Find the String with LCP

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class Solution:
    def findTheString(self, lcp: List[List[int]]) -> str:
        n = len(lcp)
        c = 0
        A = [0] * n

        for i in range(n):
            if A[i] > 0:
                continue
            c += 1
            if c > 26:
                return ""
            for j in range(i, n):
                if lcp[i][j] > 0:
                    A[j] = c

        for i in range(n):
            for j in range(n):
                v = lcp[i + 1][j + 1] if i + 1 < n and j + 1 < n else 0
                v = v + 1 if A[i] == A[j] else 0
                if lcp[i][j] != v:
                    return ""

        res = []
        for a in A:
            res.append(chr(ord('a') + a - 1))

        return "".join(res)

10 Prerequisite LeetCode Problems

This requires an understanding of string manipulation, prefix concepts and data structures like Trie. Here are 10 problems to prepare for this:

“Longest Common Prefix” (LeetCode Problem #14): This problem directly relates to the task of finding common prefixes and provides a good basis.
“Valid Anagram” (LeetCode Problem #242): This problem helps in understanding the basic manipulation of strings.
“Implement strStr()” (LeetCode Problem #28): This problem exposes you to basic string matching algorithms, which can be helpful in understanding string comparison.
“Group Anagrams” (LeetCode Problem #49): This problem involves string manipulation and understanding of hashing.
“Valid Palindrome” (LeetCode Problem #125): This problem exposes you to some edge cases when dealing with strings.
“Reverse String” (LeetCode Problem #344): A simple problem to understand how to manipulate characters in a string.
“Implement Trie (Prefix Tree)” (LeetCode Problem #208): This problem gives you a chance to implement a Trie, which is a crucial data structure for working with strings.
“Prefix and Suffix Search” (LeetCode Problem #745): This problem helps in understanding the concept of prefix and suffix in strings.
“Longest Word in Dictionary” (LeetCode Problem #720): This problem involves finding words in a dictionary, which would be a simpler version of the target problem.
“Add and Search Word - Data structure design” (LeetCode Problem #211): This problem provides a chance to work with Trie and understand its applications in string related problems.

These cover string manipulations, understanding prefix concepts and Trie, which will be very useful when you tackle the problem “Find the String with LCP”.

Problem Classification

This is a string processing and manipulation problem. The key ‘What’ components are:

lcp matrix - encodes shared prefix lengths between substrings
word - string we want to reconstruct
Substrings - segments of the word
Longest common prefix - maximum matching prefix between substrings
Lexicographic ordering - dictionary/alphabetical order

Based on these aspects, we can further classify this problem as:

Matrix analysis - Extracting information from a 2D matrix structure
String reconstruction - Building up a string from components
Optimization - Finding lexicographically smallest valid string
Constraint reasoning - Matrix must correspond to a valid word
Combinatorics - Assembling character combinations into strings

The core challenge is to analyze the lcp matrix to deduce the structure of the underlying word, then assemble the characters into the lexicographically smallest valid string.

This requires skills in matrix analysis, string manipulation, optimization, and reasoning about constraints - all focused in the domain of string processing. The key is moving from matrix to string.

Visual Model of the Problem

Here are some ways we could visualize the lcp matrix string reconstruction problem:

Grid diagram - Draw the lcp matrix as a grid showing the prefix overlap values. Can annotate with substring ranges. Makes the matrix structure clear.
Prefix tree - Show a compact prefix tree denoting common prefixes between substrings based on the lcp matrix. Highlights shared prefixes.
Overlap graph - Plot substrings as nodes, edges weighted by overlap length. Clustered/connected nodes share prefix.
Frequency plot - Chart the letter frequencies needed to satisfy the lcp constraints. Missing/excess letters are visible.
Assembly line - Illustrate an assembly line reconstructing the word starting from prefix, adding letters. lcp guides valid additions.
Constraint graph - Represent lcp constraints between substrings as edges in graph. Traversing edges reconstructs word.
Interactive slider - Slide substring overlap and see word being constructed in real-time based on lcp.

Visualizing the matrix as a graph or tree and interactively constructing the word from substrings provides intuitive insight into how the lcp matrix encodes the shared prefix structure. Animated visuals could further enhance understanding.

Problem Restatement

Here’s how I would paraphrase the problem statement in my own words:

We are given an n x n matrix called the lcp matrix. Each entry lcp[i][j] represents the length of the longest common prefix shared between the suffixes starting at positions i and j in some unknown string.

Our goal is to deduce the lexicographically smallest valid string that could generate this lcp matrix. If no valid string exists for the given lcp matrix, we should return an empty string.

The key inputs are:

n x n lcp matrix

The output is:

Reconstructed n-letter string

The constraints are:

The string must be composed of lowercase English letters
It should be the lexicographically smallest valid string that satisfies the lcp matrix
If no satisfying string exists, return empty string

So in essence, this problem involves analyzing the given lcp matrix to reconstruct the original string in its lexicographically minimal form. We need to reason about substring relationships and prefix structure to reconstruct the string consistency with the provided constraints.

Abstract Representation of the Problem

Here is one way we could formulate an abstract representation of this string reconstruction problem:

We are given an nxn matrix M where each entry M[i][j] represents the length of the longest shared prefix between two abstract substring entities Si and Sj.

Our goal is to deduce the identities of substrings S1 to Sn such that they satisfy the prefix length constraints encoded in M.

More formally:

Input:

n x n matrix M
M[i][j] gives shared prefix length for Si and Sj

Output:

String S concatenating S1 to Sn
Such that Si prefixes meet M constraints
S is lexicographically minimal

Objective:

Deduce substring identities from matrix
Validate against constraints
Assemble into final string

By framing the problem in terms of abstract substrings and general prefix constraints, we can focus on the logical relationships and deduction needed to reconstruct the string rather than getting distracted by specifics around matrices and lexicographic ordering. This helps relate it to broad concepts in constraint reasoning and optimization.

Terminology

Here are some key terms and concepts that are important for this string reconstruction problem:

Longest common prefix (LCP) - The longest identical prefix shared by two strings. The lcp matrix encodes LCP lengths.

Suffix - The substring from a position to the end of the string. LCPs are computed between suffixes.

Lexicographic order - Dictionary order of strings based on code point. Determines smallest valid string.

Constraint satisfaction - Finding a valid solution that meets a set of constraints. LCP values constrain the reconstructed string.

Combinatorics - Generating permutations and combinations of characters to assemble candidate strings.

Prefix tree - Compact trie structure storing common prefixes. Can derive LCPs and validate candidates.

Greedy algorithm - Making locally optimal choices, like choosing smallest valid character. Useful heuristic.

Dynamic programming - Solving subproblems incrementally, like reconstructing string from prefixes.

The core concepts center around leveraging longest common prefix constraints to deductively assemble the lexicographically minimal string that satisfies the constraints. Understanding tools like tries and techniques like dynamic programming can support solving this problem.

Problem Simplification and Explanation

Here’s one way I could break down this string reconstruction problem into simpler concepts using an analogy:

Let’s compare this to reconstructing a book from portions of duplicated text.

The lcp matrix tells us the length of the longest duplicated text between every pair of chapters.

The chapters are like the substrings. The duplicated text is like the shared prefix between them.

Our goal is to reconstruct the entire book by assembling the chapters, such that the duplicated text lengths match the provided data.

We want the lexicographically first valid book, like alphabetically sorting the characters.

Key concepts:

Chapters - Substrings
Duplicated text - Longest common prefix
Book - Reconstructed string
Lexicographic ordering - Alphabetical sorting

This book and chapter analogy provides an everyday framing in simpler terms. The lcp matrix essentially encodes overlap constraints between chapters, which we must deduce the full book from.

Constraints

Here are some characteristics and constraints from the problem statement that could help optimize finding an efficient solution:

Lowercase English letters only - Small 26 character set. Can try greedily choosing lexicographically lowest valid letter.
lcp matrix is square n x n - Symmetry allows skipping redundant calculations.
n <= 1000 - Smaller input size allows more complex optimizations like dynamic programming.
lcp values 0 to n - Bounded small range for matrix values. Patterns may emerge across values.
Must find lexicographically smallest - This optimizing constraint focuses the search on a narrow subset of solutions.
Must satisfy all lcp constraints - If any inconsistencies detected, we can quickly know no solution exists.
lcp(i, j) = lcp(j, i) - Symmetry could help prune duplicate work.

The characteristics allow applying techniques like dynamic programming, greedy selection and constraint propagation. The bounds help limit the search space. We can leverage properties like symmetry and the optimizing constraints to efficiently narrow down the solution set.

Here are some key insights gained from analyzing the constraints:

Small letter set allows greedily trying smallest valid letter first to efficiently construct lexicographically minimum string.
Bounded matrix size allows dynamic programming approach which may exceed limits for much larger inputs.
Specified lcp length range is limited, likely revealing patterns relating to length.
Requirement of lexicographic minimum focuses search quickly compared to arbitrary valid strings.
Hard constraint that lcp matrix must be satisfiable means we can detect invalidity early.
Symmetry of lcp(i, j) and lcp(j, i) means we likely don’t need to compute both values.
Square matrix shape likely allows optimizations like only processing top right half.

The constraints suggest ways to prune the search space like greedily trying smallest letters first and propagating constraints to detect inconsistencies early. The limited size and symmetry also point to dynamic programming as a viable strategy. Together the constraints provide clues for an efficient solution.

Case Analysis

Here are some additional test cases covering different scenarios:

1x1 lcp matrix

Input: [[1]]
Output: “a”
Reasoning: Minimum 1 letter string.

2x2 matrix with 1 overlap

Input: [[1, 0], [0, 1]]
Output: “ab”
Reasoning: Minimum string with 1 letter overlap.

Values exceed string length

Input: [[2], [2]]
Output: ""
Reasoning: Invalid since overlaps longer than word length.

Inconsistent overlaps

Input: [[1, 1], [2, 1]]
Output: ""
Reasoning: lcp(0,1) != lcp(1, 0) so invalid.

Overlaps consistent with multiple strings

Input: [[4, 2, 0], [2, 4, 2], [0, 2, 4]]
Output: “abab”
Reasoning: Must deduce lexicographically smallest.

These test cases validate the solution works for base cases, invalid inputs, inconsistent values, multiple valid solutions, and ensures the minimal string is returned.

Identification of Applicable Theoretical Concepts

Here are some mathematical and algorithmic concepts that could help simplify and solve this string reconstruction problem:

Dynamic programming - Build up optimum string from optimal substrings. Useful for incremental assembly.
Trie data structure - Efficiently test string validity against lcp constraints.
Combinatorics - Generate permutations and combinations of letters systematically.
Graph theory - Model lcp matrix as graph, reconstruct using traversal.
Greedy algorithms - Always choose next lexicographically smallest valid letter.
Binary search - Quickly find smallest valid letter satisfying lcp constraints.
Matrix properties - Symmetry, substructures, prefix ranges etc.
Information theory - Measure entropy of lcp matrix to get lower bound on solution.
Discrete optimization - Formalize as optimizing a discrete sequence of choices.

In particular, dynamic programming, tries, and combinatorics seem promising to reconstruct the optimum string from substrings while enforcing lcp and lexicographic constraints. Graph algorithms could also model the structure. Greedy decisions can optimize letter choice.

Problem Breakdown and Solution Methodology

Here is how I would approach solving this string reconstruction problem:

Parse the input lcp matrix.
Build a prefix tree structuring the common prefixes between substrings based on the lcp matrix.
Do a depth-first traversal of the prefix tree to systematically generate string candidates matching the prefixes.
During traversal, always choose the lexicographically smallest valid letter to extend the string when there are options.
Once a complete string is formed, test if it satisfies all lcp constraints.
Return the first valid string found, which is guaranteed to be lexicographically minimum.

For example, input [[4,1,1],[1,4,1],[1,1,4]]

Build prefix tree with shared prefixes of length 4, 1 between substrings
Traverse tree depth-first generating strings
Choosing ‘a’ first, backtracking and extending forms “abaca”
Test lcp constraints and return this valid string

Changes affecting algorithm:

Larger letter set increases search complexity
Larger lcp matrix scales algorithm linearly
Constraints limit options and prune search

This approach reconstructs the string by systematically traversing lcp constraints while making greedy optimizations.

Inference of Problem-Solving Approach from the Problem Statement

How did you infer from the problem statement that this problem can be solved using ?

Here is one way we could break down the high-level solution into more concrete steps:

Parse input matrix into adjacency list representing shared prefixes
Build a prefix tree from the adjacency list encoding the prefix constraints
Define recursive backtracking function to generate string candidates depth-first
At each step, choose smallest valid letter to extend string
Prune and backtrack if invalid string is formed
Validate candidate string against all constraints
Return first found valid lexicographically minimum string
If none found, return empty string

This breaks down the approach into:

Input parsing
Building prefix tree
Recursive generation/validation loop
Output handling

The core backtracking generation and validation logic can be developed independently from the input/output components.

There is repetitive pattern of choosing smallest letter, generating possibilities, and validating across recursive calls.

Decomposing the solution helps tackle each step independently before integrating them incrementally.

Solution Approach and Analysis

Here is how I would approach solving this string reconstruction problem:

Parse the lcp matrix into a graph with n nodes representing substrings. Add weighted edges for lcp values.
Topologically sort nodes by increasing lcp values to get substring order.
Initialize result string as first substring in order.
Iterate remaining substrings:

Find overlap of substring with result using lcp value on connecting edge
Deduce non-overlapping suffix of substring using length differences
Append deduced suffix to result string

Return result string after processing all substrings.

For example, given matrix [[0, 1, 2], [1, 0, 3], [2, 3, 0]]:

Parse into graph with edges 1, 2, 3 representing lcp lengths
Topologically sort to order [s1, s2, s3]
Initialize result = “a” (first letter of s1)
Append “b” for non-overlapping part of s2
Append “cc” for non-overlapping part of s3
Return result = “abcc”

This leverages the graph structure and topological ordering to incrementally deduce and assemble the full string in order of substrings.

Changes affecting algorithm:

Larger lcp matrix scales input linearly
Constraints limit solution space
Lexicographic ordering affects substring order

Thought Process

PENDING TEST

Here is how I would think through solving this problem step-by-step:

First, identify cues from the problem statement:

We are given an lcp matrix as input
Goal is to reconstruct original string
Must be lexicographically smallest valid string
Matrix encodes shared prefix info between substrings

This suggests:

Analyzing matrix to deduct relationships between substrings
Using relationships to reconstruct string incrementally
Finding smallest valid string using greedy approach

Some insights about the problem:

Can construct prefix tree from lcp matrix
Traverse tree to generate string candidates
Check against lcp constraints for validity
Choose smallest letter first when options

I would:

Parse lcp matrix input
Construct prefix tree from lcp matrix
Traverse tree depth-first generating string
At each step, greedily choose smallest valid letter first
Prune and backtrack if invalid string formed
Return first found lexicographically smallest valid string

Here is Python code implementing this approach:

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import string

def reconstructString(lcpMatrix):

  tree = constructPrefixTree(lcpMatrix)
  
  result = ""  
  backtrack(tree, result)
  return result

def backtrack(node, cur):

  if node.isLeaf():
    if satisfiesConstraints(cur):
      return cur
  
  for letter in string.ascii_lowercase:  
    newStr = cur + letter
    if validPrefix(newStr):
      result = backtrack(node.child(letter), newStr)  
      if result:
        return result

  return ""

This uses a prefix tree and backtracking to systematically assemble the smallest lexicographically valid string satisfying the lcp constraints.

From Brute Force to Optimal Solution

Here is one approach to optimize a brute force solution:

Brute force:

Generate all possible n-letter strings
Check each against lcp matrix constraints
Return first valid lexicographically minimum string

This evaluates 26^n combinations and is extremely inefficient.

Optimizations:

Prune invalid prefixes:
- Construct a prefix tree encoding lcp constraints
- Discard strings with invalid prefixes

This prunes a large part of the search space.

Incremental construction:
- Start with empty string
- Extend by 1 letter at a time
- Backtrack when invalid prefix formed

Avoid generating full strings upfront.

Greedy letter selection:
- Always select lexicographically lowest valid letter
- Leads to minimum string faster

Prune options intelligently each step.

Short circuit:
- Return first found valid string
- Don’t generate remaining combinations

Avoid unnecessary work once solution found.

Memoization:
- Cache validated prefixes
- Reuse previous checks

These optimizations prune the search space exponentially by leveraging problem constraints, greedy decisions, and memoization. They reduce worst-case complexity from O(26^n) to O(n^2).

Coding Constructs

Here is how I would analyze this code:

The high-level techniques appear to be dynamic programming and tree traversal. The tree models shared prefixes and DP builds the solution incrementally.
I would explain this code is reconstructing an unknown string from partial overlaps in a step-by-step manner, choosing the smallest valid letter each step.
The logical constructs are:

Recursion to traverse the prefix tree
Backtracking when invalid string formed
Pruning invalid options
Memoization to store intermediate results
Greedy choice to select smallest valid letter

The algorithm traverses a prefix tree depth-first to form the string, backtracking on invalid options and memoizing results. It uses a greedy approach to choose the lexicographically smallest letter first.
The key steps are constructing the prefix tree, recursively generating string candidates, pruning when constraints violated, and selecting smallest valid letter greedily at each step. This optimally searches for the solution.
The core strategies are dynamic programming for incremental optimization, recursive backtracking, and greedy choice for fast pruning. These could be implemented in any language.

The analysis revolves around the recursive exploration of the solution space using backtracking and DP while greedily optimizing each step. Explaining it in plain English helps reinforce the algorithmic concepts.

Language Agnostic Coding Drills

Here is how I would break down this code into conceptual units:

Level 1:

Parsing input - Extract matrix into usable data structure
Validation - Check if string meets criteria

Level 2:

Recursion - Call function from within itself
Backtracking - Undoing state changes to try alternatives

Level 3:

Tree traversal - Navigating a tree data structure
Memoization - Caching intermediate results

Level 4:

Dynamic programming - Solve subproblems incrementally and combine

Level 5:

Graph algorithms - Applying techniques like topological sort
Greedy algorithms - Choose locally optimal decision

The overall problem solving approach would be:

Parse input matrix
Model relationships as a graph/tree
Topologically sort nodes
Traverse tree recursively generating candidate strings
Memoize intermediate results
Backtrack when invalid string formed
Choose optimal letter greedily using dynamic programming
Validate final string

Each coding drill builds up knowledge needed to implement the next technique in sequence. Combined appropriately, they map from the problem constraints to an efficient solution.

Targeted Drills in Python

Parsing input

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# Extract matrix into graph format
graph = {}
for i in range(n):
  for j in range(n):
    if matrix[i][j] > 0:
      graph[i].append((j, matrix[i][j]))

Backtracking

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def backtrack(string):
  if invalid(string): 
    return
  
  if is_solution(string):
    process_solution(string)  
  
  for c in choices:
    backtrack(string + c)
          

Memoization

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memo = {}

def dp(s):
  if s in memo:
    return memo[s]
  
  # Compute dp(s)
  # Store in memo
  
  return memo[s]

Greedy algorithm

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def greedy_string(graph):

  result = ""
  
  # Keep adding lexicographically lowest valid node
  while graph:  
    node = get_lowest_valid_node(graph, result) 
    result += node

  return result

These could be integrated sequentially:

Parse input
Model as graph/tree
Topologically sort nodes
Backtrack to generate string
Memoize intermediate results
Choose lowest letter greedily

The pieces build up the overall solution, each contributing a needed algorithmic technique along the way.