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Below is a quick‑reference guide that expands on each of the topics you listed. Think of it as an "at‑a‑glance" cheat sheet you can flip to the right section when you need a refresher or a starting point for deeper study.

| # | Topic | What It Is / Why It Matters | Key Points / Quick Tips |
|---|-------|-----------------------------|------------------------|
| 1 | **Data Structures** | The way data is organized and stored in memory (arrays, linked lists, trees, graphs, hash tables, etc.). | • Choose based on access patterns.
• Consider time/space trade‑offs. |
| 2 | **Algorithms** | Step‑by‑step procedures that solve problems (searching, sorting, graph traversal, DP). | • Measure *time* and *space* complexity (Big‑O).
• Use algorithm design paradigms: divide & conquer, greedy, DP. |
| 3 | **Sorting Algorithms** | Efficiently order elements: quicksort, mergesort, heapsort, radix sort, etc. | • Quicksort average O(n log n); mergesort worst‑case O(n log n).
• Stability matters for some applications. |
| 4 | **Search Algorithms** | Finding an element in data structures: binary search (sorted), linear search. | • Binary search O(log n) on sorted arrays; hash tables give expected O(1). |
| 5 | **Dynamic Programming (DP)** | Optimal sub‑structure + overlapping subproblems → bottom‑up or memoization. | Example: Fibonacci numbers in O(n); Knapsack, edit distance, LCS. |
| 6 | **Greedy Algorithms** | Make a locally optimal choice at each step; works when the greedy choice leads to a global optimum. | Examples: Huffman coding, interval scheduling (activity selection), Dijkstra’s shortest path. |
| 7 | **Graph Algorithms** | Represent relationships and perform searches or optimizations on networks. |
| | • Breadth‑first search (BFS) – level‑order traversal, shortest path in unweighted graphs. |
| | • Depth‑first search (DFS) – pre/post‑order, connected components, topological sorting. |
| | • Dijkstra’s algorithm – single‑source shortest paths with non‑negative weights. |
| | • Floyd‑Warshall – all‑pairs shortest paths in dense graphs. |
| | • Minimum spanning tree: Kruskal’s (edge‑sorted) and Prim’s (vertex‑grown). |
| | • Max‑flow/min‑cut: Ford–Fulkerson, Edmonds–Karp, Dinic’s algorithm. |
| | • Bipartite matching: Hungarian algorithm for assignment; Hopcroft–Karp for
maximum cardinality matching. |
| **Graph representation** | Adjacency list (vector\) for sparse graphs; adjacency matrix or
bitset\ for dense graphs or when frequent edge queries are needed.
Edge‑struct vector for weighted graphs. |
| **Key properties to exploit** | *Planarity*: use planar embedding and bounded degree to apply linear‑time
planar algorithms.
*Tree‑width / tree‑decomposition*: dynamic programming on the tree‑decomposition
for problems that are NP‑hard in general but tractable for bounded tree‑width.
*Degree constraints*: small maximum degree allows branching factor reduction,
use of adjacency bitsets, or specialized matching/flow algorithms. |
| **Typical algorithmic strategies** | 1. *Preprocessing* – compute connected components, bridges,
biconnected components (Tarjan), tree‑decomposition, planar embedding.
2. *Reduction / Simplification* – contract series/parallel edges, apply kernel
size bounds for specific problems (e.g., Vertex Cover kernel on bounded degree).
3. *Exact algorithms with branching* – e.g., FPT algorithm for Feedback Vertex Set
on graphs of maximum degree 3 runs in O(1.47^k) time; use measure‑and‑conquer.
4. *Approximation / Heuristics* – greedy set cover, local search on bounded degree
graphs gives better guarantees (e.g., 2‑approx for Vertex Cover).
5. *Use of combinatorial structures* – e.g., matchings in bipartite graphs,
network flows to solve assignment problems efficiently due to the
structural constraints imposed by the graph’s degree bound.

In all cases, recognizing that a problem instance lives on a graph with bounded
degree allows us to apply specialized algorithms or tighter analysis. The
presence of such structural restrictions often leads to improved time
complexities and approximation ratios that would otherwise be unattainable in
general graphs.

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