No. 57 - Integer Identical to Index

Problem: Integers in an array are unique and increasingly sorted. Please write a function/method to find an integer from the array who equals to its index. For example, in the array {-3, -1, 1, 3, 5}, the number 3 equals its index 3.

Analysis: If we scan all integers in the array from beginning to end, we may check whether every element equals its index. Obviously, this solution costs O(n) time.

Since numbers are sorted in the array, let's try to utilize the binary search algorithm to optimize. Supposing we reach the ith element in the array at some step. If the value of element is also i, it is a target integer and let's return it.

What would happen when the value m is greater than the index i? For any k (k>0), the value of element with index i+k should be greater than or equal to m+k, because integers are unique and increasingly sorted in the array. Additionally because m>i, m+k>i+k. Therefore, every element on the right side of index i should be greater than its index in such a case.

Similarly, when the value of element with index i is less than i, every integer on the left side should be less than its index. Please prove it if you are interested.

Therefore, we could reduce the search scope to half for the next step, and it is a typical process for binary search. The solution can be implemented with the following Java code:

public static int getNumberSameAsIndex(int[] numbers) {

    if(numbers == null || numbers.length == 0) {

        return -1;

    }

       

    int left = 0;

    int right = numbers.length - 1;

    while(left <= right) {

        int middle = left + ((right - left) >>> 1);

        if(numbers[middle] == middle) {

            return middle;

        }

           

        if(numbers[middle] > middle) {

            right = middle - 1;

        }

        else {

            left = middle + 1;

        }

    }

       

    return -1;

}

The source code with unit test cases is shared at http://ideone.com/ZSd9kG.

More coding interview questions are discussed in my book< Coding Interviews: Questions, Analysis & Solutions>. You may find the details of this book on Amazon.com, or Apress.

The author Harry He owns all the rights of this post. If you are going to use part of or the whole of this article in your blog or webpages, please add a reference to http://codercareer.blogspot.com/. If you are going to use it in your books, please contact the author via zhedahht@gmail.com . Thanks.

No. 56 - Maximal Value of Gifts

Question: A board has n*m cells, and there is a gift with some value (value is greater than 0) in every cell. You can get gifts starting from the top-left cell, and move right or down in each step, and finally reach the cell at the bottom-right cell. What’s the maximal value of gifts you can get from the board?

For example, the maximal value of gift from the board above is 53, and the path is highlighted in red.

Analysis: It is a typical problem about dynamic programming. Firstly let’s analyze it with recursion. A function f(i, j) is defined for the maximal value of gifts when reaching the cell (i, j). There are two possible cells before the cell (i, j) is reached: One is (i - 1, j), and the other is the cell (i, j-1). Therefore, f(i, j)= max(f(i-1, j), f(i, j-1)) + gift[i, j].

Even though it’s a recursive equation, it’s not a good idea to write code in recursion, because there might be many over-lapping sub-problems. A better solution is to solve is with iteration. A 2-D matrix is utilized, and the value in each cell (i, j) is the maximal value of gift when reaching the cell (i, j) on the board.

The iterative solution can be implemented in the following Java code:

public static int getMaxValue(int[][] values) {

    int rows = values.length;
    int cols = values[0].length;
    int[][] maxValues = new int[rows][cols];

    for(int i = 0; i < rows; ++i) {
        for(int j = 0; j < cols; ++j) {
            int left = 0;
            int up = 0;

            if(i > 0) {
                up = maxValues[i - 1][j];
            }

            if(j > 0) {
                left = maxValues[i][j - 1];
            }

            maxValues[i][j] = Math.max(left, up) + values[i][j];
        }
    }

    return maxValues[rows - 1][cols - 1];
}

Optimization

The maximal value of gifts when reaching the cell (i, j) depends on the cells (i-1, j) and (i, j-1) only, so it is not necessary to save the value of the cells in the rows i-2 and above. Therefore, we can replace the 2-D matrix with an array, as the following code shows:

public static int getMaxValue(int[][] values) {

    int rows = values.length;

    int cols = values[0].length;

    int[] maxValues = new int[cols];

    for (int i = 0; i < rows; ++i) {

        for (int j = 0; j < cols; ++j) {

            int left = 0;

            int up = 0;

            if (i > 0) {

                up = maxValues[j];

            }

            if (j > 0) {

                left = maxValues[j - 1];

            }

            maxValues[j] = Math.max(left, up) + values[i][j];

        }

    }

    return maxValues[cols - 1];

}

The source code with unit test cases is shared at http://ideone.com/2vLRMk.

More coding interview questions are discussed in my book< Coding Interviews: Questions, Analysis & Solutions>. You may find the details of this book on Amazon.com, or Apress. 

The author Harry He owns all the rights of this post. If you are going to use part of or the whole of this ariticle in your blog or webpages, please add a reference to http://codercareer.blogspot.com/. If you are going to use it in your books, please contact the author via zhedahht@gmail.com . Thanks.