#756

Medium

auto_awesomeBit Manipulation plus Depth-First Search

LeetCode Problem Workspace

Pyramid Transition Matrix

The Pyramid Transition Matrix problem requires determining whether a pyramid can be formed with given blocks and valid patterns using bit manipulation and DFS.

bit manipulation depth first search breadth first search

Problem Statement

You are given a bottom row of blocks in the form of a string, and a list of allowed triangular patterns. Each triangular pattern is made up of three blocks: two bottom blocks and one top block. Your task is to determine if it's possible to build a pyramid starting from the bottom row using the allowed patterns. At each level, a block must be formed by combining two blocks directly beneath it. The goal is to use the given patterns to progressively form the layers of the pyramid.

The problem involves iterating through the base of the pyramid, starting from the bottom row, and using a depth-first search (DFS) strategy to explore all valid transitions. If you can build a complete pyramid, return true. Otherwise, return false. Note that the blocks and patterns are constrained to specific letters and the pyramid must be aesthetically pleasing according to the given rules.

Examples

Example 1

Input: bottom = "BCD", allowed = ["BCC","CDE","CEA","FFF"]

Output: true

The allowed triangular patterns are shown on the right. Starting from the bottom (level 3), we can build "CE" on level 2 and then build "A" on level 1. There are three triangular patterns in the pyramid, which are "BCC", "CDE", and "CEA". All are allowed.

Example 2

Input: bottom = "AAAA", allowed = ["AAB","AAC","BCD","BBE","DEF"]

Output: false

The allowed triangular patterns are shown on the right. Starting from the bottom (level 4), there are multiple ways to build level 3, but trying all the possibilites, you will get always stuck before building level 1.

Constraints

2 <= bottom.length <= 6
0 <= allowed.length <= 216
allowed[i].length == 3
The letters in all input strings are from the set {'A', 'B', 'C', 'D', 'E', 'F'}.
All the values of allowed are unique.

Solution Approach

Bit Manipulation for Efficient Pattern Lookup

By encoding the allowed patterns as a bitmask, we can quickly check if a pair of adjacent blocks from the bottom row can form a valid top block. This allows us to minimize the search space and reduce the complexity of the solution. Bit manipulation helps efficiently store and retrieve valid transitions between blocks.

Depth-First Search (DFS) to Build the Pyramid

Using DFS, we recursively attempt to build each level of the pyramid by validating if each pair of blocks can form a valid top block from the allowed patterns. If at any point a valid configuration cannot be found, the DFS backtracks. This strategy explores all possible configurations from bottom to top.

Pruning Invalid Paths Early

During the DFS process, if at any point no valid top block can be formed from the current pair of blocks, we can prune that path and return early, saving time. This avoids unnecessary exploration of invalid pyramid configurations and optimizes the solution.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time and space complexity of the solution depends on the number of patterns and the bottom row length. In the worst case, the solution explores all possible combinations of blocks across the pyramid layers. The bitmask representation helps reduce the complexity compared to brute force methods, especially in larger test cases with many allowed patterns.

What Interviewers Usually Probe

Candidate demonstrates understanding of how bit manipulation can optimize pattern lookup.
Candidate uses depth-first search (DFS) correctly to explore all valid configurations.
Candidate identifies opportunities to prune invalid paths early, improving efficiency.

Common Pitfalls or Variants

Common pitfalls

Failing to implement an efficient lookup mechanism for checking valid transitions between blocks.
Incorrectly handling the base case of the DFS when no valid configuration exists.
Not pruning invalid paths early, leading to excessive recursion and slower performance.

Follow-up variants

A variant could involve different sets of allowed patterns, requiring the candidate to adapt the DFS or bit manipulation strategy.
Another variant could involve an increased number of rows or larger input sizes, pushing the candidate to focus on optimizing space and time complexity.
A possible variation could involve an additional constraint on the allowed characters, changing the way patterns are represented and checked.

FAQ

How do I approach solving Pyramid Transition Matrix?

The key to solving this problem is to combine bit manipulation and DFS. First, represent the allowed patterns as bitmasks for fast lookups. Then, use DFS to recursively attempt building the pyramid, pruning invalid paths early.

What is the main challenge in the Pyramid Transition Matrix problem?

The main challenge is efficiently managing the allowed patterns and transitioning between blocks while ensuring that the pyramid can be built correctly. Bit manipulation and DFS help manage this complexity.

Can DFS be used to solve Pyramid Transition Matrix?

Yes, DFS is a core part of the solution. It helps recursively explore all possible pyramid configurations by checking the allowed transitions between blocks at each level.

How does bit manipulation help in the Pyramid Transition Matrix problem?

Bit manipulation allows for efficient representation and fast lookup of valid transitions between pairs of blocks, making it possible to check if a pair can form a valid top block.

What are common pitfalls in solving Pyramid Transition Matrix?

Common pitfalls include failing to implement an efficient lookup for valid transitions, incorrectly handling the DFS base case, and not pruning invalid paths early enough, which leads to slower performance.

terminal

Solution

Solution 1: Memoization

We define a hash table $d$ to store the allowed triangular patterns, where the key is a pair of two characters and the value is the corresponding list of characters, indicating that the two characters can form a triangular pattern with each item in the value list being the top of the triangle.

class Solution:
    def pyramidTransition(self, bottom: str, allowed: List[str]) -> bool:
        @cache
        def dfs(s: str) -> bool:
            if len(s) == 1:
                return True
            t = []
            for a, b in pairwise(s):
                cs = d[a, b]
                if not cs:
                    return False
                t.append(cs)
            return any(dfs("".join(nxt)) for nxt in product(*t))

        d = defaultdict(list)
        for a, b, c in allowed:
            d[a, b].append(c)
        return dfs(bottom)

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