Cross-Reference Guide — Pattern → Implementation Map

Purpose: Master lookup for mapping problem descriptions, patterns, and real-world scenarios to specific implementations in this repo. Start here when facing a new problem.

Section 1: Problem Pattern → Implementation Map

Arrays & Hashing

When You See...	Use This Pattern	Implementation	Complexity
"Find two values that sum to X"	Hash map lookup	`src/algo/arrays/two_sum.py`	O(n)
"Group items by property"	Hash map grouping	`src/algo/arrays/group_anagrams.py`	O(n*k log k)
"Top K / most frequent"	Bucket sort or heap	`src/algo/arrays/top_k_frequent.py`	O(n)
"Product without self"	Prefix/suffix arrays	`src/algo/arrays/product_except_self.py`	O(n)

Two Pointers

When You See...	Use This Pattern	Implementation	Complexity
"Three values sum to zero"	Sort + two pointers	`src/algo/two_pointers/three_sum.py`	O(n^2)
"Maximum area / container"	Two pointers converging	`src/algo/two_pointers/container_with_most_water.py`	O(n)
"Water trapped between bars"	Two pointers + max tracking	`src/algo/two_pointers/trapping_rain_water.py`	O(n)

Sliding Window

When You See...	Use This Pattern	Implementation	Complexity
"Minimum window containing chars"	Variable sliding window	`src/algo/sliding_window/min_window_substring.py`	O(n)
"Longest substring without repeat"	Sliding window + hash	`src/algo/sliding_window/longest_substring_no_repeat.py`	O(n)

Stacks & Queues

When You See...	Use This Pattern	Implementation	Complexity
"Valid brackets/parentheses"	Stack matching	`src/algo/stacks_queues/valid_parentheses.py`	O(n)
"Min/max in O(1)"	Auxiliary stack	`src/algo/stacks_queues/min_stack.py`	O(1)
"Next greater/warmer element"	Monotonic stack	`src/algo/stacks_queues/daily_temperatures.py`	O(n)

Binary Search

When You See...	Use This Pattern	Implementation	Complexity
"Find target in sorted array"	Binary search	`src/algo/searching/binary_search.py`	O(log n)
"Search in rotated array"	Modified binary search	`src/algo/searching/search_rotated_array.py`	O(log n)
"Find min in rotated array"	Binary search variant	`src/algo/searching/find_minimum_rotated.py`	O(log n)

Linked Lists

When You See...	Use This Pattern	Implementation	Complexity
"Reverse a linked list"	Pointer manipulation	`src/algo/linked_lists/reverse_linked_list.py`	O(n)
"Merge sorted lists"	Two-pointer merge	`src/algo/linked_lists/merge_two_sorted.py`	O(n+m)
"Cache with eviction"	Hash map + doubly linked list	`src/algo/linked_lists/lru_cache.py`	O(1)

Trees

When You See...	Use This Pattern	Implementation	Complexity
"Tree depth / height"	DFS recursion	`src/algo/trees/max_depth.py`	O(n)
"Mirror / flip tree"	Recursive swap	`src/algo/trees/invert_tree.py`	O(n)
"Is it a valid BST?"	DFS with bounds	`src/algo/trees/validate_bst.py`	O(n)
"Level-by-level traversal"	BFS with deque	`src/algo/trees/level_order_traversal.py`	O(n)

Graphs

When You See...	Use This Pattern	Implementation	Complexity
"Connected components / islands"	BFS/DFS flood fill	`src/algo/graphs/number_of_islands.py`	O(m*n)
"Deep copy a graph"	BFS/DFS + visited map	`src/algo/graphs/clone_graph.py`	O(V+E)
"Dependency ordering"	Topological sort (Kahn's)	`src/algo/graphs/topological_sort.py`	O(V+E)
"Can finish all tasks? (cycles)"	Topological sort / DFS	`src/algo/graphs/course_schedule.py`	O(V+E)
"Shortest path (positive weights)"	Dijkstra's algorithm	`src/algo/graphs/dijkstra.py`	O((V+E) log V)
"Signal delay to all nodes"	Dijkstra variant	`src/algo/graphs/network_delay_time.py`	O((V+E) log V)
"Shortest transformation chain"	BFS level-by-level	`src/algo/graphs/word_ladder.py`	O(n*m^2)
"Grid pathfinding with heuristic"	A* search	`src/algo/graphs/a_star_search.py`	O(E log V)
"Shortest path (negative weights)"	Bellman-Ford	`src/algo/graphs/bellman_ford.py`	O(V*E)
"Minimum cost to connect all"	Kruskal's / Prim's	`src/algo/graphs/minimum_spanning_tree.py`	O(E log E)
"Maximum flow through network"	Edmonds-Karp	`src/algo/graphs/network_flow.py`	O(V*E^2)
"Spatial proximity queries"	Geohash encoding	`src/algo/graphs/geohash_grid.py`	O(precision)
"Nearest neighbor in k dimensions"	KD-tree	`src/algo/graphs/kd_tree.py`	O(log n) avg

Dynamic Programming

When You See...	Use This Pattern	Implementation	Complexity
"Count ways to reach target"	DP (Fibonacci-like)	`src/algo/dp/climbing_stairs.py`	O(n)
"Minimum cost to reach amount"	DP (bottom-up)	`src/algo/dp/coin_change.py`	O(amount*coins)
"Longest increasing subsequence"	DP + binary search	`src/algo/dp/longest_increasing_subseq.py`	O(n log n)
"String edit distance"	2D DP	`src/algo/dp/edit_distance.py`	O(m*n)
"0/1 Knapsack"	DP (1D optimized)	`src/algo/dp/knapsack.py`	O(n*W)
"Longest common subsequence"	2D DP	`src/algo/dp/longest_common_subseq.py`	O(m*n)
"Shortest route visiting all cities"	Bitmask DP (TSP)	`src/algo/dp/traveling_salesman_dp.py`	O(n^2 * 2^n)
"Constraint satisfaction (Sudoku)"	Backtracking + propagation	`src/algo/dp/constraint_satisfaction.py`	varies

Heaps

When You See...	Use This Pattern	Implementation	Complexity
"Kth largest element"	Min-heap of size k	`src/algo/heaps/kth_largest.py`	O(n log k)
"Merge k sorted lists"	Heap merge	`src/algo/heaps/merge_k_sorted_lists.py`	O(n log k)
"Task scheduling with cooldown"	Max-heap + queue	`src/algo/heaps/task_scheduler.py`	O(n)

Backtracking

When You See...	Use This Pattern	Implementation	Complexity
"Generate all subsets"	Backtracking	`src/algo/backtracking/subsets.py`	O(n * 2^n)
"Generate all permutations"	Backtracking	`src/algo/backtracking/permutations.py`	O(n * n!)
"Combinations summing to target"	Backtracking (w/ reuse)	`src/algo/backtracking/combination_sum.py`	O(2^target)
"N queens placement"	Backtracking + pruning	`src/algo/backtracking/n_queens.py`	O(n!)

Greedy

When You See...	Use This Pattern	Implementation	Complexity
"Merge overlapping intervals"	Sort + merge	`src/algo/greedy/merge_intervals.py`	O(n log n)
"Can reach end? Min jumps?"	Greedy max-reach	`src/algo/greedy/jump_game.py`	O(n)
"Max non-overlapping intervals"	Greedy by end time	`src/algo/greedy/interval_scheduling.py`	O(n log n)

Bit Manipulation

When You See...	Use This Pattern	Implementation	Complexity
"Find unique element (others x2)"	XOR all	`src/algo/bit_manipulation/single_number.py`	O(n)
"Count set bits 0..n"	DP: dp[i] = dp[i>>1] + (i&1)	`src/algo/bit_manipulation/counting_bits.py`	O(n)
"Reverse bits of integer"	Bit-by-bit or D&C	`src/algo/bit_manipulation/reverse_bits.py`	O(1)

Sorting

When You See...	Use This Pattern	Implementation	Complexity
"Find kth smallest"	Quickselect	`src/algo/sorting/quickselect.py`	O(n) avg
"Count inversions"	Merge sort variant	`src/algo/sorting/merge_sort_inversions.py`	O(n log n)

Section 2: Data Structure Selection Guide

Need	Use	Why	Example Implementations
O(1) lookup by key	`dict` / hash map	Hash-based amortized O(1)	`two_sum.py`, `group_anagrams.py`
O(1) membership test	`set`	Hash-based	`number_of_islands.py` (visited set)
Ordered iteration	sorted `list`	Timsort O(n log n)	`three_sum.py` (sort first)
FIFO queue	`collections.deque`	O(1) appendleft/popleft	BFS in `level_order_traversal.py`, `word_ladder.py`
Priority queue	`heapq`	O(log n) push/pop	`dijkstra.py`, `kth_largest.py`, `merge_k_sorted_lists.py`
Stack (LIFO)	`list` with append/pop	O(1) amortized	`valid_parentheses.py`, `daily_temperatures.py`
Sorted container	`bisect` module	O(log n) search	`longest_increasing_subseq.py`, `hypothesis_patterns.py`
Disjoint sets	Union-Find class	~O(alpha(n)) amortized	`minimum_spanning_tree.py`
Spatial index	KD-tree / geohash	O(log n) nearest neighbor	`kd_tree.py`, `geohash_grid.py`
Cache with eviction	OrderedDict / DLL+dict	O(1) get, put, evict	`lru_cache.py`

When to pick what

Need key -> value?          --> dict
Need "is X in the set?"     --> set
Need min/max repeatedly?    --> heapq
Need FIFO?                  --> collections.deque
Need LIFO?                  --> list (append/pop)
Need sorted insert + search --> bisect / SortedList
Need merge/find groups?     --> Union-Find
Need nearest in 2D/3D?      --> KD-tree or geohash

Section 3: Concept Modules -- When to Study Each

Concept Module	Study When...	Key Functions / Classes	Cross-References
`t_strings.py`	Learning Python 3.14 features, preventing injection	`sql_safe()`, `html_safe()`, `structured_log()`, `render()`	PEP 750, ref sheet 09 (Python 3.14)
`advanced_typing.py`	Writing typed Python, understanding Protocol vs ABC	`Drawable` Protocol, `Stack[T]`, `Container[T]`, `@overload`, `TypeGuard`	PEP 544/612/647/695/696, ref sheet 09
`hypothesis_patterns.py`	Learning property-based testing, finding edge cases	`SortedList`, `BoundedCounter`	Hypothesis docs, ref sheet 09, `tests/concepts/test_hypothesis_patterns.py`
`fft_dct.py`	Signal processing, frequency analysis, sensor data	`compute_fft()`, `filter_signal()`, `compute_dct()`, `spectral_analysis()`	Radar, telemetry, radio, and other sensor streams
`modern_flask.py`	Web API patterns, app factory, testing	`create_app()`, `api_bp` Blueprint, `ValidationError`	System design round, API questions
`validation.py`	Data validation, API boundaries, Pydantic v2 vs Zod	`User` model, `Address`, `Shape` discriminated union, `serialize_user()`	Practical problem solving round

Concept -> Algorithm connections

t_strings demonstrate the template pattern -- same concept as parameterized queries in sql_safe(), analogous to how DP builds solutions from templates
advanced_typing shows Stack[T] -- the same LIFO structure used in valid_parentheses.py and daily_temperatures.py
hypothesis_patterns uses bisect -- the same binary search strategy in longest_increasing_subseq.py
fft_dct is pure signal processing -- relevant for sensor data pipelines
modern_flask is the API layer -- pairs with validation.py for full request lifecycle
validation with Pydantic is the runtime counterpart to advanced_typing's static type system

Section 4: Quick Decision Tree

START: What does the problem ask for?
|
+-- "Find / search for something"
|   |
|   +-- Input is sorted?
|   |   +-- YES --> Binary search (searching/binary_search.py)
|   |   |           Rotated? --> search_rotated_array.py, find_minimum_rotated.py
|   |   +-- NO  --> Hash map O(n) lookup (arrays/two_sum.py pattern)
|   |
|   +-- Find in a graph?
|       +-- Unweighted shortest path --> BFS (graphs/number_of_islands.py pattern)
|       +-- Weighted, positive       --> Dijkstra (graphs/dijkstra.py)
|       +-- Weighted, negative       --> Bellman-Ford (graphs/bellman_ford.py)
|       +-- With heuristic           --> A* (graphs/a_star_search.py)
|
+-- "Optimize a value" (min cost, max profit, count ways)
|   |
|   +-- Overlapping subproblems?
|   |   +-- YES --> Dynamic programming (dp/)
|   |   |          1D: climbing_stairs.py, coin_change.py
|   |   |          2D: edit_distance.py, longest_common_subseq.py
|   |   |          Bitmask: traveling_salesman_dp.py
|   |   +-- NO  --> Greedy (prove greedy choice property)
|   |               merge_intervals.py, jump_game.py, interval_scheduling.py
|   |
|   +-- Need top-K or priority ordering?
|       +-- Kth element     --> Heap (heaps/kth_largest.py) or Quickselect (sorting/quickselect.py)
|       +-- Merge K streams --> Heap merge (heaps/merge_k_sorted_lists.py)
|
+-- "Generate all / enumerate"
|   |
|   +-- All subsets          --> Backtracking (backtracking/subsets.py)
|   +-- All permutations     --> Backtracking (backtracking/permutations.py)
|   +-- Combinations to sum  --> Backtracking with reuse (backtracking/combination_sum.py)
|   +-- Placement with rules --> Backtracking + pruning (backtracking/n_queens.py)
|
+-- "Process a sequence" (subarray, substring)
|   |
|   +-- Contiguous subarray/substring?
|   |   +-- YES --> Sliding window (sliding_window/)
|   |   |          Fixed size:    pattern in patterns/sliding_window.py
|   |   |          Variable size: min_window_substring.py, longest_substring_no_repeat.py
|   |   +-- NO  --> Subsequence: DP (dp/longest_increasing_subseq.py, longest_common_subseq.py)
|   |
|   +-- Next greater/smaller element? --> Monotonic stack (stacks_queues/daily_temperatures.py)
|   +-- Valid nesting?                --> Stack (stacks_queues/valid_parentheses.py)
|
+-- "Graph structure" (explicit or implicit)
|   |
|   +-- Connected components  --> DFS/BFS (graphs/number_of_islands.py) or Union-Find
|   +-- Dependency ordering   --> Topological sort (graphs/topological_sort.py)
|   +-- Cycle detection       --> Course schedule pattern (graphs/course_schedule.py)
|   +-- Clone / copy          --> BFS/DFS + visited map (graphs/clone_graph.py)
|   +-- Minimum spanning tree --> Kruskal's (graphs/minimum_spanning_tree.py)
|   +-- Maximum flow          --> Edmonds-Karp (graphs/network_flow.py)
|
+-- "Design a data structure"
    |
    +-- O(1) access + eviction --> LRU Cache (linked_lists/lru_cache.py)
    +-- O(1) min/max retrieval --> Min Stack (stacks_queues/min_stack.py)
    +-- Sorted insert + search --> SortedList with bisect (concepts/hypothesis_patterns.py)
    +-- Generic typed container -> Stack[T] (concepts/advanced_typing.py)

Section 5: Complexity Quick-Reference by Category

Use this to sanity-check your solution's complexity during an interview.

O(1) solutions

Problem	Implementation
Min stack operations	`stacks_queues/min_stack.py`
LRU cache get/put	`linked_lists/lru_cache.py`
Reverse bits	`bit_manipulation/reverse_bits.py`

O(n) solutions

Problem	Implementation
Two sum	`arrays/two_sum.py`
Product except self	`arrays/product_except_self.py`
Top K frequent	`arrays/top_k_frequent.py`
Container with most water	`two_pointers/container_with_most_water.py`
Trapping rain water	`two_pointers/trapping_rain_water.py`
Min window substring	`sliding_window/min_window_substring.py`
Longest substring no repeat	`sliding_window/longest_substring_no_repeat.py`
Valid parentheses	`stacks_queues/valid_parentheses.py`
Daily temperatures	`stacks_queues/daily_temperatures.py`
Reverse linked list	`linked_lists/reverse_linked_list.py`
Max depth of tree	`trees/max_depth.py`
Invert tree	`trees/invert_tree.py`
Validate BST	`trees/validate_bst.py`
Level order traversal	`trees/level_order_traversal.py`
Climbing stairs	`dp/climbing_stairs.py`
Single number	`bit_manipulation/single_number.py`
Counting bits	`bit_manipulation/counting_bits.py`
Task scheduler	`heaps/task_scheduler.py`
Jump game	`greedy/jump_game.py`
Quickselect (average)	`sorting/quickselect.py`

O(n log n) solutions

Problem	Implementation
Three sum	`two_pointers/three_sum.py` (sort dominates, then O(n^2) scan)
Longest increasing subseq	`dp/longest_increasing_subseq.py`
Merge intervals	`greedy/merge_intervals.py`
Interval scheduling	`greedy/interval_scheduling.py`
Merge sort inversions	`sorting/merge_sort_inversions.py`

O(log n) solutions

Problem	Implementation
Binary search	`searching/binary_search.py`
Search rotated array	`searching/search_rotated_array.py`
Find minimum rotated	`searching/find_minimum_rotated.py`

O(V+E) graph solutions

Problem	Implementation
Clone graph	`graphs/clone_graph.py`
Topological sort	`graphs/topological_sort.py`
Course schedule	`graphs/course_schedule.py`

O((V+E) log V) graph solutions

Problem	Implementation
Dijkstra	`graphs/dijkstra.py`
Network delay time	`graphs/network_delay_time.py`
A* search	`graphs/a_star_search.py`

Exponential solutions (backtracking)

Problem	Implementation
Subsets O(n * 2^n)	`backtracking/subsets.py`
Permutations O(n * n!)	`backtracking/permutations.py`
Combination sum O(2^target)	`backtracking/combination_sum.py`
N queens O(n!)	`backtracking/n_queens.py`
TSP O(n^2 * 2^n)	`dp/traveling_salesman_dp.py`

Section 6: Pattern Recognition Cheat Sheet

When you read a problem statement, look for these keywords and jump to the right section.

Keyword in Problem	First Thing to Try	If That Fails
"sorted"	Binary search, two pointers	--
"contiguous subarray"	Sliding window	Prefix sums, Kadane's
"substring"	Sliding window + hash map	DP
"parentheses" / "brackets"	Stack	--
"next greater" / "next warmer"	Monotonic stack	--
"shortest path"	BFS (unweighted), Dijkstra (weighted)	Bellman-Ford (negative), A* (heuristic)
"connected" / "island" / "region"	DFS/BFS flood fill, Union-Find	--
"dependency" / "prerequisite"	Topological sort	--
"cycle"	DFS coloring (directed), Union-Find (undirected)	--
"minimum cost" / "count ways"	Dynamic programming	--
"all subsets" / "all combinations"	Backtracking	Bitmask enumeration
"merge" / "overlapping intervals"	Sort + greedy sweep	--
"kth largest" / "top k"	Heap of size k	Quickselect
"design" / "implement"	Choose data structures, define API	--
"cache" / "eviction"	LRU = hash map + doubly linked list	--
"stream" / "online"	Heap, sliding window	--
"frequency" / "how many times"	Hash map / Counter	--
"edit distance" / "transform"	2D DP	BFS (word_ladder)
"knapsack" / "subset sum"	DP (1D or 2D)	--
"schedule tasks"	Heap + greedy	Topological sort if deps
"maximum flow"	Edmonds-Karp / Ford-Fulkerson	--
"nearest point" / "proximity"	KD-tree, geohash	--
"visit all cities/nodes"	TSP bitmask DP	--
"XOR" / "single unique"	Bit manipulation	--

Section 7: Study Order Recommendations

If you have 2 hours (crunch mode)

Skim this decision tree (Section 4) -- internalize the branching
Review arrays/two_sum.py (hash map pattern)
Review sliding_window/min_window_substring.py (window pattern)
Review dp/coin_change.py (DP template)
Review graphs/dijkstra.py (graph traversal)
Review backtracking/subsets.py (enumeration template)

If you have 1 day

Add to the above:

All of Section 1 (scan for recognition, don't memorize)
trees/validate_bst.py + level_order_traversal.py
greedy/merge_intervals.py
linked_lists/lru_cache.py
concepts/modern_flask.py + concepts/validation.py

If you have 1 week

Work through every implementation in order:

Arrays & hashing (day 1)
Two pointers + sliding window (day 2)
Stacks, searching, linked lists (day 3)
Trees + graphs (day 4)
DP + backtracking (day 5)
Heaps, greedy, bit manipulation, sorting (day 6)
Concept modules (day 7)

Cross-Reference to Other Sheets

Sheet	Use it for...
`01-python-stdlib.md`	Python built-in functions, data types, standard library
`02-data-structures.md`	Detailed data structure theory and operations
`03-algorithm-templates.md`	Pseudocode templates for each algorithm family
`04-big-o-complexity.md`	Complexity analysis, input size -> acceptable complexity
`05-common-patterns.md`	Problem signal -> pattern mapping (compact version)
`06-system-design.md`	System design interview framework
`07-interview-day-guide.md`	Day-of logistics, communication framework, timing
08-cross-reference-guide.md	You are here -- the master lookup
`09-python-314-and-modern-patterns.md`	Python 3.14 features, t-strings, typing, Hypothesis