ST: Introduction to Graph Algorithms This Class Website and Contact - PowerPoint PPT Presentation

ST: Introduction to Graph Algorithms

This Class

Website and Contact Website ◮ www.cs.kent.edu/ ∼ aleitert/iga/ ◮ Important information ◮ Slides (maybe ;) ◮ Announcements Email ◮ aleitert@cs.kent.edu Office hours ◮ Monday and Wednesday, 1.00 – 2.00 p.m. Room 352, Math and CS Building ◮ Or appointment via email 3 / 26

Requirements Completely new class from scratch! ◮ Experimental. ◮ No textbook. ◮ Feedback is highly appreciated. Requirements ◮ Quizzes 50 % ◮ Project 50 % 4 / 26

Quizzes Questions ◮ 5 questions, each 20 % ◮ Mostly given in advance ◮ Usual case: Given a graph, run a certain algorithm on it. Dates (may change) ◮ March 2. during class ◮ April 6. during class ◮ May 10. Wednesday, 12:45 – 3:00 p. m. (“Final”) 5 / 26

Project Selection ◮ Pick 1 out of 4. ◮ 2 theory, 2 coding projects (more details next week) Groups ◮ Up to 4 students per group. (Some exceptions are allowed) ◮ One grade for the whole group. ◮ I don’t care about your group drama. ◮ Report groups until Feb 3. 6 / 26

Project-Reports Report progress during the semester ◮ 2 small reports Feb 24. and Apr 6. ◮ 1 larger reports Mar 16. Small Reports (1 or 2 sentences per question) ◮ What have you done so far? ◮ What do you plan to do next? ◮ Any blockers? Larger Report ( ≈ 1 page) ◮ What is the status of your project? ◮ What is you approach for solving it? 7 / 26

Alternative to Project Give a presentation ◮ ≥ 20 minutes ◮ 1 or 2 students per talk ◮ Only if you can convince me in advance that you give a good presentation! Topic ideas ◮ Shortest path beyond Dijkstra ◮ RT + Walker algorithms for drawing trees ◮ Reconfiguration problems ◮ Dominating Set and variants 8 / 26

Not in this Class but Useful to Know

Array Based Lists Add( e ) (amortised) O ( 1 ) ◮ Adds an element e to the end. AddAt( e , i ) O ( n ) ◮ Adds an element e at index i . Other elements are shifted. Get( i ) / Set( i , e ) O ( 1 ) ◮ Reads or overrides the element at index i . Find( e ) O ( n ) ◮ Finds the first element equal to e . 10 / 26

Trees 11 / 26

Priority Queues Enqueue O ( log n ) ◮ Adds an element to the queue. Dequeue O ( log n ) ◮ Removes the smallest element in the queue. Min O ( 1 ) ◮ Return the smallest element in the queue without removing it. Dequeue Enqueue 5 2 4 1 3 Min 12 / 26

Sorting Array based ◮ In O ( n log n ) time. ◮ Requires O ( log n ) additional space. ◮ Not stable. 2 a 6 a 6 b 2 b 5 4 3 1 Input 2 b 2 a 6 a 6 b 1 3 4 5 Output 13 / 26

Hash Tables Insert( k , v ) O ( 1 ) ◮ Inserts a key-value pair ( k , v ) . Delete( k ) O ( 1 ) ◮ Deletes a value with the given key k . Find( k ) O ( 1 ) ◮ Finds a value with the given key k . Note: Elements cannot be sorted within the table. 14 / 26

Graphs

Graph Graph A graph G = ( V , E ) is a set V of vertices connected by an edge set E . 16 / 26

Variations Multi-Graph: Multiple edges between two vertices. Directed: Edges have a direction. Weighted: Vertices and/or edges have weights. Simple: No multiple edges, no loops. Simple Undirected Graph A simple undirected graph G = ( V , E ) is a set V of vertices connected � � by an edge set E ⊆ { u , v } | u , v ∈ V , u � = v . An edge { u , v } is also written as uv . (More definitions later.) 17 / 26

Implementation: Adjacency List For each vertex, there is an array storing pointers † to all neighbours. 0 1 3 4 1 1 2 5 2 5 2 1 3 0 3 0 2 4 3 4 4 0 3 5 G 5 1 4 † Usually, the vertex index is sufficient. 18 / 26

Example Problem

Example Problem Let G = ( V , E , ω ) be an undirected weighted graph and let A ⊂ V be a non-empty proper subset of V (i. e., A � = V ). The separation sep ( A ) is defined as sep ( A ) = min { ω ( ab ) | a ∈ A , b ∈ V \ A } . Given a graph G , find a subset A with maximum possible separation. 20 / 26

Understanding the Problem Separation Let A ⊂ V be a non-empty. The separation sep ( A ) is defined as sep ( A ) = min { ω ( a , b ) | a ∈ A , b ∈ V \ A } . 5 7 A 3 G Problem: Find the set A for which sep ( A ) is maximal. 21 / 26

Finding a First Solution Naive approach ◮ Test all subsets A ◮ Problem: Too many subsets. Observation ◮ A solution is defined by an edge. 22 / 26

Finding a First Solution Better approach ◮ For each edge e , check if there is a set A with sep ( A ) = ω ( e ) . ◮ How do we check this? Observation ◮ If sep ( A ) = ω ( e ) , then, for each edge uv with ω ( uv ) < ω ( e ) , u ∈ A if and only if v ∈ A , i. e., both are in A or bot are not in A . Lemma There is a set A with sep ( A ) = ω ( uv ) if and only if there is no path from u to v where each edge has a lower weight than uv . 23 / 26

Finding a First Solution Algorithm idea ◮ For each edge uv , check if there is path from u to v only using edges with less weight than uv . ◮ If there is no such path, store uv as potential solution. ◮ Out of all these edges uv , pick the one with the largest weight. Runtime ◮ O ( | E | 2 ) This is an acceptable solution. However, can we do better? 24 / 26

Improve a Solution A defines a partition of the graph and sep ( A ) is represented by the smallest edge connecting bots sets. What else do we know about this edge? 25 / 26

Improve a Solution A defines a partition of the graph and sep ( A ) is represented by the smallest edge connecting bots sets. What else do we know about this edge? Lemma If e is in { ab | a ∈ A , b ∈ V \ A } and ω ( e ) = sep ( A ) , then there is a minimum spanning tree containing e . This gives us a new algorithm: ◮ Compute a minimum spanning tree T . ◮ Find the edge e of T with the largest weight. ◮ Runtime (using Prim’s algorithm): O ( | E | log | V | ) Question: Do we really need a MST? 25 / 26

Improve a Solution Minimum Spanning Tree ◮ The tree with the smallest sum of edges. Observation ◮ We only want the largest edge e of an MST. ◮ If we remove all edges e ′ from G with ω ( e ) ≤ ω ( e ′ ) , G is disconnected. Can we find e without finding the MST first? 26 / 26

Improve a Solution Minimum Spanning Tree ◮ The tree with the smallest sum of edges. Observation ◮ We only want the largest edge e of an MST. ◮ If we remove all edges e ′ from G with ω ( e ) ≤ ω ( e ′ ) , G is disconnected. Can we find e without finding the MST first? ◮ Related problem: Minimum Bottleneck Spanning Tree ◮ Can be implemented in (expected) linear time. ◮ Uses quick-select algorithm as basic strategy. ◮ Not easy to implement correctly. 26 / 26