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Home AI News Codeforces Problem B Solution: Finding the Spruce Trees Dynamically

Codeforces Problem B Solution: Finding the Spruce Trees Dynamically

Updated on Nov 02,2025

Table of Contents

This article delves into a solution for Problem B, 'Find the Spruce,' from the recent Codeforces Round 689. We will break down the problem statement, explain the core concepts, and present a dynamic programming approach to efficiently count the number of spruce trees within a given grid. This is perfect for competitive programmers looking to sharpen their skills.

Key Points

Understanding the Spruce Tree Definition: A spruce tree is defined recursively based on a set of asterisk characters.

Dynamic Programming Approach: Utilizing a DP table to efficiently count spruce trees.

Matrix Traversal: Implementing bottom-up traversal to compute spruce tree counts.

Edge Case Handling: Correctly identifying and handling edge cases near the boundaries of the grid.

Problem Statement: Find the Spruce

What is a Spruce Tree?

The problem presents a grid filled with asterisks ('*') and dots ('.'). The goal is to count how many 'spruce trees' exist within the grid.

A spruce tree of height k is defined as a set of asterisk cells centered at a point (x, y). Each row (z) from x to x + k -1 must contain a set of asterisks. The width of each row should be increasing and consist of consecutive asterisks, forming a traditional spruce tree shape. Essentially, you're searching for triangles of asterisks.

The key here is that a valid spruce must be a perfect, symmetrical shape. Irregular arrangements, like missing elements or asymmetrical structures, do not count.

Input and Output

The input consists of several test cases. Each test case includes the dimensions of the grid (n x m) and the grid itself. The matrix contains characters ' * ' or ' . ' matrix contents.

The output for each test case should be a single integer: the total number of spruce trees in the matrix. We will employ dynamic programming for efficient computation.

Here is example of valid and invalid spruce trees:

Valid Spruce Trees Invalid Spruce Trees
Valid Spruce Trees Invalid Spruce Trees

Understanding the Code Implementation

Setting up the DP Table

First, we initialize a 2D vector called dp of dimensions n+1 x m+1, filled with zeros. This table will store the maximum possible height of the spruce tree ending at each cell (i, j) . The vector is initialized like this:

vector<vector<int>> dp(n+1, vector<int>(m+1));

We will go through each item in dp matrix. The values with * will be converted to 1.

Implementing the Bottom-Up Approach

After that, we can begin to fill the dp table based on our observations. We want to check each table. Please note there are two for loops going from bottom right corner, not top left corner. In case there are points out of range, skip those points and continue to the next one. If current dp point is equal to *, we find the minimum value of those three points (a point on left, a point on right, a point on center). And set dp[i][j] to one plus the minimum of those three points. Then, we loop will complete checking from bottom to top, left to right.

for (int i=n-1; i>=0; i--)
{
   for (int j=m-2; j>=1; j--)
   {
      if (dp[i][j] == 1)
      {
         dp[i][j] += min({dp[i+1][j-1], dp[i+1][j], dp[i+1][j+1]});
      }
   }
}

Counting Total Spruce Trees

After completing dp table, we can count every point by looping through dp table, the total spruce trees are accumulated based on height of each point.

for (int i=0; i<n; i++)
{
   for (int j=0; j<m; j++)
   {
      ans += dp[i][j];
   }
}

Advantages and Disadvantages of the Solution

👍 Pros

Efficient Time Complexity: O(n*m) makes it viable for large grids.

Clear Dynamic Programming Implementation: Easy to understand and debug.

Space Optimized: Only requires O(n*m) space for the DP matrix.

👎 Cons

Space Usage: While linear, it still uses a significant amount of space for large grids.

Dynamic Programming Solution

DP Matrix Initialization

The first step in the algorithm is initializing a DP matrix of same size as the input grid. Each DP[i][j] should ideally hold information about the maximum height spruce tree that can be formed with (i, j) as a peak. Initially, each cell containing asterisk should have DP[i][j] = 1 . We are essentially building solution from smaller subproblems.

Bottom-Up Computation

The computation of the DP table proceeds bottom-up. Starting from the second-to-last row and going upward, for each cell (i, j), we check three conditions:

  1. Is the cell an asterisk?
  2. Are its three bottom neighbors also within the grid boundaries?
  3. All these bottom neighbors contain value of ‘1’?

If these three conditions are satisfied, we take the minimum of the DP values among its three bottom neighbors: DP[i+1][j-1], DP[i+1][j], and DP[i+1][j+1]. Adding one to it, and we can save the number in the dp table.

Counting Spruce Trees

To count all spruce trees, we simply iterate through the DP matrix and add all the non-zero DP table entries to a total sum. Each entry represents a spruce tree having the specific height. Each dp[i][j] represents height, so we only need to accumulate each dp table after constructing it.

FAQ

What is the time complexity of this solution?
The time complexity is O(n*m), where n and m are the dimensions of the grid, due to the double-nested loops used to populate and iterate through the DP table.
What is the space complexity of this solution?
The space complexity is O(n*m), corresponding to the space required to store the DP matrix.
How can I verify my solution?
Test your solution with various test cases, particularly those with boundary conditions and complex configurations, to ensure robustness. You can also compare the output with a brute-force approach for smaller grids to validate the results.

Related Questions

Are there other approaches to solve this problem?
While dynamic programming provides an efficient solution, other approaches such as divide and conquer or even backtracking (with careful pruning) might be applicable for smaller grids, although they are unlikely to scale as well as dynamic programming for larger inputs. The dynamic programming method is a practical approach for solving this problem. It combines efficient iteration through the grid with careful handling of boundary conditions. This approach results in clean, optimized solution, which helps the problem clear easily. There are not much space to optimize because the whole grid must be checked.
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