Problem Description

You have an array called nums, and each element in the array is associated with an index from 0 to the length of the array minus 1. Your goal is to calculate the sum of elements whose indexes have a specific number of bits set to 1 when viewed in binary. This specific number of set bits is given by the integer k.

For instance, if k is 3 and you're looking at the index 7 (which is 0111 in binary), since 7 has three set bits, you would include the value of nums[7] in your sum. You repeat this process for all indices in the nums array and return the total sum.

Intuition

To solve this problem efficiently, you can iterate over each index of the nums array while simultaneously checking how many set bits are in the binary representation of that index. If the count of set bits is equal to k, then you add the value at that index to your running total sum.

For the implementation:

1. You iterate through nums array using enumerate to get both index (i) and value (x).
2. For each index i, you check how many set bits it has by using the built-in Python method bit_count().
3. If i.bit_count() is equal to k, you include the value x in your sum.
4. You aggregate these values to determine the final answer, which is the sum of all the numbers at indices with k set bits.

This solution is direct and leverages Python's built-in functions to handle binary operations efficiently, avoiding the need for manual bit-counting or more complex bitwise manipulation.

Solution Approach

The solution uses a simple linear scan algorithm which is quite effective for this problem. No complex data structures or patterns are necessary.

Here is a step-by-step breakdown of the implementation using the provided solution code as a reference:

1. We use a for loop in combination with enumerate to iterate over the nums array. This gives us access to both the index and the value at that index.
2. For each index i, we check the number of set bits in its binary representation. This is done using the bit_count() method.
3. We include the value x if the number of set bits (1s in the binary form) of the index i is equal to k. The condition i.bit_count() == k evaluates to True or False based on whether the number of set bits matches k.
4. We use the built-in sum function to add up all the values x for which the corresponding index i satisfies the condition mentioned above.

In terms of algorithms and data structures, this solution can be categorized as a brute-force approach since it evaluates the condition for every element in the array without using a more nuanced algorithm or data structure to optimize the process. However, since the problem has to do with checking each index individually, the brute-force method is appropriate and efficient in this context, especially since Python's bit_count() method is optimized internally.

The sum expression employs a generator expression, which is more memory-efficient than building an intermediary list of elements to sum, as it calculates the sum on the fly.

In short, the solution is elegant, taking advantage of Python's language features to achieve the goal with minimal code.

Example Walkthrough

Let's take an example where nums = [0, 1, 2, 4, 8, 16, 3, 5] and k = 2. We want to find the sum of elements in nums where the indices have exactly two bits set to 1 when viewed in binary.

1. We start with the first element, nums[0] which is 0. The index 0 has 0 bits set to 1 when viewed in binary (0b0), so it does not match k.
2. Move to the second element, nums[1] which is 1. The index 1 has 1 bit set to 1 in binary (0b1), so it also does not match k.
3. Continue to nums[2] which is 2. The index 2 in binary is 0b10, which has 1 bit set, not matching k.
4. Next is nums[3] at index 3, with a value of 4. Binary 3 is 0b11, which has 2 bits set. This matches k, so we include 4 in our sum.
5. Move to nums[4], value 8 and index 4(0b100). Index 4 has only 1 bit set, so we do not add 8 to our sum.
6. For nums[5], value 16 at index 5(0b101), the binary representation has 2 set bits, matching k. We include 16.
7. At nums[6], index 6 (0b110) has 2 bits set. Since k is 2, we add value 3 to our sum.
8. Lastly, nums[7] is 5, with index 7 (0b111), which has 3 set bits, so it's not included.

Now, we sum up the included numbers: 4 + 16 + 3 = 23.

So the final sum of the values in the nums array at indices with exactly k set bits is 23.

Using the provided solution approach:

1. We iterate over nums with the index and value.
2. For each index, we use bit_count() to count set bits.
3. We check if the count matches k
4. We conclude with the sum of values that match the condition.

In this example, we can see how the algorithm steps through each element and uses the bit_count() function to filter and sum the correct elements efficiently.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the provided method sumIndicesWithKSetBits primarily depends on the number of elements in nums and the computation of bit_count for each index.

• Iterating over all indices of the nums list: If there are n elements in nums, this requires O(n) time.
• For each index i, the bit_count() operation is performed, which takes O(log(i)) time since it counts the number of set bits in the binary representation of the index.

Since i ranges from 0 to n-1, in the worst-case scenario, bit_count() will be called with i = n-1, which takes O(log(n-1)) time. Therefore, the combined time complexity for the computation over all indices would be O(n log(n)).

Overall, the Time Complexity is O(n log(n)).

Space Complexity

The space complexity is fairly straightforward to analyze:

• The sum function generates a temporary generator expression and does not store values, thus it uses constant space.
• No additional data structures are used that scale with the input size.

Therefore, the Space Complexity is O(1) — constant space complexity, since extra space does not depend on the input size.

Learn more about how to find time and space complexity quickly using problem constraints.