Problem Description

In this problem, we are given a binary string s and an array of queries, where each query is represented by a pair of integers [first_i, second_i]. The goal is to find the shortest substring within s that, when converted to a decimal value val and XORed with first_i, yields second_i. In simpler terms, for each query, we want to find a substring of s such that val ^ first_i == second_i.

The result of each query should be an array [left_i, right_i] indicating the 0-indexed start and end positions of the substring within s. If there is more than one possible substring, we select the one with the smallest left_i. If no such substring exists, we return [-1, -1] for that query.

A substring, in this context, is defined as a sequential chunk of characters from the string s without altering the order or skipping any characters in between.

Intuition

The intuition behind the solution is to use a dictionary to keep track of all possible substrings and their decimal values. We iterate over the string s, and at each position, we try to build substrings of different lengths (up to 32-bits, considering the problem constraints) while keeping track of their decimal values.

We use the following steps:

1. Create an empty dictionary d to store the substrings.
2. Iterate over the binary string s (using index i for the start of the substring).
3. Within each iteration of i, we construct substrings and their decimal values by appending one bit at a time (using index j), shifting the previously calculated value to the left, and adding the new bit.
4. If the decimal value x of the current substring has not been seen before, we store it in the dictionary with its starting and ending indexes [i, i + j].
5. If the value is zero, we can break the inner loop early since zero XORed with any number gives that number, and we can't find any smaller substring with a value of zero.
6. Now that we have precomputed the dictionary with substrings and their decimal representations, we can answer each query by looking for the decimal value that, when XORed with first_i, gives second_i. We use XOR in the query because of the property that if a ^ b = c, then a = b ^ c.

For every query [first, second], we perform the XOR operation first ^ second and look up the result in the dictionary. If there's a match, we return the corresponding substring's start and end indexes; otherwise, we return [-1, -1] for that query.

The benefit of this approach is that it avoids recalculating decimal values of the same substrings for different queries, leveraging the properties of XOR and the lookup efficiency of a dictionary.

Solution Approach

The solution approach utilizes a lookup table (Python dictionary) to store the decimal values of all possible substrings and their corresponding starting and ending positions. Here's the step-by-step breakdown of the implemented algorithm:

1. Initialization: A Python dictionary d is created to serve as the lookup table.

2. Outer Loop - Iterating over String s:

   • An outer for-loop is set up with variable i which iterates from 0 to the length of string s minus 1. Each i represents the starting position of the potential substrings.

3. Inner Loop - Building the Substrings:

   • For each position i, an inner for-loop with variable j is used to try to build substrings of increasing size, sequentially appended by one character at a time (up to a maximum of 31 iterations for a 32-bit limit).

   • During construction, a temporary variable x is used to calculate the decimal representation of the substring using bit manipulation. At each iteration, x is left-shifted by one bit (multiplied by 2) and the next bit of s is taken into account by using a bitwise OR (denoted by |) with the result. This is equivalent to appending the next character of s and converting the new binary string to a decimal value.

     x = x << 1 | int(s[i + j])

4. Updating the Lookup Table:

   • If the current decimal value x does not already exist in the dictionary d, it is added, with its value being a list [i, i + j], representing the start and end indexes of the current substring.
   • If x is 0, we can break from the loop early because no smaller substring than the already found will have the same value, due to the nature of XOR.

5. Handling Queries:

   • After populating the lookup table, we can efficiently answer each query using list comprehension.

   • For each query [first, second] in the queries list, perform the XOR operation first ^ second and look it up in the dictionary d:

     d.get(first ^ second, [-1, -1])

   • If the result of the XOR operation is found in d, the starting and ending positions are returned. If the XOR result is not found, [-1, -1] is returned, indicating no such substring exists.

By converting substrings to their decimal values once and storing them in the dictionary, the algorithm avoids redundant calculations for different queries. Bit manipulation and a lookup table are central to the algorithm's efficiency.

This algorithm greatly optimizes the process of finding the required substrings, especially when dealing with multiple queries, as it leverages the constant time complexity (average case) of dictionary lookups in Python.

Example Walkthrough

Let's walk through a small example to illustrate the solution approach:

Assume our binary string s is "1011" and we have a single query queries = [[2, 3]]. We want to find the shortest substring that, when converted to decimal and XORed with 2, results in 3.

Following the solution approach:

1. Initialization: We start by initializing an empty Python dictionary d.

2. Outer Loop - Iterating over String s:

   • We iterate over string s. In our example, our string is "1011", hence i will run from 0 to 3.

3. Inner Loop - Building the Substrings:

   • For i = 0, we try to build substrings starting at s[0].
     • For j = 0, our substring is "1" (in binary), which equals 1 in decimal, so we store {1: [0, 0]} in our dictionary.
     • For j = 1, our substring is "10", which equals 2 in decimal, so {2: [0, 1]} is added to the dictionary.
     • Continuing, we get "101" (5 in decimal) and "1011" (11 in decimal), resulting in {5: [0, 2]} and {11: [0, 3]} added to the dictionary.
   • We repeat this for i = 1, 2, 3, each time constructing new substrings and converting them to decimal numbers and adding to the dictionary if they are not yet present.
     • For i = 1 (and j = 0), the substring is "0", which as per step 4 of the implementation does not need to be stored since 0 XORed with any number will not help us find a unique answer we do not already have.
     • For i = 2 (and j = 0), the substring is "11" (3 in decimal), so {3: [2, 2]} is added to the dictionary.
     • i = 3 only gives us the substring "1" again, which is already in the dictionary, so we do not add it.

   The final dictionary d after processing will look like this:

   d = {
     1: [0, 0],
     2: [0, 1],
     5: [0, 2],
     11: [0, 3],
     3: [2, 2]
   }

4. Handling Queries:

   • Now, we answer our query [2, 3]. We compute 2 ^ 3 which equals 1. We look up the value 1 in our dictionary.
   • We find it as d[1], which is [0, 0].

Therefore, for the query [2, 3], the answer is [0, 0]. The shortest substring from s starting at index 0 and ending at index 0 when converted to decimal and XORed with 2 results in 3.

This illustrates how the algorithm prepares a lookup table for efficient query handling based on substring decimal values. It leverages the dictionary data structure to preemptively store all possible substrings and their indexes, significantly speeding up the processing of each query.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given code can be broken down as follows:

1. The outer loop runs n times where n is the length of the string s.

2. The inner loop, which is responsible for constructing the different XOR substrings and storing them into the dictionary d, runs for at most 32 iterations (since we're looking at a maximum of a 32-bit integer representation).

3. Within the inner loop, there is a constant-time operation of left shifting (x << 1) and OR-ing (| int(s[i + j])), as well as dictionary insertion or check which can be considered O(1) on average.

4. After the loops, the code iterates through each query to look up the XOR result in the dictionary, which can be done in O(1) time on average for each query.

Assuming m queries, the total time complexity is:

O(n * 32 + m) = O(n + m)

This assumes dictionary operations take constant time.

However, we must bear in mind that if there were a lot of collisions and the Python dictionary had to handle these, the worst-case time complexity for insertions and lookups could degrade to O(n). But this is highly unlikely with a good hash function and is not considered in average case analysis.

Space Complexity

1. The space complexity mainly comes from the dictionary d where all unique XOR results of substrings starting at different positions are stored.

2. In the worst case, every possible starting index and length could yield a unique XOR value, which means the size of d could grow to n * 32.

3. The space complexity also includes the output list of lists, but this only adds space linearly proportional to the number of queries m.

Thus, the space complexity is O(n * 32 + m) = O(n + m), with the same assumption about average case behavior of dictionary storage.

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