1 CPSC 320: Intermediate Algorithm Design and Analysis July 14, 2014

2 Course Outline Introduction and basic concepts Asymptotic notation Greedy algorithms Graph theory Amortized analysis Recursion Divide-and-conquer algorithms Randomized algorithms Dynamic programming algorithms NP-completeness

3 Maximum Flow Problem

4 Maximum Flow Assume a directed graph where each edge has a capacity Assume two nodes (source and sink) Problem: find a flow from source to sink that uses maximum possible value In any node (except source and sink) incoming flow equals outgoing flow Application: Flow of materials from a system to a client (e.g., water pipes) Flow of electrical current through a network Transportation of goods

5 Maximum Flow – Formal Definitions

6 Ford-Fulkerson Method

7 Residual Network

8 Flow augmentation

9 Augmenting path

10 Ford-Fulkerson Algorithm

11 Minimum Cut Cut: partition of the nodes of a graph such that two nodes are in different subsets Capacity of the cut: sum of the capacities of all cut edges in one direction Minimum cut: cut with minimum capacity Applications: Network reliability evaluation

12 Minimum Cut – Formal Definitions

13 Relationship between flow and cut

14 Minimum Spanning Tree

15 Spanning Tree

16 Example

17 Strategies Prim’s method: Start with a randomly selected node Pick smallest cost edge connecting visited node to a new node Add it to the tree Repeat until tree is complete Kruskal’s method: Start with all nodes isolated and no edges Pick smallest cost edge left that doesn’t create a cycle Add it to the tree Repeat until tree is complete

18 Kruskal’s Algorithm

19 Time Complexity

20 Union-Find Data Structure

21 Union-Find Data Structure

22 Array implementation

23 Tree implementation

24 Tree implementation (cont.) Path compression: When find is performed, every node in the path is updated to point to root What is the bound for find now?

25 Cost claims

26 Amortized Analysis

27 Amortized cost

28 Array Implementation of Union-Find

29 Push/Multipop Stack

30 Push/Multipop Stack

31 Potential Method (aka Accounting Method)

32 Example

33 Binary Counter

34 Union-Find with Path Compression