Find a Value of a Mysterious Function Closest to Target

Hard

American Express

2 views

Topics:

ArraysSliding WindowsBit Manipulation

Winston was given the above mysterious function func. He has an integer array arr and an integer target and he wants to find the values l and r that make the value |func(arr, l, r) - target| minimum possible.

Return the minimum possible value of |func(arr, l, r) - target|.

Notice that func should be called with the values l and r where 0 <= l, r < arr.length.

Example 1:

Input: arr = [9,12,3,7,15], target = 5
Output: 2
Explanation: Calling func with all the pairs of [l,r] = [[0,0],[1,1],[2,2],[3,3],[4,4],[0,1],[1,2],[2,3],[3,4],[0,2],[1,3],[2,4],[0,3],[1,4],[0,4]], Winston got the following results [9,12,3,7,15,8,0,3,7,0,0,3,0,0,0]. The value closest to 5 is 7 and 3, thus the minimum difference is 2.

Example 2:

Input: arr = [1000000,1000000,1000000], target = 1
Output: 999999
Explanation: Winston called the func with all possible values of [l,r] and he always got 1000000, thus the min difference is 999999.

Example 3:

Input: arr = [1,2,4,8,16], target = 0
Output: 0

Constraints:

1 <= arr.length <= 10⁵
1 <= arr[i] <= 10⁶
0 <= target <= 10⁷

Try coding it in LeetCode

Find a Value of a Mysterious Function Closest to Target

Hard

American Express

2 views

Topics:

ArraysSliding WindowsBit Manipulation

Winston was given the above mysterious function func. He has an integer array arr and an integer target and he wants to find the values l and r that make the value |func(arr, l, r) - target| minimum possible.

Return the minimum possible value of |func(arr, l, r) - target|.

Notice that func should be called with the values l and r where 0 <= l, r < arr.length.

Example 1:

Input: arr = [9,12,3,7,15], target = 5
Output: 2
Explanation: Calling func with all the pairs of [l,r] = [[0,0],[1,1],[2,2],[3,3],[4,4],[0,1],[1,2],[2,3],[3,4],[0,2],[1,3],[2,4],[0,3],[1,4],[0,4]], Winston got the following results [9,12,3,7,15,8,0,3,7,0,0,3,0,0,0]. The value closest to 5 is 7 and 3, thus the minimum difference is 2.

Example 2:

Input: arr = [1000000,1000000,1000000], target = 1
Output: 999999
Explanation: Winston called the func with all possible values of [l,r] and he always got 1000000, thus the min difference is 999999.

Example 3:

Input: arr = [1,2,4,8,16], target = 0
Output: 0

Constraints:

1 <= arr.length <= 10⁵
1 <= arr[i] <= 10⁶
0 <= target <= 10⁷

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What is the range of values for integers within the input array `arr` and the target integer?
Is the input array guaranteed to be non-empty?
If multiple pairs (x, y) result in the same minimum absolute difference, can I return any one of those f(x, y) values, or is there a tie-breaking criteria?
Could you provide examples of what the expected output should look like given a sample input?
Are there any memory constraints I should be aware of, considering the potential size of the input array?

Brute Force Solution

Approach

We are given a function that's hard to understand directly, and a target value. The brute force approach involves computing the function for every possible input and comparing the results to the target. We want to find the function's output value that is closest to our target.

Here's how the algorithm would work step-by-step:

Imagine you have a list of all possible things you can put into the mysterious function.
Now, one by one, take each item from the list and plug it into the function to get a result.
Compare each result to the target value and figure out the difference (how far apart they are).
Keep track of which result had the smallest difference from the target. This is the closest value we've found so far.
After you've gone through every item on the list, the 'closest value' you've been tracking is the answer.

Code Implementation

def find_closest_value_brute_force(possible_inputs, mysterious_function, target_value):

    closest_value = None
    smallest_difference = float('inf')

    # Iterate through all possible inputs to the mysterious function
    for current_input in possible_inputs:
        function_result = mysterious_function(current_input)

        # Calculate the difference between the function result and the target value
        current_difference = abs(function_result - target_value)

        # Check if the current difference is smaller than the smallest difference seen so far
        # If so, update the closest value and smallest difference
        if current_difference < smallest_difference:

            #Update smallest difference
            smallest_difference = current_difference

            #Update closest function value found so far
            closest_value = function_result

    # Return the closest value found
    return closest_value

Big(O) Analysis

Time Complexity

O(n) – The algorithm iterates through a list of n possible inputs to the mysterious function. For each input, it computes the function's output and compares it to the target, a constant-time operation. Therefore, the algorithm performs a constant amount of work for each of the n inputs, resulting in a linear time complexity of O(n).

Space Complexity

O(1) – The algorithm iterates through a list of possible inputs, calculating the function's output for each. It keeps track of only the closest value found so far and the minimum difference encountered. Since we only store a few variables representing the closest value and the smallest difference, and these do not depend on the size of the input space (N, the number of possible inputs), the auxiliary space used remains constant. Therefore, the space complexity is O(1).

Optimal Solution

Approach

The key to solving this problem efficiently lies in exploiting the function's properties. By understanding how the function behaves, we can use a process of elimination to quickly narrow down the search area and pinpoint the value closest to the target, without exhaustively checking every possibility. This method is like a guessing game where each guess cleverly informs the next.

Here's how the algorithm would work step-by-step:

First, understand how the mysterious function changes when you give it different inputs. Is it always increasing, always decreasing, or does it sometimes go up and sometimes go down?
Then, try giving the function a 'test' value. This gives you a starting point to see if you are above or below the target.
If the function is always increasing or always decreasing, you can use 'binary search'. Imagine slicing the possible inputs in half each time. If your test value is too low, you know the answer is in the upper half. If it's too high, it's in the lower half.
Repeat the process of choosing the middle value of your remaining range until the range is tiny. At this point, you've found the input that gives the closest output to your target.
If the function doesn't always increase or decrease, you might need a slightly different approach. You can still start with a test value and see if the function's output is higher or lower than your target.
Based on the function's behavior around that test value, try to determine a direction (bigger or smaller inputs) that is likely to get you closer to your target.
Keep adjusting your inputs in that direction, checking the function's output each time, until you either find a value that is very close to your target, or you see the function start to move away from the target. If you're moving away, you know you've passed the best value, and you can go back to the previous input.
Finally, compare the outputs of the last two or three inputs you tested to determine which one is closest to the target. That input is your answer.

Code Implementation

def find_best_value(function, target, low_range, high_range):
    best_value = low_range
    closest_difference = abs(function(low_range) - target)

    while low_range <= high_range:
        # Calculate midpoint to narrow search space
        midpoint = (low_range + high_range) // 2

        function_value = function(midpoint)
        difference = abs(function_value - target)

        # Update best value if current difference is smaller
        if difference < closest_difference:
            closest_difference = difference
            best_value = midpoint

        # Adjust search range based on function value relative to target
        if function_value < target:
            low_range = midpoint + 1

        else:
            high_range = midpoint - 1

    return best_value

def find_closest_value(function, target, initial_guess):
    current_input = initial_guess
    step_size = 1

    best_input = initial_guess
    min_difference = abs(function(initial_guess) - target)

    # Determine initial direction to search
    if function(initial_guess) < target:
        direction = 1
    else:
        direction = -1

    # Iterate until function moves away from the target
    while True:
        next_input = current_input + direction * step_size
        current_value = function(next_input)
        current_difference = abs(current_value - target)

        # Update best input if this input is better
        if current_difference < min_difference:
            min_difference = current_difference
            best_input = next_input
            current_input = next_input

        else:
            # Step size is reduced when we move away
            if step_size == 1:
                return best_input
            else:
                return best_input

def mysterious_function_solver(function, target, is_monotonic):
    if is_monotonic:
        # If function is monotonic, use binary search
        best_value = find_best_value(function, target, -100, 100)
        return best_value
    else:
        # If function is non-monotonic, use iterative search
        initial_guess = 0
        best_value = find_closest_value(function, target, initial_guess)

        # Compare output to target to find input that yields closest output
        return best_value

Big(O) Analysis

Time Complexity

O(log n) – The provided solution uses a binary search approach when the function is monotonically increasing or decreasing. In this case, the search space is halved in each iteration. The number of iterations required to find the closest value is therefore logarithmic with respect to the size of the possible input range, n. Thus, the time complexity is O(log n).

Space Complexity

O(1) – The algorithm, based on the plain English explanation, primarily involves iterative adjustments and comparisons. It uses a few variables to store the current test value, the function's output at that value, the target value, and potentially a few previous values for comparison. The number of these variables remains constant regardless of the input size N (which could represent the range of possible inputs to the mysterious function), leading to constant auxiliary space usage. Therefore, the space complexity is O(1).

Edge Cases

Case	How to Handle
Empty or null input array	Return positive infinity or throw an exception as no valid AND value exists.
Array with a single element	Return the absolute difference between the element and the target (f(x,x) = x).
Array with all elements being zero	Return absolute difference between 0 and target, as all AND results will be zero.
Large input array causing performance issues	Optimize by using prefix AND array or pruning based on bitwise properties to reduce comparisons.
Target value is very large	The bitwise AND will be smaller than any input number, so it is safe to only check smaller numbers and return absolute difference to the target.
Integer overflow during bitwise AND operation	The bitwise AND operation should not cause an overflow since the result will always be less than or equal to both operands.
Array contains negative numbers	Bitwise AND works with negative numbers but must consider how negative numbers are represented (two's complement) and their impact on proximity to the target.
Array contains a large number of duplicate elements	The number of duplicate elements can be reduced by pre-processing using set data structure.