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Course Outline

Dive deep into data structures and algorithm analysis, including hash tables, heaps, search trees, and graph algorithms. Learn about spanning trees, MST, shortest-path algorithms, B-trees, and kD-trees. Understand the cut theorem and how "greed works" in Prim's and Kruskal's algorithms.

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Course Outline

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  1. Course Outline • Introduction and Algorithm Analysis (Ch. 2) • Hash Tables: dictionary data structure (Ch. 5) • Heaps: priority queue data structures (Ch. 6) • Balanced Search Trees: general search structures (Ch. 4.1-4.5) • Union-Find data structure (Ch. 8.1–8.5) • Graphs: Representations and basic algorithms • Topological Sort (Ch. 9.1-9.2) • Minimum spanning trees (Ch. 9.5) • Shortest-path algorithms (Ch. 9.3.2) • B-Trees: External-Memory data structures (Ch. 4.7) • kD-Trees: Multi-Dimensional data structures (Ch. 12.6) • Misc.: Streaming data, randomization

  2. Minimum spanning tree 2 2 1 2 10 4 1 3 7 2 4 6 1 8 4 5 6 1 weight of tree = 16 • Connected, undirected graph with weights on edges • Spanning tree: connected, hits all nodes, has no cycles • MST = spanning tree of minimum total edge weight

  3. Cut theorem 2 2 10 4 1 1 3 2 7 2 4 8 4 5 6 6 1 1 • For any cut in the graph, the lowest-weight edge crossing the cut is in a MST. • (If ties, each lowest-weight edge is in some MST) • “Greed works.”

  4. Prim’s algorithm 2 2 1 2 10 4 1 3 7 2 4 6 1 8 4 5 6 1 T = one vertex, no edges for i = 1 to n-1 add the cheapest edge joining T to another vertex • very similar to Dijkstra’s shortest-path algorithm • implementation: priority queue (binary heap) of edges from known to unknown vertices • time: O( |E| log |V| )

  5. Kruskal’s algorithm 2 2 1 2 10 4 1 3 7 2 4 6 1 8 4 5 6 1 for i = 1 to n-1 add the cheapest edge that doesn’t create a cycle • implementation: priority queue (binary heap) of all edges • disjoint set union to detect cycles • time: O( |E| log |V| )

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