I was practicing on Leetcode and came across this problem.
Problem statement (link):
You are given an array nums consisting of non-negative integers.
You are also given a queries array, where queries[i] = [xi, mi].
The answer to the ith query is the maximum bitwise XOR value of xi and any element of nums that does not exceed mi.
In other words, the answer is max(nums[j] XOR xi) for all j such that nums[j] <= mi.
If all elements in nums are larger than mi, then the answer is -1.
Return an integer array answer where answer.length == queries.length and answer[i] is the answer to the ith query.
Constraints:
1 <= nums.length, queries.length <= 10^5
queries[i].length == 2
0 <= nums[j], xi, mi <= 10^9
I solved this question using the trie approach and went to discuss section to see other's solutions. There, I came across this solution (link):
class Solution {
public:
vector<int> maximizeXor(vector<int>& nums, vector<vector<int>>& queries) {
const int n = nums.size(), q = queries.size();
vector<int> ans(q, -1);
sort(nums.begin(), nums.end());
for (int i = 0; i < q; i++) {
const int x = queries[i][0], m = queries[i][1];
if (m < nums[0]) continue;
int end = upper_bound(nums.begin(), nums.end(), m) - nums.begin();
int start = 0;
int k = 0, cur = 0;
for (int bit = 31; bit >= 0; bit--) {
if (x & (1 << bit)) { // hope A[i] this bit == 0
if (!(nums[start] & (1 << bit))) {
k |= 1 << bit;
end = lower_bound(nums.begin() + start, nums.begin() + end, cur | (1 << bit)) - nums.begin();
} else {
cur |= 1 << bit;
}
} else { // hope: A[i] this bit == 1
if (start <= end - 1 && (nums[end - 1] & (1 << bit))) {
k |= 1 << bit;
cur |= 1 << bit;
start = lower_bound(nums.begin() + start, nums.begin() + end, cur) - nums.begin();
}
}
}
ans[i] = k;
}
return ans;
}
};
Unfortunately, I'm not able to understand this solution. I would appreciate it if someone can give a proper explanation for this solution (mainly while looping through bits).
There are some issues with this implementation. start and end should remain iterators, they could be used directly without adding/subtracting nums.begin() all the time. We're talking about non-negative integers, so provided they fit into normal int first bit is 0 anyway, so we should start with int bit = 30 to skip one needless iteration. For integers right as for iterators as well, start <= end - 1 is better compared as start < end. The code consists of one single function, there's absolutely no need for a class then, so one should prefer a namespace. Applying these changes, the code would look like this:
namespace Solution
{
// as EXACTLY two values, std::pair is more appropriate
// we are not modifying queries, so should be accepted as const
std::vector<int> maximizeXor
(
std::vector<int>& nums, std::vector<std::pair<int, int>>const& queries
)
{
const int q = queries.size();
std::vector<int> ans(q, -1);
sort(nums.begin(), nums.end());
// remove duplicates:
// -> less numbers to iterate over
nums.erase(unique(nums.begin(), nums.end()), nums.end());
for (int i = 0; i < q; ++i)
{
int const x = queries[i].first, m = queries[i].second;
// we have a sorted array, remember?
// if first value is larger than the query maximum, then there are no
// corresponding numbers – and as the vector is initialised to -1
// anyway, the appropriate value is there already so we can simply skip
if (m < nums[0])
{
continue;
}
// using iterators pointing at the appropriate indices
auto end = upper_bound(nums.begin(), nums.end(), m);
auto start = nums.begin();
int /*k = 0,*/ cur = 0;
// intention is to check each bit of x
// modifying the loop!
//for (int bit = 30; bit >= 0; bit--)
int const MaxBit = 1 << sizeof(int) * CHAR_BIT - 2;
for (int bit = MaxBit; start != prev(end); bit >>= 1)
{
// OK; fixing an issue and adding some tricks to handle the loop
// a bit cleverer...
// sizeof(int) * CHAR_BIT: int is NOT guaranteed to have exactly
// 32 bits! if you want to be on the safe side, either calculate
// as above or use int32_t instead
// changed abort condition:
// I modified the algorithm slightly such that we can break early
// unique'ing the vector allows us to drop the original
// condition bit >= 0 entirely, this will be explained later
// I store the bit-MASK in bit now, now we do not have to
// calculate it again and again (1 << bit)
if (x & bit)
{
// so x has a 1-bit at bit index 'bit'
// in the range yet to be considered we have two groups of
// numbers:
// 1. those having a 0-bit at bit-index 'bit'
// 2. those having a 1-bit
// if we compare single bits, we get:
// x = *1***
// num = *0*** XOR: *1***
// num = *1*** XOR: *0***
// IF now there are numbers with a zero bit at all, then one
// of these will produce the maximum, whereas those with a
// 1-bit cannot asnumbers are sorted, we can just check very
// first value of the range:
// any number having a 1-bit at the same bit index will produce
// a zero-bit – thus these numbers CANNOT produce the maximum
if (!(*start & bit))
{
// bits differ, remember?
// thus the XOR will have a one-bit we store right now
// actually, we do NOT need that, we can handle that cleverer
//k |= 1 << bit;
// instead, I handle this with the NEW loop condition
// fine – there ARE numbers with zero-bits, so remove all
// numbers with 1-bit from range; as they all are at the end
// of, we simply move this one towards front:
end = lower_bound(start, end, cur | bit);
// cur contains those bits of the number producing the
// maximum that have been evaluated so far, it is a
// lower bound for – we do NOT modify it, but we can
// calculate a new upper bound from!
}
else
{
// well, there is no such number with a 0-bit
// we cannot move end or start position
cur |= bit;
}
}
else
{
// analogously:
// x = *0***
// num = *0*** XOR: *0***
// num = *1*** XOR: *1***
// so all members having a 1-bit are of interest – IF there
// are – and we can skip those numbers with 0-bit at the
// beginning
// if there are, then they are at the very end
// 'end' iterator points to one past, so we need predecessor
if (/*start < end &&*/ *prev(end) & bit)
{
// first condition is handled in the loop now
// as above: we can handle that cleverer
//k |= 1 << bit;
// now current mask NEEDS the one-bit
cur |= bit;
start = lower_bound(start, end, cur);
}
}
// with unchanged loop it was not possible to break early as k still
// needed to be calculated
//ans[i] = k;
// with or without early break, we can always:
ans[i] = *start ^ x;
// with every iteration, we extend the bit mask 'cur' the numbers
// have to match with by one bit (either the 0 gets confirmed
// or replaced by a 1).
// After 31 iterations (sign bit is ignored as we only have
// positive integers), *all* bits are defined (if we had omitted
// the early breaks we could have calculated
// ans[i] = cur ^ x; as well...).
// so all numbers that yet might have remained in the valid range
// must match this pattern, i. e. be equal. However as unique-ing,
// there is exactly one single value left...
}
}
return ans;
}
} // namespace Solution
Be aware that std::lower_bound has (random access iterators provided, as is with std::vector) a complexity of O(log(n)), so executing one single query has O(log(n)) with n being the amount of numbers. Adding the overhead of sorting and querying m times, we get a total complexity of O(n*log(n) + m*log(n)) = O((n+m)*log(n)) compared to 'naive' iteration with complexity of O(m*n). If m is of similar magnitude as n or larger we have a complexity advantage (already in original variant, my adjustments just trim the constants a bit, but do not change complexity).