该算法的时间复杂度是多少？

Question

Good day, I'm trying to improve my abilities with Big-O, and I wrote a Java algorithm to print a tree to the console.

public static void printTree(Tree tree){
    int height = getHeight(tree);
    int maxWidth = (int)Math.pow(2, height) - 1;
    HashMap<Coordinate, Integer> coordinates = getTreeCoordinates(tree, maxWidth/(int)Math.pow(2, 1) + 1, 1, maxWidth);
    printCoordinatesToConsole(coordinates, maxWidth, height);
}

static void printCoordinatesToConsole(HashMap<Coordinate, Integer> coordinates, int width, int height){
    for (int j = 1; j <= height; j++){
        for (int i = 1; i <= width; i++){
            if (coordinates.containsKey(new Coordinate(i, j)))
                System.out.print(coordinates.get(new Coordinate(i, j)));
            else
                System.out.print(' ');
        }
        System.out.print("n\n");
    }
}

static HashMap<Coordinate, Integer> getTreeCoordinates(Tree tree, int x, int y, int maxWidth){
    HashMap<Coordinate, Integer> result = new HashMap<>();
    result.put(new Coordinate(x, y), tree.data);
    if (tree.left == null && tree.right == null){
        return result;
    }
    else if (tree.left == null){
        result.putAll(getTreeCoordinates(tree.right, x+maxWidth/(int)Math.pow(2, y+1) + 1, y+1, maxWidth));
        return result;
    }
    else if (tree.right == null){
        result.putAll(getTreeCoordinates(tree.left, x-maxWidth/(int)Math.pow(2, y+1) - 1, y+1, maxWidth));
        return result;
    }
    else{
        result.putAll(getTreeCoordinates(tree.right, x+maxWidth/(int)Math.pow(2, y+1) + 1, y+1, maxWidth));
        result.putAll(getTreeCoordinates(tree.left, x-maxWidth/(int)Math.pow(2, y+1) - 1, y+1, maxWidth));
        return result;
    }
}

As far as I can tell, the complexity would be based on:

1.Finding tree height O(n)

2.Storing the coordinates for all elements in the hashmap O(n)

3.Printing the coordinates to the screen O(width*height)

Now, since width is 2^height, which in the worst case would be n, does this mean the time complexity is O(n*2^n)? Also, if I were to ignore the printing (or if I were to print directly to the coordinates on the console rather than iterating through all of the width/height), would the complexity then be O(n)

Thank you!

Answer 1

1. Finding tree height is O(n).

If getHeight is more or less the following, it will be O(n), where n is the number of nodes in the tree:

static int getHeight(Tree tree) {
    if (tree == null) {
        return 0;
    }

    return 1 + max(getHeight(tree.left, tree.right));
}

2. Storing the coordinates for all elements in the hash map O( getTreeCoordinates ) = O( height * n) right now, but can be improved to O(n).

O( HashMap.putAll ) technically depends on implementation, but is almost certainly linear in the number of elements! Instead of using HashMap.putAll , you can pass the HashMap down to the recursive calls, eg:

static HashMap<Coordinate, Integer> getTreeCoordinates(Tree tree, int x, int y, int maxWidth){
    HashMap<Coordinate, Integer> result = new HashMap<>();
    return getTreeCoordinatesImpl(tree, x, y, maxWidth, result);
}

static void getTreeCoordinatesImpl(Tree tree, int x, int y, int maxWidth, HashMap<Coordinate, Integer> result){
    result.put(new Coordinate(x, y), tree.data);
    if (tree.right != null){
        getTreeCoordinatesImpl(tree.right, x+maxWidth/(int)Math.pow(2, y+1) + 1, y+1, maxWidth, result);
    }
    if (tree.left != null){
        getTreeCoordinatesImpl(tree.left, x-maxWidth/(int)Math.pow(2, y+1) - 1, y+1, maxWidth, result);
    }
}

3. Printing the coordinates to the screen is O( height * width ). It will be O(n), where n is the number of nodes is the tree, if you iterate over the HashMap instead of over all (x, y) coordinates.

Both O( HashMap.containsKey ) and O( HashMap.get ) are 1 (constant time). To be precise, they're amortized constant in time - on average, they take a constant time, but a single run can be linear in the number of elements of the hash map in the rare worst case.

In big O notation all constants are equivalent (O(1) is equivalent to O(2)). So:

O( printCoordinatesToConsole ) =

O( height ) * O( width ) * (O( HashMap.containsKey ) + O( HashMap.get )) =

O( height ) * O( width ) * O(1) =

O( height * width )

---

Now, since width is 2^height, which in the worst case would be n, does this mean the time complexity is O(n*2^n)?

Let's do the math (n is the number of nodes in the tree, I assume getTreeCoordinates is edited as described above):

O( printTree ) =

O( getHeight ) + O( getTreeCoordinates ) + O( printCoordinatesToConsole ) =

O(n) + O(n) + O( height * width )

Since height * width >= n:

O( printTree ) = O( height * width )

---

Also, if I were to ignore the printing (or if I were to print directly to the coordinates on the console rather than iterating through all of the width/height), would the complexity then be O(n)

Yes, then the above equation becomes either (no printing):

O( printTree ) = O( getHeight ) + O( getTreeCoordinates ) = O(n) + O(n) = O(n)

or (printing with iteration over the hash map of nodes):

O( printTree ) = O( getHeight ) + O( getTreeCoordinates ) + O(n) = O(n)

---

It will be helpful if you include the definitions of Tree , Coordinate and getHeight , and if applicable, the contents of tree . You can also use an online playground like ideone.com to provide a runnable example.