Problem Description

The problem requires us to display a given string text on a screen with specific width w and height h. We are given a list of possible font sizes fonts sorted in ascending order, and we need to find out the largest font size we can use to display the entire text on the screen.

We are provided with a FontInfo interface that contains two methods:

• getWidth(fontSize, ch): which returns the width of character ch when fontSize is used.
• getHeight(fontSize): which provides the height of any character when fontSize is used.

We must ensure that the total width of text, calculated by summing up the width of each character in text using a given fontSize, does not exceed the screen width w. Additionally, the height of the text for the chosen fontSize, given by getHeight(fontSize), cannot be greater than the screen height h. The text is displayed in a single line.

Constraints are applied to both font width and height to ensure that they are non-decreasing with the increase in font size. If no font size can make the text fit on the screen, we are to return -1.

Intuition

This problem can naturally be approached with a binary search strategy, since the font sizes are sorted and we need to find the maximum fontSize that fits the constraints. The key steps when applying binary search in this context are:

• Define a function check() that determines if a given fontSize can be used to fit the text within the dimensions of the screen (w and h).
• Run a binary search over the array of font sizes. At each step, the middle font size is tested using the check() function.
• If check() returns true, indicating the current font size fits the text within the screen, we search in the higher (larger font size) half of the remaining fonts, since our goal is to find the maximum size.
• If check() returns false, we search in the lower half, discarding the current and larger font sizes as they will also not fit.
• We keep narrowing our search range until we pinpoint the maximum size that fits, or we determine that no font size is suitable, in which case we return -1.

The binary search concludes either when we have successfully found the largest fontSize that allows the text to fit within the given w and h, or when we have checked all possible font sizes, and none can fit the text.

Learn more about Binary Search patterns.

Solution Approach

The provided solution in Python implements the binary search strategy to identify the maximum usable font size as follows:

1. A helper function check(size) is defined that accepts a font size and determines if the text can be displayed on the screen using that particular font size. The function checks two conditions:

   • Whether the height of the text using the specified fontSize exceeds the available height h of the screen.
   • Whether the total width of text at the specified fontSize can fit within the width w of the screen by summing the widths of all characters in text with getWidth(fontSize, c).

2. The binary search algorithm is used by initializing two pointers: left and right. left starts at 0, and right starts at the last index of the fonts array.

3. The main loop of the binary search continues until left and right converge. In each iteration of the loop, a variable mid is calculated to find the middle index of the current search range (left and right). We calculate mid using the expression (left + right + 1) >> 1, which is a bit manipulation technique to calculate the average of left and right and shift one bit to the right, effectively dividing by 2.

4. The check(fonts[mid]) function call tests if the font size indicated by the mid index along with text will fit both width and height constraints on the screen. Two scenarios are possible:

   • If true, it means the text fits the screen with fonts[mid], so the search continues in the right half (including mid) to see if a larger font size could also fit.
   • If false, the text does not fit with fonts[mid], and hence all larger font sizes will not fit either, so the search is confined to the left half, excluding mid.

5. Upon completion of the loop, if the check function returns true for fonts[left], that means fonts[left] is the largest font size that the text can be displayed with on the screen. If check fails for fonts[left], then no available font sizes can display the text within the screen limits, and -1 is returned.

The solution leverages the sorted nature of the fonts array and the binary search algorithm to efficiently zone in on the maximum font size that satisfies the display constraints. The time complexity of the binary search algorithm is O(log n), which significantly reduces the number of checks needed compared to a linear search, especially for larger font arrays.

Example Walkthrough

Let's consider a small example to illustrate the solution approach using a binary search strategy mentioned in the problem description.

Suppose we have the following scenario:

• The given text is "LeetCode".
• The screen dimension, width (w), is 100 units, and the height (h), is 20 units.
• We have a sorted array of possible font sizes fonts = [10, 12, 14, 16, 18, 20].
• A FontInfo interface with two hypothetical results:
  • getWidth(fontSize, ch) returns fontSize * 1.2 units for each ch.
  • getHeight(fontSize) returns fontSize units.

Let's walk through the steps of the binary search strategy using this example:

1. We need to define a helper function check(size). This function will determine whether or not the text can be displayed using a specific font size without exceeding the screen dimensions.

2. Our binary search starts with setting left to 0 and right to the last index of the fonts array, which is 5 in this case.

3. We enter a loop that will continue until left and right converge. Initially, left = 0 and right = 5.

4. Calculate mid. Let's begin with the first iteration:

   • mid = (0 + 5 + 1) >> 1 evaluates to 3, so fontSize = fonts[3] is 16.

5. We call check(16) to see if we can use a font size of 16 units. The height at this size is 16 units, which does not exceed our screen height of 20 units. The width would be the sum of each character's width using getWidth(16, ch), which is 16 * 1.2 * len("LeetCode"). If this sum is less than or equal to 100 units, check(16) returns true.

6. Depending on the result, we adjust our left and right:

   • If check(16) is true, it means we can potentially use an even larger font size, so we set left = mid.
   • If check(16) is false, then we need to try smaller font sizes, and we set right = mid - 1.

7. We then repeat this process until left and right converge on the largest font size that can be used without exceeding the screen dimensions.

8. Eventually, suppose left converges on index 2, with check(fonts[2]) returning true and check(fonts[3]) returning false. This would indicate that 14 is the largest font size we can use.

If during this iterative process no fontSize passes the check function, then no font size will fit the text on the screen and we return -1.

In our example, let's say after iterating we have found that 14 is the maximum size we can use to fit "LeetCode" within the screen dimensions, so that would be our function's return value. If every font size tested returns false in our check function, then we would return -1, indicating no suitable font size is available.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of this binary search algorithm within a bounded list of font sizes needs to be analyzed based on two main operations:

1. The binary search itself.
2. The check function which is called at each step of the binary search.

Performing binary search on the sorted list of font sizes has a time complexity of O(log n), where n is the number of font sizes in the list fonts.

The check function gets called once for every step of the binary search. Inside the check function, there's a sum operation that iterates through every character c in the string text.

The time taken to get the width and height for a given fontSize is not specified, so we assume it to be O(1) for each call to fontInfo.getWidth and fontInfo.getHeight.

The total number of characters in text is denoted as m. Therefore, the complexity for checking all characters in the text at any font size is O(m).

For every binary search step, the check function is invoked once which would result in the time complexity of O(m) for each step of the binary search. Therefore, the combined time complexity for the binary search and the check function is O(m * log n).

Space Complexity

The space complexity of the algorithm is the additional space required outside of the inputs. The algorithm uses a constant amount of space for variables such as left, right, mid, and ans. Hence, the space complexity is O(1).

We are not considering the space taken up by the input text, font list, or the FontInfo object, as these are not part of the algorithm's own space requirements.

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