Note. Out of town Thursday afternoon. Willing to meet before 1pm, me if you want to meet then so I know to be in my office

Transcription

1 raphs and Trees

2 Note Out of town Thursday afternoon Willing to meet before pm, me if you want to meet then so I know to be in my office

3 few extra remarks about recursion If you can write it recursively you can also write it iteratively, and vice versa hurch-turing Thesis ut why pick one method over the other?

4 xample: ibonacci numbers int fib(int n) { if(n == n == 0) return ; else return fib(n-)+fib(n-) } int fib(int n) { int current = ; int previous = ; int temp; if(n == 0 n == ) return ; for(int i = ; i < n; i++) { temp = current + previous; previous = current; current = temp; } return current; }

5 Results... Recursive may be easier, but may not be the best way Work out the best way before moving forward

6 raphs and Trees Represents a lot of things ormally a graph is a pair of sets, = (V,) V - The set of vertices - the set of edges ach edge connects vertices together

7 Terminology Lots of terminology surrounding graphs, tons of types, specific patterns, etc Tip: lways read the description of graphs carefully - they describe a lot in just a few words

8 raph terminology Undirected - you can travel along an edge in either direction irected (igraph) - One-way travel only (but still possible to have edges!) Note: ll undirected graphs can be converted into directed graphs, but not the other way around! Unweighted - dges have no value associated with them Weighted - dges have a value associated with them xamples on board

9 Yet more terminology ycle - set of edges that forms a loop yclic - raph contains a cycle cyclic - No cycles allowed in the graph - irected cyclic raph Self-loop: n edge from a vertex to itself (yes, this counts as a cycle) Simple raph - Shorthand for Undirected Unweighted No-self-loops graph

10 Yes, there s still more egree (undirected only): The number of edges connected to a vertex In/Out-degree (directed only): The number of edges entering/leaving a graph Regular - ll vertices have the same degree (Simple) Path - sequence of (distinct) edges going from u to v onnected - or any two vertices there exists a path between them irected graphs - strongly connected components

11 Special graphs lique (K n ) - graph will all possible edges on n vertices ipartite - graph with two sets of vertices. dges only allowed between sets omplete ipartite (K n,m ) - bipartite graph with all possible edges mpty graph - No vertices or edges ycle raph - One giant cycle, no other edges allowed Path raph - ll linked in a line

12 nd... Tree: ny acyclic graph with n vertices and n- edges. Or.. n acyclic directed graph where there is only one path between any two vertices orest - collection of trees

13 raph representations djacency Matrix - a V x V matrix where [u][v] represents the existence and weight of an edge from u to v ood for dense graphs, easy to work with O(n ) memory djacency list - list of lists. ach list represents a vertex, and contains the vertices it is connected to ood for sparse graphs, harder to work with O(n + m) memory xample on board

14 Trees Used in a lot of more complex data structures Often a single vertex is designated as the root Root is on top. Not on bottom llows a natural direction for edges

15 Tree terminology Root - The top of the tree hildren - Those connected one layer below escendant - an get to these nodes by only moving downward ncestor - Node that can be reached by following parents

16 Minimum Spanning Trees onsider a weighted graph ind a set of edges that forms a tree (or forest, if is not connected) within in such a way that the sum of the edges is minimized ssentially, find the cheapest possible spanning tree that you can

17 Prim s lgorithm Idea: Take a greedy approach to making a MST Two versions: dj. Matrix and dj. List Matrix - O(n ) List - O(m log n) [O( log V)] ist: Pick a starting point. Repeatedly add the closest edge to the current tree to the tree until you re done

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19 Start with. ind closest vertex to tree

20 Start with. ind closest vertex to tree

21 Repeatedly find the closest vertex to the tree

22 Repeatedly find the closest vertex to the tree

23 Repeatedly find the closest vertex to the tree

24 Repeatedly find the closest vertex to the tree

25 Repeatedly find the closest vertex to the tree

26 Repeatedly find the closest vertex to the tree rbitrary tiebreaker: lphabetical order

27 Repeatedly find the closest vertex to the tree

28 Repeatedly find the closest vertex to the tree

29 Repeatedly find the closest vertex to the tree

30 ll N nodes in the tree. one!

31 Kruskals algorithm Repeatedly pick the smallest edge that has not been visited If it does not make a cycle, add it to the tree Initialized as a forest of singletons Tree is stored as a Union ind. ives us the runtimes we want

32 Pick edges, if there is no cycle, add to tree

33 Pick edges, if there is no cycle, add to tree

34 Pick edges, if there is no cycle, add to tree

35 Pick edges, if there is no cycle, add to tree

36 Pick edges, if there is no cycle, add to tree

37 Pick edges, if there is no cycle, add to tree

38 ll vertices connected. one

39 ycle case

40 ycle case

41 ycle case

42 ycle case