dp-pattern-library

dp-pattern-library

A specialized skill for dynamic programming pattern recognition, matching problems to known DP patterns, generating template code, and providing optimization guidance for DP solutions.

Purpose

Assist with dynamic programming by:

• Matching problems to 50+ classic DP patterns
• Generating template code for matched patterns
• Detecting problem variants (knapsack variants, LCS variants, etc.)
• Providing state design recommendations
• Suggesting optimization techniques

Capabilities

Core Features

1. Pattern Recognition

• Analyze problem statement for DP indicators
• Match to known pattern categories
• Identify problem variants and transformations
• Suggest state representation

1. Pattern Categories

• Linear DP (1D array)
• Grid/Matrix DP (2D paths)
• String DP (LCS, edit distance)
• Interval DP (ranges, parenthesization)
• Tree DP (subtree problems)
• Bitmask DP (subset enumeration)
• Digit DP (number counting)
• Knapsack variants
• DP with state machine

1. Code Generation

• Template code for recognized patterns
• Multiple language support (Python, C++, Java)
• Comments explaining state and transitions
• Space-optimized variants

1. Optimization Guidance

• Rolling array technique
• Convex hull trick
• Divide and conquer optimization
• Monotonic queue/stack optimization
• Knuth optimization

Pattern Library

Linear DP Patterns

| Pattern | State | Transition | Example Problems |

|---------|-------|------------|------------------|

| Fibonacci | dp[i] = answer for position i | dp[i] = dp[i-1] + dp[i-2] | Climbing Stairs, House Robber |

| Min/Max Path | dp[i] = best answer ending at i | dp[i] = opt(dp[j]) + cost(j,i) | Minimum Path Sum |

| Counting | dp[i] = ways to reach state i | dp[i] = sum(dp[j]) | Unique Paths, Decode Ways |

| LIS | dp[i] = LIS ending at i | dp[i] = max(dp[j]) + 1 where j < i, a[j] < a[i] | Longest Increasing Subsequence |

String DP Patterns

| Pattern | State | Example Problems |

|---------|-------|------------------|

| Edit Distance | dp[i][j] = distance for s1[0..i], s2[0..j] | Edit Distance, One Edit Distance |

| LCS | dp[i][j] = LCS of s1[0..i], s2[0..j] | Longest Common Subsequence |

| Palindrome | dp[i][j] = is s[i..j] palindrome | Longest Palindromic Substring |

| Regex Match | dp[i][j] = s[0..i] matches p[0..j] | Regular Expression Matching |

Knapsack Patterns

| Variant | State | Transition |

|---------|-------|------------|

| 0/1 Knapsack | dp[i][w] = max value with items 0..i, capacity w | dp[i][w] = max(dp[i-1][w], dp[i-1][w-wt[i]] + val[i]) |

| Unbounded | dp[w] = max value with capacity w | dp[w] = max(dp[w], dp[w-wt[i]] + val[i]) |

| Bounded | dp[i][w] = max value with limited items | Use binary representation or deque |

| Subset Sum | dp[i][s] = can reach sum s with items 0..i | dp[i][s] = dp[i-1][s] or dp[i-1][s-a[i]] |

Grid DP Patterns

| Pattern | State | Example Problems |

|---------|-------|------------------|

| Path Count | dp[i][j] = ways to reach (i,j) | Unique Paths, Unique Paths II |

| Path Min/Max | dp[i][j] = best path to (i,j) | Minimum Path Sum |

| Multi-path | dp[i][j][k][l] = two paths simultaneously | Cherry Pickup |

Interval DP Patterns

| Pattern | State | Example Problems |

|---------|-------|------------------|

| MCM | dp[i][j] = cost for range [i,j] | Matrix Chain Multiplication |

| Burst | dp[i][j] = max coins from balloons[i..j] | Burst Balloons |

| Merge | dp[i][j] = cost to merge range [i,j] | Minimum Cost to Merge Stones |

Tree DP Patterns

| Pattern | State | Example Problems |

|---------|-------|------------------|

| Subtree | dp[v] = answer for subtree rooted at v | Binary Tree Maximum Path Sum |

| Rerooting | dp[v] = answer when v is root | Sum of Distances in Tree |

| Parent-Child | dp[v][0/1] = answer with constraint | House Robber III |

Bitmask DP Patterns

| Pattern | State | Example Problems |

|---------|-------|------------------|

| TSP | dp[mask][last] = min cost visiting mask cities ending at last | Traveling Salesman Problem |

| Assignment | dp[mask] = min cost assigning tasks to subset | Task Assignment |

| SOS | dp[mask] = sum over subsets | Subset Sum over Subsets |

Usage

Pattern Matching

# Match problem to DP pattern
dp-pattern-library match --problem "Given an array of integers, find the longest increasing subsequence"

# Output:
# Pattern: Linear DP - Longest Increasing Subsequence (LIS)
# State: dp[i] = length of LIS ending at index i
# Transition: dp[i] = max(dp[j] + 1) for all j < i where arr[j] < arr[i]
# Time: O(n^2) naive, O(n log n) with binary search
# Space: O(n)

Template Generation

# Generate template code
dp-pattern-library template --pattern "lis" --language python

# Output:
def lengthOfLIS(nums):
    if not nums:
        return 0

    n = len(nums)
    # dp[i] = length of LIS ending at index i
    dp = [1] * n

    for i in range(1, n):
        for j in range(i):
            if nums[j] < nums[i]:
                dp[i] = max(dp[i], dp[j] + 1)

    return max(dp)

Optimization Suggestions

# Get optimization recommendations
dp-pattern-library optimize --pattern "lis"

# Output:
# Current: O(n^2) time, O(n) space
# Optimizations:
# 1. Binary Search: O(n log n) time
#    - Maintain sorted list of smallest tail elements
#    - Binary search for insertion point
# 2. Segment Tree: O(n log n) time
#    - For coordinate compression + range max query

Output Schema

{
  "match": {
    "pattern": "Linear DP - LIS",
    "confidence": 0.95,
    "category": "linear",
    "variants": ["LIS", "LDS", "LNDS"]
  },
  "state": {
    "description": "dp[i] = length of LIS ending at index i",
    "dimensions": 1,
    "meaning": "LIS length ending at position i"
  },
  "transition": {
    "formula": "dp[i] = max(dp[j] + 1) for j < i, arr[j] < arr[i]",
    "baseCase": "dp[i] = 1 for all i",
    "order": "left to right"
  },
  "complexity": {
    "time": "O(n^2)",
    "space": "O(n)",
    "optimized": {
      "time": "O(n log n)",
      "technique": "binary search on patience sort"
    }
  },
  "template": {
    "python": "...",
    "cpp": "...",
    "java": "..."
  },
  "similarProblems": [
    "Longest Increasing Subsequence",
    "Number of Longest Increasing Subsequence",
    "Russian Doll Envelopes",
    "Maximum Length of Pair Chain"
  ]
}

Integration with Processes

This skill enhances:

• dp-pattern-matching - Core pattern matching workflow
• dp-state-optimization - State space optimization
• dp-transition-derivation - Deriving transitions
• leetcode-problem-solving - DP problem identification
• classic-dp-library - Building a personal DP library

Pattern Recognition Indicators

| Indicator | Likely Pattern |

|-----------|----------------|

| "maximum/minimum" + "subarray/subsequence" | Linear DP |

| "number of ways" | Counting DP |

| "can reach/achieve" | Boolean DP |

| "edit/transform string" | String DP |

| "merge/combine intervals" | Interval DP |

| "tree/subtree" | Tree DP |

| "select subset" + small n | Bitmask DP |

| "count numbers with property" | Digit DP |

| "items + capacity" | Knapsack |

References

• Dynamic Programming Patterns
• DP Visualization Tools
• LeetCode DP Patterns
• CP Algorithms - DP
• CSES DP Section

Error Handling

| Error | Cause | Resolution |

|-------|-------|------------|

| NO_PATTERN_MATCH | Problem doesn't fit known patterns | Consider greedy or other approaches |

| AMBIGUOUS_MATCH | Multiple patterns could apply | Provide more problem details |

| COMPLEX_STATE | State too complex for templates | Manual state design needed |

Best Practices

1. Start with brute force - Understand recurrence before optimizing
2. Draw state diagram - Visualize transitions
3. Verify base cases - Most DP bugs are in base cases
4. Check state uniqueness - Each state should be uniquely defined
5. Consider space optimization - Often can reduce dimension
6. Test with small inputs - Trace through by hand