Algorithms and Data Structures Lesson 4

Transcription

1 Algorithms and Data Structures Lesson 4 Michael Schwarzkopf weimar.de/de/medien/professuren/medieninformatik/grafische datenverarbeitung Bauhaus University Weimar June 13, 2018

2 Overview...of things you should definetely know about if you want a very good grade Depth search (Stack) Breadth search (Queue) Spanning tree Prim Kruskal Shortest Path (Dijkstra) Flux in a Network Sorting based Mathematical... Range searching Lecture Devide and Conquer Closest Points Multiplication of Polynomials Fast Fourier Transform

3 Depth search using a Stack Recursive: Take a Vertex, output it, mark as visited. Look in Adjacency list and do it again recursively for all non visited neighbours. Gives us: A, F, C, B, D, E

4 Depth search using a Stack Using Stack: Take a Vertex & push in Stack, mark as visited. While there are Elements in the Stack: If a neighbour of Vertex on the Stack is not visited: put neighbour in the Stack. else (if all neighbours are visited): pop Vertex and output it.

5 Depth search using a Stack Until here, every Vertex we visited had at least one unvisited neighbour. So we just pushed Now look what happenes if we visit D now

6 Depth search using a Stack Those who have no unvisited neighbour get popped out: D, B, C. F However has still E as neighbour. so we push it

7 Depth search using a Stack After looking at E, every vertex was visited once so we pop until the Stack is empty D, B, C, E, F, A

8 Breath search using a Queue Similar: Enqueue Startvertex, read lowest entry in queue. If it has unvisited neighbour: enqueue them, mark them as visited if not: dequeue. Here: A

9 Breath search using a Queue F has unvisited neighbours, so enqueue them After that everything is marked as visited. Empty the queue: A, F, B, C, D, E

10 Spanning Tree Subset of edges of G, which has no circles in it connects all vertices sum of weights is Smallest possible. media/file:minimum_spanning_tree.svg Prim: similar to depth search Kruskal: Take all small edges in raising order without creating a circle Let s do in detail!

11 Prim Algorithm Take a vertex, mark as visited, while not all vertices are visited: for all visited vertices: check all connected edges if smallest of them does not form a circle: add to min. spantree. mark adjacent vertex as visited else: exclude edge from min span tree Well explained in lecture slides. (22 35)

12 Prim Complexity Each vertex has to be visited: O( V ) Each edge has eighter to be removed from or added to the min. TreeO( E ) O( V + E ) In most efficient implementation, finding smallest edge takes O(log V ) time, so Prim runs in O(( V + E ) * log V )

13 Kruskal Algorithm Sort Edges from smallest to biggest while graph is not connected and not all vertics are in min tree: if smallest unchecked egde doesn t form circle: add to min. spantree else: remove edge Sorting: O( E log E ) Adding & excluding circles: negligible small. Kruskal runs in O( E log E )

14 Dijkstra Shortest path in a graph from one certain vertex to another (e.g. A to F) Model a World map as a graph, do Dijkstra and you have a Navigation system!

15 Dijkstra Algorithm Start Vertex gets value 0, mark as visited while not all vertices are visited: for all visited vertices: all reachable vertices get minimal sum of values of edges to reach them Mark edge to vertex with smallest value Mark this vertex as visited This gives us shortest paths from start to EACH vertex. So we can just choose this from start to target node.

16 Dijkstra Algorithm Starting at S, which value is 0; connected vertices get 0 + weight of edge as value.

17 Dijkstra Algorithm As 1 is smaller than 7, we choose the edge in between S and the Node with 1. Next: Value of each neighbour is value + weight

18 Dijkstra Algorithm 1+2=3;1+5=6 3 is smaller than 6 and 7 Connect the path

19 Dijkstra Algorithm We can reach z in 12 steps or less. But first, we choose another one as 4 is smaller than 6, 7 and 12

20 Dijkstra Algorithm is 9, as 6 and 7 are smaller, we connect them first. Pay attention to what happenes now:

21 Dijkstra Algorithm As is smaller than 9, the Vertex gets a lower value now.

22 Dijkstra Algorithm Bacause of this, the value at z shrinks and we found our shortest path which is only 11 units long!

23 Flow network Directed, weighted graph Weights are called capacities. The flux can t be greater than capacity. 2 Special vertices: source and drain Q: How can we maximize the flux from source to drain? ki/fl %C3%BCsse_und_Schni tte_in_netzwerken#/medi a/file:fluss-in-graph2.png

24 Ford Folkerson Initialize each flux as 0 for all paths from source to drain: search for the smallest capacity a; add a to each flux on the path. Note: If you go against a direction of an edge, the value has to be taken as negative. This explaines well: Ford Fulkerson in 5 minutes Step by step example

25 Ford Folkerson Init with 0 1 more Capacity left Max. flux is 2 Total flux is 4

26 Devide and Conquer Closest Point pair: devide set in two halves, assume, c. p. p. is eiter here or here

27 Devide and Conquer Closest Point pair: Recursively repeat on subset: here or here

28 Devide and Conquer Closest Point pair: Do this until you get to following situation: If there are only 3 Points in interval, calculate the 3 distances and choose the smallest If there are only 2, just take their distance as the smallest

29 Devide and Conquer Closest Point pair: Combining the devided intervals we compare the closest pairs each and take the smallest BUT: What if a distance between intervals is smaller than one inside of them?

30 Devide and Conquer Closest Point pair: That s why wo do this: the red lines are as far away to each side as the smallest distance found yet. We have to check each point pair in between, each step!

31 Devide and Conquer Closest Point pair: Works fastest with sorted sets Sorting: O( n log n) Deviding until smallest Interval: O(log n) Sorting after each devision: O( n log n * log n) = O(n log² n)

32 Polynomial Multiplication (8x³ 7x² + x + 2) (3x³ + 6x² + 3x + 1) = 8x³ (3x³ + 6x² + 3x + 1) 7x² (3x³ + 6x² + 3x + 1) + x (3x³ + 6x² + 3x + 1) + 2 (3x³ + 6x² + 3x + 1) Easy to see: Quadratic runtime; n = degree Runs in O(n²) Let s do faster!

33 Polynomial Multiplication (8x³ 7x² + x + 2) (3x³ + 6x² + 3x + 1) p(x) = (8x³ 7x² + x + 2) q(x) = (3x³ + 6x² + 3x + 1) Split each in the middle: p(x) = x² (8x 7) + (x + 2) q(x) = x² (3x + 6) + (3x + 1) Set terms in brackets: ph(x) := (8x 7) pl(x) := (x + 2) qh(x) := (3x + 6) ql(x) := (3x + 1)

34 Polynomial Multiplication ph(x) = (8x 7) qh(x) = (3x + 6) pl(x) = (x + 2) ql(x) = (3x + 1) Calculate the r s like this: rl(x) = pl(x) ql(x) = (3x² + 7x + 2) rh(x) = ph(x) qh(x) = (24² + 27x 42) rm(x) =( ph(x) + pl(x) ) ( qh(x) + ql(x) ) = (54x² + 33x 35)

35 Polynomial Multiplication rl(x) = (3x² + 7x + 2) rm(x) = (54x² + 33x 35) rh(x) = (24x² + 27x 42) p(x) q(x) = rl(x) + ( rm(x) rh(x) rl(x) ) x² + rh(x) x⁴ = (24x⁶ + 27x⁵ 42x⁴) + (27x⁴ x³ + 5x²) + (3x² + 7x + 2) = (24x⁶ + 27x⁵ 15x⁴ x³ + 8x² + 7x + 2)

36 Polynomial Multiplication Try yourself if you need. In this example we got from 4 x 4 multiplications to 3 x 2 and a few polynomial additions. (which are very efficiently calculated) In bigger polynomials we don t devide just once. As we devide per Step into 3 polynomials of half size: O( n log2( 3 ))

37 Assignment Write a program which takes a graph as an input & calculates the shortest path between two given vertices using Dijkstras algorithm and output it as an adjacency list as well as the length of it: A D D G List could look like this. G H...

38 Assignment There is an adjacency matrix provided on the website. thats what the program should run on. Calculate the Path from A to T, where A is the first line and T is the last This time its up to you whether you read the text file as an input or initialize the matrix in the code. (Because reading the file as input will be easier) Good luck!

39 Assignment Conditions Code comes from nowhere else than your brain!.java /.c /.cpp NO.docx.pdf etc!! Good comments make the difference between alright and very good! Put Matriculation number as comment above Deadline: 26 June 2018, 23:59 Mail: weimar.de