algorithms (2il15) – lecture 13 wrap-up lecturetu/e algorithms (2il15) – lecture 13 15 geometric...

TU/e Algorithms (2IL15) – Lecture 13

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Algorithms (2IL15) – Lecture 13

Wrap-up lecture


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Part I of the course: Optimization Problems we want to find valid solution minimizing cost (or maximizing profit) always start by investigating structure of optimal solution backtracking: just try all solutions (pruning when possible)

− extra exercises: backtracking algorithms for NP-hard problems Vertex Cover, 3-SAT, Hamiltonian Cycle, …

greedy algorithms: make choice that seems best, recursively solve remaining

subproblem − proof is crucial; make sure you know proof structure by heart − extra exercises: greedy algorithms explained in book

dynamic programming: give recursive formula, fill in table

− make sure you know steps to design dynamic-programming algorithm − extra exercises: dynamic-programming algorithms explained in book


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Part II of the course: optimization problems on graphs single-source shortest paths

Find shortest path from source vertex to all other vertices

all-pairs shortest paths

Find shortest paths between all pairs of vertices

maximum flow

Find maximum flow from source vertex to target vertex

maximum bipartite matching

Find maximum number of disjoint pairs of vertices with edge between them

3 1

8

5

3

3 2

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2.3

− 2

2 1

4


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single-source / all-pairs shortest paths

− you should know and understand the various algorithms SSSP: Bellman-Ford, Dijkstra, algorithm on DAGs APSP: relation to matrix multiplication, Johnson’s reweighting

− extra exercises: give algorithms, give proofs of important lemmas

maximum flow + maximum bipartite matching

− definition of flow network, flow − Ford-Fulkerson method + Edmonds-Karp implementation (concept of augmenting flow, residual path, etcetera) − relation between flows and cuts (e.g. max flow = min cut) − integer capacities: there is always a max flow that is integral basis for many applications, e.g. for max bipartite matching)


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Part III of the course: miscellaneous topics

Part III of the course: Miscellaneous topics

NP-completeness

− definitions and concepts: P, NP, reductions − list of NP-complete problems you should be able to reproduce NP-completeness proofs … (they also teach you tricks for other NP-completeness proofs)

approximation algorithms − definitions and concepts: approximation factor, (F)PTAS − how to prove approximation ratio: compare to lower (upper) bound

linear programming

− definitions and concepts: LP, standard form integer LP, 0/1-LP − how to model a problem as an LP


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The exam most exercises are of the same type as the homework exercises

about 1/3 of exercises is (almost) identical to homework exercises

exam may also have some questions where the answer can be found

literally on the slides or in the book (reproduce algorithm or proof, definitions)

How to prepare 1. study the slides (only look into the book when there are things in the slides

that you do not understand)

2. solve all the homework exercises again !

3. reproduce algorithms / proofs from the book


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What we did not do in the course … … lots, in particular … algorithms for big data geometric algorithms algorithms for geographic data


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Algorithms for Big Data

disk

internal memory L3 cache

L2 cache

L1 cache

ALU

registers

Data, data, everywhere. Economist 2010


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I/O- and caching behavior analysis of running time: (asymptotically) determine number of elementary operations as a function of input size basic assumption: elementary operations take roughly same amount of time is this justified? Not when data is stored on disk:

disk operations can be factor 100,000 (or more) slower than operations on data in main memory

− how can we design I/O-efficient algorithms? − how should we analyze such algorithms? − can / should we also worry about caching ?

disk

internal memory L3 cache

L2 cache

L1 cache

ALU

registers


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streaming algorithms data stream model: data is read sequentially in one (or few) passes size of the working memory is much smaller than size of data example: traffic going through high speed routers (data can not be revisited) − how can we design algorithms for data streams?

(e.g. maintain approximate median of stream of numbers) − how should we analyze such algorithms?

take Advanced Algorithms (big data + more approximation algorithms)


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Geometric algorithms


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computer-aided design and manufacturing (CAD/CAM)

3D model of power plant (12.748.510 triangles)

motion planning

virtual walkthroughs


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3D copier

surface reconstruction

3D scanner 3D printer

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3 4 5 6 7 8 9

http://www.cyberware.com/products/msInfo.html


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many areas where geometry plays important role:

geographic information systems

computer-aided design and manufacturing

robotics and motion planning

computer graphics and virtual reality

databases

structural computational biology

and more …

30 45 age

salary

30.000

50.000

http://images.google.nl/imgres?imgurl=http://ftp.sv.vt.edu/pub/projects97/goel_narasimhan/goel_narasimhan_project/assets/images/molecule.gif&imgrefurl=http://ftp.sv.vt.edu/pub/projects97/goel_narasimhan/goel_narasimhan_project/assets/images/&h=823&w=909&sz=159&tbnid=aReC3v_zmG0J:&tbnh=132&tbnw=145&start=1&prev=/images%3Fq%3Dmolecule%26hl%3Dnl%26lr%3D%26sa%3DG


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Geometric data structures

store geometric data in 2-, 3- or higher dimensional space such that certain queries can be answered efficiently applications: GIS, graphics and virtual reality, … Examples:

range searching

nearest-neighbor searching

point location

want to know more about algorithms and data structures for spatial data?

take Geometric Algorithms


Algorithms for Geographic Data


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geographic information systems (GIS)

Mississippi delta

Memphis Memphis

Memphis Memphis

Arlington Arlington

30 cities:

430 = 1018 possibilities


automated cartography examples: map labeling scaling and simplification

schematization

maps for various types of data, e.g., flows

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trajectory data data of animals, humans, vehicles, … Examples of analysis tasks: How similar are two trajectories? How to find clusters? How to represent a cluster? How to simplify a trajectory? … take Algorithms for Geographic Data


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THE END

well, except for the exam …

comments / suggestions for improvements are welcome

algorithms (2il15) – lecture 13 wrap-up lecturetu/e algorithms (2il15) – lecture 13 15 geometric...

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