lecture 17

CS 332: Algorithms Graph Algorithms

Interval Trees ,[object Object],[object Object],10 7 11 5 8 4 18 15 23 21 17 19 i = [7,10]; i  low = 7; i  high = 10

Review: Interval Trees ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],10 7 11 5 8 4 18 15 23 21 17 19 i = [7,10]; i  low = 7; i  high = 10

Review: Interval Trees ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Review: Interval Trees [17,19] 23 [5,11] 18 [21,23] 23 [4,8] 8 [15,18] 18 [7,10] 10 int max Note that:

Review: Searching Interval Trees ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Review: Correctness of IntervalSearch() ,[object Object],[object Object],[object Object],[object Object],[object Object]

Correctness of IntervalSearch() ,[object Object],[object Object],[object Object],[object Object],[object Object],while (x != NULL && !overlap(i, x->interval)) if (x->left != NULL && x->left->max  i->low) x = x->left; else x = x->right; return x;

Review: Correctness of IntervalSearch() ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],while (x != NULL && !overlap(i, x->interval)) if (x->left != NULL && x->left->max  i->low) x = x->left; else x = x->right; return x;

Next Up: Graph Algorithms ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Graphs ,[object Object],[object Object],[object Object],[object Object]

Graph Variations ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Graph Variations ,[object Object],[object Object],[object Object],[object Object],[object Object]

Representing Graphs ,[object Object],[object Object],[object Object]

Graphs: Adjacency Matrix ,[object Object],1 2 4 3 a d b c 4 ?? 3 2 1 4 3 2 1 A

Graphs: Adjacency Matrix ,[object Object],1 2 4 3 a d b c 0 1 0 0 4 0 0 0 0 3 0 1 0 0 2 0 1 1 0 1 4 3 2 1 A

Graphs: Adjacency Matrix ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Graphs: Adjacency List ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],1 2 4 3

Graphs: Adjacency List ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Graph Searching ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Breadth-First Search ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Breadth-First Search ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Breadth-First Search ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],What does v->p represent? What does v->d represent?

Breadth-First Search: Example         r s t u v w x y

Breadth-First Search: Example   0      r s t u v w x y s Q:

Breadth-First Search: Example 1  0 1     r s t u v w x y w Q: r

Breadth-First Search: Example 1  0 1 2 2   r s t u v w x y r Q: t x

Breadth-First Search: Example 1 2 0 1 2 2   r s t u v w x y Q: t x v

Breadth-First Search: Example 1 2 0 1 2 2 3  r s t u v w x y Q: x v u

Breadth-First Search: Example 1 2 0 1 2 2 3 3 r s t u v w x y Q: v u y

Breadth-First Search: Example 1 2 0 1 2 2 3 3 r s t u v w x y Q: u y

Breadth-First Search: Example 1 2 0 1 2 2 3 3 r s t u v w x y Q: y

Breadth-First Search: Example 1 2 0 1 2 2 3 3 r s t u v w x y Q: Ø

BFS: The Code Again ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],What will be the running time? Total running time: O(V+E) Touch every vertex: O(V) u = every vertex, but only once ( Why? ) So v = every vertex that appears in some other vert’s adjacency list

BFS: The Code Again ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],What will be the storage cost in addition to storing the tree? Total space used: O(max(degree(v))) = O(E)

Breadth-First Search: Properties ,[object Object],[object Object],[object Object],[object Object],[object Object]

lecture 17

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lecture 17