# REBXOR - Editorial

Author: Yurii Rebryk
Tester: Kevin Atienza
Editorialists: Pushkar Mishra and Suhash Venkatesh

Medium

### PREREQUISITES:

Tries, Properties of Bitwise XOR

### PROBLEM:

Given is an array A containing numbers in the range 0 to 10^9. We have to find l_1, r_1, l_2 and r_2 (l_1 \leq r_1 < l_2 \leq r_2) such that (\,\,A[l_1] ⊕ A[l_1 + 1] ⊕ \dots ⊕ A[r_1]\,\,) + (\,\,A[l_2] ⊕ A[l_2 + 1] ⊕ \dots ⊕ A[r_2]\,\,) is maximised. Here x ⊕ y refers to Bitwise XOR of x and y.

### EXPLANATION:

Let’s start off by thinking of the simplest algorithm that we can. We we will then optimise it step by step by making certain observations.
So, the simplest thing to do is to implement what the problem says. We can run four loops, one each for l_1, r_1, l_2 and r_2. This gives us an \mathcal{O}(N^4) algorithm which will exceed the time limit for all the subtasks.

How can we optimise this? Let’s make an observation. If we know for each i = 1 to N, the maximum value we can get for (\,\,A[l_1] ⊕ A[l_1 + 1] ⊕ \dots ⊕ A[r_1]\,\,) and (\,\,A[l_2] ⊕ A[l_2 + 1] ⊕ \dots ⊕ A[r_2]\,\,) such that r_1 \leq i and i < l_2, then we can tell the answer in \mathcal{O}(N) by simply taking the maximum of the sum of the two terms over all i.

What remains now is to calculate efficiently for each i the max (\,\,A[l_1] ⊕ A[l_1 + 1] ⊕ \dots ⊕ A[r_1]\,\,) and (\,\,A[l_2] ⊕ A[l_2 + 1] ⊕ \dots ⊕ A[r_2]\,\,) such that r_1 \leq i and i < l_2. Let’s call the two terms lbest[i] and rbest[i] respectively. For calculating lbest[i], we iterate from j = i to 1 and find the j such that val = (\,\,A[j] ⊕ A[j + 1] ⊕ \dots ⊕ A[i]\,\,) is maximised. Then, lbest[i] = max(\,lbest[i-1],\, val\,)\,\,. Similarly, we can fill the array rbest.

Calculating the arrays lbest and rbest require \mathcal{O}(N^2) time. After that, the answer can be computed in \mathcal{O}(N). For this subtask, an \mathcal{O}(N^2) solution will pass.

We need to somehow speed up the calculation of arrays lbest and rbest. Up till now, we haven’t utilised any property of XOR operation in our solution. Let’s try to optimise our algorithm using the following property:
Let cumul[i] = (\,\,A[1] ⊕ A[2] ⊕ \dots ⊕ A[i]\,\,).
Then for some j\leq i, cumul[j-1] ⊕ cumul[i]\, = (\,\,A[j] ⊕ A[j + 1] ⊕ \dots ⊕ A[i]\,\,).
The proof behind this is left to the reader as a simple exercise.

We can now say that lbest[i] = max(\,lbest[i-1],\, val\,)\,\, where val = maximum of (cumul[j-1] ⊕ cumul[i]\,\,) for j = 1 to i.

Calculating lbest this way still takes \mathcal{O}(N^2). For i = 1 to N, we need to somehow optimise finding the index j such that (cumul[j-1] ⊕ cumul[i]\,\,) is maximum. If we can do it in \mathcal{O}(\log A[i])\,, then we can compute the array lbest (and analogously, rbest too) in \mathcal{O}(N\log A_{max})\,.

We can use Tries to get the required complexity. We will keep a trie which holds the binary representation of elements in the array cumul. While processing for i from 1 to N, we will first use the trie to get the value which is maximum of (cumul[j-1] ⊕ cumul[i]\,\,) for 1 \leq j \leq i, and then insert cumul[i] into the trie. This will allow us to calculate lbest in the required complexity. Similarly, we can calculate rbest also.

Inserting into a trie is a standard operation. Let us look at how we do the other operation, i.e., for a value x, finding the value y in the trie such that x ⊕ y is maximised. Since, the trie stores binary representations of numbers, we first convert x to its binary representation. Then, we can go down the trie and for each bit in the representation of x, we try to find whether bit of opposite value exists or not. It is easy to see why we seek opposite; because 1 ⊕ 1 and 0 ⊕ 0 = 0 while 0 ⊕ 1 = 1.

### COMPLEXITY:

\mathcal{O}(N\log A_{max})

### SAMPLE SOLUTIONS:

11 Likes

Really nice problem. These are the steps I followed.

For first subtask, it was clear a brute force approach will work. That is exactly what I did. I preprocessed the array and got prefix-xor array. From that, an O(n^2) algorithm checking all subsets worked. But got TLE and SIGSEGV for all the test cases of subtask 2.

Here’s my code

Then while surfing, I got this link . From this I got the idea. But I had never coded a trie before. So I googled and studied it. I then implemented it using a tree. But apparently this approach was slow. Got TLE in three test files in subtask 2. Here is my code

Can anyone tell me why it got TLE and how I could have improved it?

Then I studied how to implement it using three and two arrays from this link hoping it would be faster. But I could not understand them. Can anyone explain them please?

Then I read a codeforces blog, that explained how to implement trie using a 2D array. I understood that and implemented it quickly. But I didn’t know what the array size should be. I set it to 1000000 (10^6) and got SIGSEGV.
My code

Then I tried increasing the array size by 10 times and got AC with 100 points.
My final code

Can anyone plz explain what the array size should be and why? Also, shouldn’t array size 10^7 * 2 give SIGSEGV?

4 Likes

+1 to the problem setter for this problem and +5 for the awesome test data.
Learned a new way of creating binary trees without using dynamic memory allocation.

Used dynamic memory allocation and this happened : https://www.codechef.com/viewsolution/8082903
Used a different approach and Voila! https://www.codechef.com/viewsolution/8145313

This is one of the best problems i solved till date.

2 Likes

Each insertion in Trie creates O(log(Max(A[i]))) New Nodes . So here N is 4 * 10 ^ 5 and log(Max[A[i]]) is 30 hence you need about 1.2 * 10 ^ 7 space

1 Like

@grebnesieh Thanks a lot for teaching me this wonderful way. I always used dynamic memory allocation. This approach is new for me.

@rajat1603 Thank you. I get it now. Just one thing. 1.2*10^7 is for worst case right? I mean do we always need O(log(Max(A[i]))) space?

Can anybody help me finding where am I going wrong ?
https://www.codechef.com/viewsolution/8164680

Providing failed test case will be much appreciated.

Hi. I implement a Trie using a struct node and traversed it recursively while inserting and querying. I passed 18/19 tests but got TLE in one. Can anyone explain how can I improve this method to pass all the tests? I think the reason I’m getting TLE is because complexity is O(120 * N) = 1.2 * 10^7 But can anyone explain how can I improve on this?
Here is my solution


[1] Why it is giving TLE as time complexity is same as editorial.

[1]: https://www.codechef.com/viewsolution/8011026

Your problem is the dynamic memory allocation (new). It’s to slow when called for each node seperately.

Allocate an vector of nodes and take nodes from this array, keeping track with a counter

I think, I got it Thanks!

@anishshah I faced exactly the same problem. Dynamic memory allocation is costly. There is a better way of doing it.
Struct node {int x,y};
Maintain a array of node(of size 410^530 say) and a array index initialized to 1. Now whenever u need a new node just increment the array index and store the updated value in x or y variable of the current node. This way you can avoid dynamic allocation. See this https://www.codechef.com/viewsolution/8145313
I hope the code is clear enough.

Can you please explain your idea here.Thanks!

Here in your code few things can be optimized , try static allocation instead of dynamic memory allocation, static memory allocation is faster , and wont result in segmentation fault as there are only two leaves, and padding can be till 30 bits ,30 is enough eventhough anything above that shouldnt make much difference

1 Like

Constant factors do matter

@dkrdharmesh Instead of using lists that dynamically allocate with pointers, he used so-called “statically allocated list”. Let’s take a simpler example:

Say you have to program a linked list. You can do it in a classic way, with pointers and allocations, or you cat allocate an array of “nodes”, and instead of pointers you use indices to that array. Take this example:

2 Likes

CODE:

struct Node { int inf, nxt };

Node Nodes[MAXNODES];
int nodes_count, head = 0;

void insert(int head, int value) {

Nodes[++nodes_count] = Node(value, head) // Create a new node with info field value and next field equal to head of list

head = nodes_count // Redirect the head pointer (index) to the new node -- this inserts at front of list

}


To iterate, you just do something like:

for(int p = head; p; p = Nodes[p].nxt) {
//Nodes[p].inf will be the value, for example
}


You can do this with any dyn-alloc data structure. It’s faster.

3 Likes

I researched / problem solve the approach described in the editorial. Still, TLE on task 2. Times for task 1 suggest I was WAY over time on task 2. I am using python. Is this an impossible task on python or is there a much faster way to implement tries in that language? Thanks so much for any input.

https://www.codechef.com/viewsolution/8161088

I used the same approach with dynamic memory allocation and got TLE in 4 tasks of subtask 2.
my solution is : https://www.codechef.com/viewsolution/8083878
After that I realised an optimisation that maximum posible XOR for any subarray till i 0<=i<n is the value when all the k bits are one, where k is the number of bits in the maximum number in the array.
So whenever this value is encountered, maximum xor subarray after that index will be that value and their is no need to play with the tries further, I used this fact and execution time was nearly halved.
My final AC’ed solution is : https://www.codechef.com/viewsolution/8089627

I used this reference to study the concept of tries : http://threads-iiith.quora.com/Tutorial-on-Trie-and-example-problems

3 Likes

my solution complexity : N+2Nlog Amax+2Nlog Amax ~ O(N log Amax)**

same approach I have I also implemented but I don’t know why test case : 18 gave TLE??

any genuine region??