSum(vector<int>& nums) {
4 // Initialize the result accumulator
5 int result = 0;
6
7 // Traverse the array, tracking each element's index
8 for (int index = 0; index < nums.size(); ++index) {
9 // Elements at even indices are added,
10 // while elements at odd indices are subtracted
11 if (index % 2 == 0) {
12 result += nums[index];
13 } else {
14 result -= nums[index];
15 }
16 }
17
18 // Return the final alternating sum
19 return result;
20 }
21};
22```
23
24**Explanation of changes:**
25
261. **Naming standardization**: Renamed `ans` to `result` and `i` to `index` for clarity, while keeping the method name `alternatingSum` unchanged.
27
282. **Readability improvement**: Replaced the ternary expression with an explicit `if-else` branch, making the add/subtract logic easier to follow at a glance.
29
303. **Comments**: Added English comments describing the accumulator initialization, the alternating add/subtract rule based on index parity, and the return value.
31
32**Alternative perspective**: If you prefer a more compact style, the original ternary form is equally valid:
33
34```cpp
35class Solution {
36public:
37 int alternatingSum(vector<int>& nums) {
38 int result = 0;
39 // Add values at even indices, subtract values at odd indices
40 for (int index = 0; index < nums.size(); ++index) {
41 result += (index % 2 == 0) ? nums[index] : -nums[index];
42 }
43 return result;
44 }
45};
461/**
2 * Calculates the alternating sum of an array.
3 * Elements at even indices are added, while elements at odd indices are subtracted.
4 *
5 * @param nums - The input array of numbers
6 * @returns The alternating sum of the array
7 */
8function alternatingSum(nums: number[]): number {
9 // Initialize the result accumulator
10 let ans: number = 0;
11
12 // Iterate through each element of the array
13 for (let i = 0; i < nums.length; ++i) {
14 // Add the element if its index is even, otherwise subtract it
15 ans += i % 2 === 0 ? nums[i] : -nums[i];
16 }
17
18 // Return the final alternating sum
19 return ans;
20}
21```
22
23**Explanation of the approach:**
24
251. **Initialization**: A variable `ans` starts at `0` to accumulate the result.
262. **Iteration**: A standard `for` loop traverses every index of the input array.
273. **Sign determination**: The ternary expression `i % 2 === 0 ? nums[i] : -nums[i]` checks the parity of the index — even indices contribute positively, odd indices contribute negatively.
284. **Return**: After processing all elements, the accumulated alternating sum is returned.
29
30**Complexity analysis:**
31
32- **Time complexity**: `O(n)`, where `n` is the length of `nums`, since each element is visited exactly once.
33- **Space complexity**: `O(1)`, as only a constant amount of extra space is used.
34
35**Alternative perspective** — the same logic can be expressed functionally using `reduce`:
36
37```typescript
38/**
39 * Functional variant: computes the alternating sum using Array.prototype.reduce.
40 *
41 * @param nums - The input array of numbers
42 * @returns The alternating sum of the array
43 */
44function alternatingSum(nums: number[]): number {
45 // Accumulate values, flipping the sign based on index parity
46 return nums.reduce((acc, val, i) => acc + (i % 2 === 0 ? val : -val), 0);
47}
48Time and Space Complexity
-
Time Complexity:
O(n), wherenis the length of the arraynums. The expressionnums[0::2]traverses the elements at even indices andnums[1::2]traverses the elements at odd indices. Together, each element of the array is visited exactly once, and eachsumcall performs a linear pass over its slice, giving a total ofO(n)work. -
Space Complexity:
O(1). The algorithm only computes two running sums and subtracts them, so no extra space that grows with the input is logically required. (Strictly speaking, the slicesnums[0::2]andnums[1::2]create temporary lists of total sizeO(n)in Python; this can be reduced to trueO(1)by using generator-based iteration, e.g.sum(nums[::2]) - sum(x for x in nums[1::2])rewritten withitertools.islice, while the asymptotic time remains unchanged.)
Pattern Learn more about how to find time and space complexity quickly.
Common Pitfalls
Pitfall 1: Inverting the Signs Due to Index-Parity Confusion
The most frequent mistake is mixing up which indices are added and which are subtracted. Some problems (or one's mental model) treat positions as 1-indexed ("the 1st element, the 2nd element, ..."), where the 1st element sits at an odd position. Carrying that thinking into this 0-indexed problem flips every sign and produces the negated answer.
Buggy code:
class Solution:
def alternatingSum(self, nums: List[int]) -> int:
ans = 0
for i, x in enumerate(nums):
# WRONG: subtracts even indices, adds odd indices
if i % 2 == 0:
ans -= x
else:
ans += x
return ans
For nums = [4, 2, 5, 3], this returns -4 instead of 4. Sneakily, this bug can pass tests where the correct answer happens to be 0 (e.g., nums = [1, 1]), so a couple of green test cases may give false confidence.
A close cousin of this bug appears in the slicing version when the slice start points are swapped:
# WRONG: odd-indexed sum minus even-indexed sum
return sum(nums[1::2]) - sum(nums[0::2])
Solution: Anchor your reasoning on a concrete fact from the problem statement: nums[0] is always added. Index 0 is even, so even indices get + and odd indices get -. Verify with the smallest non-trivial example before submitting — for [4, 2] the answer must be 4 - 2 = 2, not -2:
class Solution:
def alternatingSum(self, nums: List[int]) -> int:
ans = 0
for i, x in enumerate(nums):
ans += x if i % 2 == 0 else -x # even index -> add, odd index -> subtract
return ans
Pitfall 2: Alternating on Values Instead of Indices
Another subtle error is toggling the sign based on something other than the index — for example, resetting a sign flag inside a condition, or iterating over a filtered/modified sequence so element positions no longer match their original indices:
Buggy code:
class Solution:
def alternatingSum(self, nums: List[int]) -> int:
ans = 0
sign = 1
for x in nums:
if x != 0: # WRONG: skipping zeros desynchronizes the sign
ans += sign * x
sign = -sign
return ans
Skipping zeros looks harmless ("adding or subtracting 0 changes nothing"), but because sign is only flipped for non-zero elements, every element after a zero gets the wrong sign. For nums = [1, 0, 2] the correct answer is 1 - 0 + 2 = 3, yet this code returns 1 - 2 = -1.
Solution: Tie the sign strictly to the element's original index and never conditionally skip the sign update. Either flip the sign unconditionally on every iteration, or derive it directly from the index (i % 2), as in the corrected code above. The slicing approach sum(nums[0::2]) - sum(nums[1::2]) is also immune to this bug since positions are fixed by the slice itself.
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