Problem Transformation

Transformation in Maximum Subarray Problem

The Maximum Subarray Problem asks for the contiguous subarray with the largest sum. When applied to stock market data, it can be interpreted as finding the best time to buy and sell to maximize profit. However, when we shift focus from daily prices to daily price changes, the problem transforms into an equivalent but more straightforward problem.

Daily Prices vs. Daily Change

  • Daily Prices: These are the raw data points that tell you the price of the stock on a specific day.
  • Daily Change: This is the difference between the stock price of two consecutive days.

Why the Transformation Works

  1. Simplifies Problem: Looking at daily changes converts the problem from finding a subarray with the maximum sum of prices to a subarray with the maximum sum of changes, which directly relates to profit.

  2. Directly Tackles Profit: The objective is to maximize the difference between the selling and buying prices, which is the same as maximizing the sum of a subarray of daily changes.

  3. Preserves Original Problem: The transformation is lossless. That is, you can reconstruct one problem from the other. This makes them equivalent.

Example

Let’s consider a simplified example. Suppose the daily prices are [1, 2, 4, 2, 5]. The daily changes would then be [1, 2, -2, 3].

  • Maximum subarray in daily prices: [1, 2, 4, 2, 5] with sum = 14
  • Maximum subarray in daily changes: [1, 2, -2, 3] with sum = 4

Notice that the maximum sum of daily changes (4) is precisely the profit you’d earn if you bought at the lowest point (1) and sold at the highest point (5).

In this way, the maximum subarray problem for daily changes directly corresponds to maximizing profit in the context of daily stock prices.

This is an example of both problem transformation and problem simplification.

  1. Problem Transformation: The issue of finding the best days to buy and sell stocks to maximize profit is transformed into a maximum subarray problem involving daily changes in stock prices. The two problems are mathematically equivalent but presented in different forms.

  2. Problem Simplification: By focusing on daily changes instead of daily prices, the problem becomes easier to solve. Instead of working with raw stock prices, working with daily changes allows you to apply a standard algorithmic solution (e.g., Kadane’s algorithm) for finding the maximum subarray sum.

So, it’s a transformation because you’re looking at the problem in a new way (daily changes instead of daily prices), and it’s a simplification because this new perspective makes the problem easier to solve algorithmically.

Max Consecutive Ones III

This problem can be transformed to another equivalent problem: Find the longest subarray with at most K zeros.

Transforming the problem this way can make it more straightforward to solve. If you look for the longest subarray with at most k zeros, you can easily flip those zeros to ones and find the maximum number of consecutive ones. This is a common technique used to simplify problems and often leads to more efficient solutions.

Claude Explanation

Here is content on Problem Transformation following the specified structure:

Problem Transformation is a technique in computer science and mathematics that involves reformulating a problem into an equivalent problem that is easier to solve. The goal is to make the problem more tractable while preserving the desired outcome. Some common ways to transform problems include:

  • Simplification - Removing extraneous information from the problem description or variables to make the problem simpler.

  • Decomposition - Breaking a problem down into sub-problems that can be solved independently.

  • Reduction - Mapping the problem onto a problem that is already known to be solvable.

  • Induction - Solving special cases of the problem and generalizing a solution.

  • Inversion - Solving the inverse of the problem, which may be easier.

Example in Java:

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// Simplification 
// Original problem: Find the maximum of an array with duplicate elements
// Simplified: Find the maximum of an array with unique elements

int max(int[] array) {
  int max = Integer.MIN_VALUE;
  for (int i : array) {
    if (i > max) {
      max = i; 
    }
  }
  return max;
}

// Decomposition
// Original problem: Sort an array
// Break into subproblems: 
// - Divide array into two halves
// - Sort each half recursively 
// - Merge the sorted halves

void mergeSort(int[] array) {
  if (array.length <= 1) {
    return;
  }
  
  int mid = array.length / 2;
  
  int[] left = Arrays.copyOfRange(array, 0, mid);
  int[] right = Arrays.copyOfRange(array, mid, array.length);
  
  mergeSort(left);
  mergeSort(right);
  
  merge(left, right, array);
}

void merge(int[] left, int[] right, int[] result) {
  // merge left and right into result 
}

// Reduction
// Original problem: Shortest path between nodes in a graph 
// Reduce to: Dijkstra's algorithm

void dijkstra(Graph graph, Node source) {
  // Run Dijkstra's algorithm
  // ...
}

// Inversion
// Original problem: Validate a Sudoku board
// Inverted: Find violations of Sudoku rules

boolean isValidSudoku(int[][] board) {
  return findViolations(board).isEmpty(); 
}

List<Violation> findViolations(int[][] board) {
  // Find violations
  // ... 
  return violations;
}

Example in C++:

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// Simplification
// Original problem: Count frequency of words in a file
// Simplified: Count number of words in a file

int countWords(string filename) {
  int count = 0;
  ifstream file(filename);
  string word;
  while (file >> word) {
    count++;
  }
  return count;
}

// Decomposition 
// Original problem: Reverse a linked list
// Break into subproblems:
// - Reverse subsequent nodes recursively 
// - Append head node to end of reversed rest

Node* reverse(Node* head) {
  if (head->next == NULL) {
    return head;
  }
  
  Node* rest = reverse(head->next);
  head->next->next = head;
  head->next = NULL;
  
  return rest;
}

// Reduction
// Original problem: Travelling Salesman Problem 
// Reduce to: Dynamic programming Held-Karp algorithm

void heldKarp(Graph graph) {
  // Run Held-Karp algorithm
  // ...
} 

// Inversion
// Original problem: Check if binary tree is height balanced
// Inverted: Find height imbalance

bool isBalanced(Node* root) {
  return findImbalance(root) == -1;
}

int findImbalance(Node* node) {
  if (node == NULL) return 0;
  
  int leftHeight = findImbalance(node->left);
  if (leftHeight == -1) return -1;
  
  int rightHeight = findImbalance(node->right);
  if (rightHeight == -1) return -1;

  if (abs(leftHeight - rightHeight) > 1) {
    return -1;
  } else {
    return max(leftHeight, rightHeight) + 1; 
  }
}

Example in Python:

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# Simplification
# Original problem: Sum integers in a file
# Simplified: Count integers in a file  

def count_ints(filename):
  count = 0
  with open(filename) as f:
    for line in f:
      count += line.count(',') 
  return count

# Decomposition
# Original problem: Permute a string
# Break into subproblems:
# - Get all permutations of substring 
# - Insert each character of original string in every position

def permute(s):
  if len(s) <= 1:
    return [s]
  
  perms = []  
  for perm in permute(s[1:]):
    for i in range(len(perm)+1):
      perms.append(perm[:i] + s[0] + perm[i:])

  return perms

# Reduction
# Original problem: Shortest substring containing all chars in a string
# Reduce to: Sliding window algorithm  

def min_window(s, t):
  # Apply sliding window algorithm
  #...

# Inversion  
# Original problem: Sudoku solver
# Inverted: Sudoku checker

def is_valid_sudoku(board):
  return not find_violations(board)

def find_violations(board):
  # Find violations
  #...
  return violations

In summary, problem transformation is a key technique in computer science and mathematics that can make problems more tractable. Common transformations include simplification, decomposition, reduction, and inversion. The key is to reformulate the problem into an equivalent one that is easier to reason about and solve.