Google | Interview Question
Google | Interview Question
In this blog post, we’ll explore an intriguing algorithmic problem that was presented as an interview question by Google. The challenge involves manipulating an array of distinct integers to find the smallest positive integer that cannot be formed using various mathematical operations.
Problem Statement
You are given an array of distinct elements, and your goal is to find the minimum positive number that cannot be formed using the elements of the array. You can choose any subset of the array, reorder them, and place the following operators between them: +, -, *, /, as well as parentheses ( and ). Importantly, each element can be used only once in any expression.
Examples
Example 1:
Input: [1, 2] Output: 4
Explanation:
In this case, we can create the numbers 1 and 2 directly from the array. We can also form the number 3 by combining them (1 + 2). However, there is no combination of operations that allows us to create the number 4 using the provided elements.
Example 2:
Input: [1, 2, 3] Output: 10
Explanation:
From the elements 1, 2, and 3, we can create all numbers from 1 up to 9, but there is no way to form the number 10.
Approach to the Solution
One naive approach might involve generating all possible expressions from the elements of the array using the allowed operators. This could be achieved through permutations and combinations, leading to a vast number of potential expressions to evaluate. The valid results could then be stored in a set, from which we would find the lowest positive integer not present.
However, this brute-force method is computationally expensive and inefficient. Given the factorial growth of permutations and combinations with respect to the number of elements, we should aim for a more optimized approach.
Optimized Approach
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Subset and Combination Generation: Instead of generating all possible strings, we can focus on generating all possible subsets of the array. For each subset, we can recursively evaluate all possible combinations of the numbers using the allowed operations.
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Dynamic Programming or Backtracking: Using dynamic programming or backtracking techniques, we can keep track of the numbers that can be formed with the current subset. This way, we avoid recalculating combinations that have already been evaluated.
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Tracking Possible Values: We can maintain a boolean array or a set to track which positive integers can be formed. With every valid expression evaluated, we mark the corresponding index as true.
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Finding the Minimum Missing Positive Integer: Finally, we iterate through the boolean array or set to find the smallest index that remains false, which will represent the smallest positive integer that cannot be formed.
Complexity Analysis
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Time Complexity: The time complexity of generating all combinations and evaluating them is exponential in nature, specifically O(2^n * p(n)) where n is the number of elements in the array and p(n) is the time taken to evaluate expressions formed by n elements.
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Space Complexity: The space complexity can be O(n) for storing the boolean array or set to track the formed integers, plus the space used for recursion stack in case of backtracking.
Conclusion
This problem combines combinatorial generation with arithmetic evaluation, making it a complex yet fascinating challenge. While brute-force approaches provide a baseline, optimizing through subset generation and dynamic programming allows us to tackle larger input sizes efficiently.
If you have any further insights or optimizations, feel free to share your thoughts in the comments below! Let’s discuss and refine our approach to this intriguing problem.