Part 4: Similar Problems - Applying the Pattern
Five LeetCode problems that reinforce the same DP concepts
The Pattern
The original problem (LeetCode 1937 - Maximum Points with Cost) establishes a pattern: matrix-based DP with sequential row selection and distance-based penalties. The breakthrough insight involves using a two-pass optimization (left-to-right and right-to-left) to reduce O(m × n²) complexity to O(m × n).
Here are five problems that reinforce these exact patterns.
1. Minimum Path Cost in a Grid
Problem Number: LeetCode 2304 Difficulty: Medium
How it's similar
This is the closest match to Maximum Points with Cost. You traverse an m × n grid row-by-row, selecting one cell from each row. The key similarity is the explicit transition cost matrix: moving from cell with value grid[r][c1] to column c2 in the next row costs moveCost[grid[r][c1]][c2].
Like problem 1937, you must balance the immediate cell value against transition penalties to future rows.
Key DP Pattern
State: dp[r][c] = minimum cost to reach cell (r, c)
Transition:
dp[r+1][c2] = min(dp[r][c1] + moveCost[grid[r][c1]][c2] + grid[r+1][c2])
for all c1.
Complexity: O(m × n²) or can be optimized depending on cost structure
Markov Property: Current state depends only on previous row values plus explicit transition costs.
2. Minimum Falling Path Sum II
Problem Number: LeetCode 1289 Difficulty: Hard
How it's similar
Given an n × n grid, find the minimum sum falling path where you pick one element from each row, but cannot pick the same column in consecutive rows.
This replaces the distance-based penalty in 1937 with a hard constraint (infinite penalty for same-column selection). The optimization challenge is identical: avoid checking all n columns for each cell.
Key DP Pattern
State: dp[r][c] = minimum sum reaching (r, c)
Optimization Trick: Track first minimum and second minimum from the previous row:
- If current cell is not in the first-min column → use first-min
- Otherwise → use second-min
Complexity: O(n²) time instead of O(n³)
Key Insight: Transforms nested loops into linear passes through algebraic decomposition.
3. Minimum Falling Path Sum
Problem Number: LeetCode 931 Difficulty: Medium
How it's similar
The foundational falling path problem. Given an n × n matrix, find minimum sum path from any first-row cell to any last-row cell, where each step can move to three adjacent positions in the next row (down-left, down, down-right).
This introduces a positional constraint as the penalty mechanism—you can't jump arbitrarily between columns, only to adjacent ones.
Key DP Pattern
State: dp[r][c] = minimum sum to reach (r, c)
Transition:
dp[r][c] = matrix[r][c] + min(
dp[r-1][c-1],
dp[r-1][c],
dp[r-1][c+1]
)
Complexity: O(m × n) time, O(n) space
Key Insight: This is the entry point for understanding row-by-row DP with constrained transitions. The three-choice limitation makes penalties implicit (infinite cost for non-adjacent moves).
4. Cherry Pickup II
Problem Number: LeetCode 1463 Difficulty: Hard
How it's similar
Two robots start at the top row (positions 0 and n-1) and move down simultaneously, each picking cherries. Both robots must process each row, and each can move to one of three adjacent columns in the next row.
The complexity comes from coordinating two selections per row with movement penalties (restricted to 3 adjacent positions).
Key DP Pattern
State: dp[r][c1][c2] = maximum cherries with robot1 at column c1 and robot2 at column c2 in row r
Transition: Involves 3 × 3 = 9 combinations of moves
Complexity: O(m × n²) time (tracking all pairs of column positions)
Key Insight: The Markov property extends to multi-dimensional state space. Reinforces sequential decision-making with multiple agents and constrained transitions.
5. Triangle
Problem Number: LeetCode 120 Difficulty: Medium
How it's similar
Given a triangle structure (row 0 has 1 element, row 1 has 2, etc.), find minimum path sum from apex to base. At each row, you can move to adjacent positions in the next row: from index i, you can reach i or i+1.
This creates a matrix-like structure with variable row widths and adjacency-based movement penalties.
Key DP Pattern
State: dp[r][i] = minimum sum to reach position i in row r
Transition:
dp[r][i] = triangle[r][i] + min(
dp[r-1][i-1],
dp[r-1][i]
)
Complexity: O(n²) time, O(n) space
Key Insight: Reinforces row-by-row processing with positional constraints. The triangular structure adds complexity to boundary conditions while maintaining the core pattern.
Common Thinking Patterns Across All Five Problems
1. Markov Property
Current optimal state depends only on the previous row's states, not on the path taken to reach those states. This enables dynamic programming with state compression.
2. State Definition
Consistently use (row, column) or (row, column1, column2) to represent maximum/minimum value achievable at that position. The state space grows with the number of simultaneous selections.
3. Transition from Previous Row
All problems require computing each cell's value based on previous row values plus some penalty function:
dp[r][c] = value[r][c] + optimize(dp[r-1][c'] + penalty(c', c))
4. Handling Penalties
Problems vary in penalty types:
- Distance-based: 1937, 2304
- Constraint-based: 1289 (cannot pick same column)
- Adjacency-based: 931, 1463, 120 (can only move to adjacent positions)
- Explicit cost matrices: 2304
The optimization challenge is reducing O(n²) penalty calculations to O(n) through mathematical insight.
5. Optimization Techniques
Common optimization strategies:
- Two-pass sweep for distance penalties (left-to-right + right-to-left)
- First/second-min tracking for constraint problems
- Space optimization using 1D arrays instead of 2D matrices
- Running maximums or precomputed aggregates
These problems train you to recognize when nested loops can be eliminated through clever preprocessing.
Practice Strategy
- Start with Triangle (120) - Simplest pattern, builds foundation
- Move to Minimum Falling Path Sum (931) - Adds three-choice constraint
- Try Minimum Path Cost in a Grid (2304) - Most similar to original problem
- Challenge: Minimum Falling Path Sum II (1289) - Requires optimization insight
- Master: Cherry Pickup II (1463) - Multi-agent complexity
For each problem, ask yourself:
"What is the penalty for transitioning from position c1 in row r to position c2 in row r+1?"
Once you identify the penalty function, the DP structure becomes clear.
Summary
All five problems share the core pattern from Maximum Points with Cost:
- ✅ Sequential row processing (Markov property)
- ✅ State depends on previous row + penalty
- ✅ Optimization challenge: reduce O(n²) to O(n) per row
- ✅ Balance immediate gain vs. future flexibility
Mastering these five problems will make you proficient in matrix-based sequential DP with various penalty structures.