351. Android Unlock Patterns

Problem Description

The problem belongs to the category of backtracking problems that simulate the process of constructing the unlock patterns of an Android device, which has a distinctive 3 x 3 grid with dots. A user has to draw a pattern by connecting dots without repeating any dot. The challenge is to count all unique and valid unlock patterns that consist of at least m and at most n dots.

The constraints for a pattern to be valid are:

1. Distinct Dots: Every dot can only appear once in a pattern.
2. Line Segment Passing: If a line connecting two dots passes through the center of another dot, then that center dot must have already been used in the pattern.

The task involves finding all such unique patterns that obey the above rules and counting them, which may sound simple but can be quite complex to approach due to the possibilities and constraints involved.

Intuition

The core idea of the solution is to use backtracking, a common technique to explore all the possibilities in a systematic way. Backtracking allows us to build a solution step-by-step and backtrack as soon as we realize that the current path will not lead to a solution. This way, we efficiently explore the search space of all possible patterns.

To simplify the problem, we use the following strategies:

1. Symmetry: The 3 x 3 grid has symmetrical properties that we can exploit to reduce computation. For example, starting from corner points (1, 3, 7, 9) or starting from edge midpoints (2, 4, 6, 8) give rise to patterns that are essentially similar. Therefore, computing for one corner and multiplying the result by 4 (since there are 4 corners) and doing the same for the edge midpoints is sufficient.

2. Cross Points: To address the constraint regarding the line segment passing, we create an auxiliary 'cross' matrix that records the cross point for each pair of dots. For instance, the pair (1, 3) has the cross point 2. During the exploration of patterns, we use this information to check if we are allowed to connect two dots straightaway or if we need to have passed through certain dots first.

Approaching the problem this way breaks down the complex task into smaller steps and uses reasoning to efficiently search for all possible solutions by considering the symmetry and constraints unique to the unlock pattern challenge.

In essence, we consider each dot as a starting point and explore all possible patterns emanating from it while keeping track of the visited dots and ensuring we adhere to the rules stated. We recursively build paths and count those that fall within the m to n range.

Learn more about Dynamic Programming and Backtracking patterns.

Solution Approach

The solution's approach can be divided into the following steps, each leveraging specific algorithms, data structures, or patterns commonly used in backtracking problems:

1. Defining the Cross Matrix: We start by defining a cross matrix where cross[i][j] represents the middle point (the point that should be previously visited) when drawing a line from dot i to dot j. This matrix is precomputed and hard-coded into the solution to comply with the constraint regarding the line segments.

2. Backtracking Function dfs: A recursive backtracking function dfs is implemented to explore all unique paths from any start dot. This function takes two parameters: the current dot i and a counter cnt that tracks the number of dots in the current pattern. The backtracking ends when the pattern's length exceeds n.

3. Maintaining vis Vector: We use a vis vector which is a list of boolean values indicating whether a dot has been visited (True) or not (False). This helps avoid reusing a dot in the same pattern, ensuring all dots are distinct.

4. Exploring (DFS) and Counting: For each dot, we perform a Depth-First Search (DFS) by recursively calling the dfs function on the next unvisited dot. If the current dot count is between m and n (inclusive), we increment our count of valid patterns (ans). The number of patterns that start from a certain dot can be obtained by simply returning ans at the end of exploring from that dot.

5. Respecting the Constraints: While exploring, we check if moving from the current dot i to the next dot j directly is valid or not by using the cross matrix. If it passes over a dot that hasn't been visited yet (vis[x] is False where x is the cross point), then we skip this path.

6. Symmetric Property Exploitation: The solution uses symmetry in the grid by calling the dfs function for only three distinct starting points - a corner, an edge midpoint, and the center. The counts for the corner and edge midpoints are then multiplied by 4 because there are four symmetric instances for corners and edge midpoints on the grid. The center is considered only once as it is unique.

7. Computing the Final Result: The final answer is the sum of patterns starting from the corner (returned value from dfs multiplied by 4), patterns starting from an edge midpoint (also multiplied by 4), and patterns starting from the center.

By following this approach, we systematically explore the entire search space while diligently sticking to the problem's rules and exploiting the grid's symmetric property to avoid redundant computation. This results in a computationally efficient solution to the problem.

Example Walkthrough

Let's take a simplified example to illustrate the solution. Imagine we are interested in finding valid patterns that consist of exactly 2 dots on the 3 x 3 grid. For simplicity, we'll start with a single starting point, which is the top-left corner dot 1. Here's how the backtracking process with the aforementioned approach looks like:

1. We begin by defining the cross matrix with NULL values indicating no cross points. Since our example only considers patterns with 2 dots, we don't need to worry much about cross points here.

2. The backtracking function dfs starts with dot 1. We mark dot 1 as visited in the vis vector.

3. We call dfs with the starting dot 1 and cnt initialized to 1 since we've already included dot 1 in our pattern.

4. From dot 1, we explore the next possible dots. Let's try to move to dot 2. Since it has not been visited and does not require passing through a middle dot, we recurse with dfs(2, cnt+1).

5. Inside dfs(2, 2), we check if cnt equals n (which is 2 in our hypothetical case). As it does, we increment our pattern count ans by 1 for this valid sequence [1, 2]. There's no further recursion since we've reached the maximum dots allowed.

6. We backtrack and try the next possibility, moving to dot 3. Since it's valid and doesn't cross over any unvisited dots, we call dfs(3, cnt+1) and increment our count with the sequence [1, 3].

7. We continue this process trying all possible dots (4, 5, 6, 7, 8, 9) and counting the number of valid patterns of length 2 starting with dot 1. For each valid move, we increment ans.

8. Since we're only calculating patterns starting from dot 1 in this example, there's no need for symmetry exploitation. But in a full solution, we would multiply the ans from this step by 4 to account for patterns starting from corners (1, 3, 7, 9) due to the symmetric property.

9. The final answer for 2-dot patterns starting from the corner would be the value of ans obtained from all the recursive dfs calls starting with dot 1.

So, assuming our grid only allowed patterns to be exactly 2 dots starting from dot 1, the ans would represent the number of valid patterns, and the process highlighted here would enumerate them all. In a complete solution, patterns starting from different symmetrical points on the grid and of varying lengths between m and n would all be similarly explored to get the final count.

Solution Implementation

Time and Space Complexity

Time Complexity

The provided code employs a Depth-First Search (DFS) strategy for traversing the space of all possible patterns that can be created on a 3x3 keypad, starting from digit 1, digit 2, and digit 5. The worst-case time complexity would be calculated under the condition that we visit each digit exactly once and go through all possible patterns. Since there are 9 digits and each DFS invocation can lead up to 8 further invocations (minus the visited numbers and the cross numbers that require a middle number to be visited first), the upper bound can initially seem to be O(9!). Yet, the time complexity is better due to the constraints that cut down the search space significantly - the cross matrix prevents jumps over unvisited keys.

Given that we invoke the DFS starting from 1, 2, and 5, these are our symmetrical starting points, and the answer will be the result from calling DFS from these start points, each multiplied by the number of symmetric equivalents on the keypad (corners are 4 times, edges are 4 times, and the center is unique):

1. Invoke dfs(1) and multiply the result by 4 (for the 4 corners).
2. Invoke dfs(2) and multiply the result by 4 (for the 4 edge middle points).
3. Invoke dfs(5) (the center).

Hence, we can simplify the naive factorial complexity as each corner and edge contribute equally.

Despite the upper bound given by the factorial, the actual time complexity is significantly less due to the constraints reducing the number of candidates at each step. However, providing an exact time complexity is difficult without empirical analysis or a more detailed combinatorial examination, which typically isn't expected for interview-style coding problems.

Space Complexity

The space complexity of the algorithm is O(n) due to the recursive nature of DFS, where n in this context is the maximum depth of the recursive stack, which corresponds to the maximum length of the pattern (up to 9). We also have additional data structures - vis (used to mark visited numbers) and cross (a matrix indicating the middle number that must be visited when going from one number to another in a direct line), but these are of fixed size and hence constitute O(1) additional space. Therefore, the space complexity remains O(n) where n stands for the number of recursive calls.

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