2505. Bitwise OR of All Subsequence Sums

Problem Description

This problem requests the calculation of a special value derived from all possible subsequence sums of an integer array nums. To explain further, a subsequence is a series of numbers that come from the original array either entirely or after removing some elements, but with the original order preserved. An empty subsequence is also considered valid. The special value that the problem requires is the bitwise OR of all the unique subsequence sums.

For instance, given an array [1,2], the subsequences are [1], [2], and [1,2], plus the empty subsequence []. The sum of these subsequences are 1, 2, 3, and 0 respectively. The bitwise OR of these sums (1 | 2 | 3 | 0) equals 3.

The problem's complexity comes from the potentially large number of subsequences, which increases exponentially with the size of the array. The challenge is to find an efficient way to compute the special value without having to actually generate and sum every possible subsequence.

Intuition

To devise a solution to the problem without generating every possible subsequence (which would be computationally expensive), we observe that each bit position in the integers can be treated independently due to the properties of the bitwise OR operator.

Here's the intuition behind the solution:

1. Initialize a counter array, cnt, which will keep track of the number of times a bit is set at different positions across all numbers in the array.
2. Iterate through each number v in the array nums. For each bit position i, check if the ith bit of v is set (i.e., equals 1). If so, increment the count in cnt[i].
3. Initialize a variable ans, which will hold the final answer.
4. Iterate through the counter array cnt. If at any bit position i, cnt[i] is non-zero, it implies that this bit appeared in our subsequence sum. Thus, set the ith bit in ans to 1 by performing a bitwise OR between ans and 1 shifted i times to the left.
5. Additionally, we carry over half of the count from each bit position to the next higher position by adding cnt[i] // 2 to cnt[i + 1]. This represents the combination of bits when we add numbers together, as bits carry when there is a sum of 1 + 1 = 10 in binary.
6. The loop iterates up to 63, as it accounts for the carry that could propagate through all 32 bit positions twice (since every addition might cause a carry to the next position).
7. The variable ans now contains the bitwise OR of all the possible subsequence sums, which is the required answer.

The solution leverages the fact that we can analyze and manipulate the bits of the entire array rather than individual subsequences. It counts bit occurrences and then uses those counts to determine which bits will definitely be set in the final OR result. The carrying of half the count to the next bit position simulates how bit addition would occur across all subsequence sums.

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Solution Approach

The implementation of the solution is based on bit manipulation and the understanding of how bitwise OR and sum operations interact with each other on the binary level.

Here's a detailed walk through of the solution code:

• The solution declares a Solution class with a method called subsequenceSumOr, which accepts an array nums.

• Inside this method, a cnt array with 64 elements initialized to 0 is created. This array is used to count the occurrences of set bits in the binary representation of all numbers in nums. It reserves enough space to handle the bit carry operations that can happen as we simulate adding the numbers together.

1cnt = [0] * 64

• The ans variable is initialized to 0. This variable will hold the result of the bitwise OR of all possible subsequence sums.

1ans = 0

• The solution runs a loop over each number v in the nums array and then a nested loop over each bit position i, starting from 0 up to 30 (inclusive), which is sufficient to cover the bit-positions for a 32 bit integer.

1for v in nums:
2    for i in range(31):

• If the ith bit of v is set (i.e., v >> i bitwise right shift i and bitwise AND with 1 gives us the ith bit), the corresponding count in cnt[i] is incremented.

1if (v >> i) & 1:
2    cnt[i] += 1

• After the counts are determined, the solution iterates over all 63 possible bit positions—enough to account for carry-over during addition. For each i, if cnt[i] is non-zero, 1 << i (bitwise left shift, which effectively is 2^i) is bitwise ORed with ans. This sets the ith bit of ans because that bit will appear in at least one subsequence sum.

1for i in range(63):
2    if cnt[i]:
3        ans |= 1 << i

• The solution also ensures that the number of times a bit is carried over to the next higher position is taken into account by adding cnt[i] // 2 to cnt[i + 1]. This simulates the carry-over process in binary addition across all subsequence sums.

1    cnt[i + 1] += cnt[i] // 2

• Finally, ans is returned. It now holds the value of the bitwise OR of the sum of all possible subsequences.

1return ans

The implementation successfully leverages bit manipulation to efficiently solve a problem that would otherwise require an impractical brute-force approach due to the exponentially growing number of subsequences in the array.

Example Walkthrough

Let's use the integer array [3,5] to illustrate the solution approach:

1. We initialize a counter array cnt of 64 elements to keep track of the number of set bits at each bit position:

   1cnt = [0] * 64

2. We create ans variable, initialized to 0, which will store our answer:

   1ans = 0

3. We iterate over each value v in nums (3 and 5), and for each value, we iterate over each bit position i from 0 to 30:

   1nums = [3, 5]
   2for v in nums:
   3    for i in range(31):
   4        if (v >> i) & 1:
   5            cnt[i] += 1

4. Binary representations for 3 and 5 are 011 and 101 respectively.

   After iterating through the array, our cnt array will have the counts for each bit position:

   1index:  0  1  2  ...
   2cnt:   [0, 2, 1, ...]  # Only showing relevant bits

   Here, the bit at index 1 occurred twice (1 from both 3 and 5), and the bit at index 2 occurred once (from 5).

5. We then iterate through the counter cnt and update the ans variable with the bitwise OR operation for any non-zero counts, also handling the carry over:

   1for i in range(63):
   2    if cnt[i]:
   3        ans |= 1 << i
   4    cnt[i + 1] += cnt[i] // 2

   In this process, the ans variable will be updated as follows:

   1loop i=1: ans |= 1 << 1  # as cnt[1] had a count of 2
   2          ans now becomes 2
   3loop i=2: ans |= 1 << 2  # as cnt[2] had a count of 1
   4          ans now becomes 6

   Simultaneously, the carry over for the next position is handled:

   1cnt[2] += cnt[1] // 2
   2cnt array becomes [0, 0, 2, ...] after carry, showing we carried 1 to the next significant bit.

6. The final value of ans is 6, which is the bitwise OR of sum of all possible subsequences of the array [3,5].

The returned result from our example array [3,5] is 6, which corroborates with the computation of the subsequence sums:

• [] sum is 0,
• [3] sum is 3,
• [5] sum is 5,
• [3,5] sum is 8.

The bitwise OR of all sums (0 | 3 | 5 | 8) is indeed 6. This example walk-through shows how the bit manipulation approach efficiently calculates the special value without having to enumerate all subsequences.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the provided code is determined by the number of loops and operations within each loop:

1. The outer loop runs for each element in the nums list, thus it is O(n) where n is the length of the list.
2. The inner loop runs for a constant 31 times (assumes 32-bit integer processing), which is O(1) with respect to the input size.

Since the inner loop is nested within the outer loop, we multiply the complexities of two loops, which results in O(n) * O(1), leading to a final time complexity of O(n).

Space Complexity

The space complexity of the code consists of:

1. The space used by the cnt array, which has a fixed size of 64, thus O(1).
2. The ans variable, which also occupies a constant space and does not depend on the input size.

Considering both of these together, the overall space complexity of the provided code is O(1) since the extra space required does not scale with the size of the input (n).

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