DIRECTED GRAPHS BBM 201 DATA STRUCTURES DEPT. OF COMPUTER ENGINEERING

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Juliet Agatha Rogers
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1 Acknowledgement: he course slides are adapted from the slides prepared by R. Sedgewick and K. Wayne of Princeton University. BBM DAA SRUCURES DEP. O COMPUER ENGINEERING DIRECED GRAPHS

2 Directed Graphs Digraph API Digraph search ODAY

3 Directed graphs Digraph. Set of vertices connected pairwise by directed edges.

4 Road network Vertex = intersection; edge = one-way street.

5 Digraph applications digraph vertex directed edge transportation street intersection one-way street web web page hyperlink food web species predator-prey relationship WordNet synset hypernym scheduling task precedence constraint financial bank transaction cell phone person placed call infectious disease person infection game board position legal move citation journal article citation object graph object pointer inheritance hierarchy class inherits from control flow code block jump

6 Some digraph problems Path. Is there a directed path from s to t? s Shortest path. What is the shortest directed path from s to t? opological sort. Can you draw the digraph so that all edges point upwards? t Strong connectivity. Is there a directed path between all pairs of vertices? ransitive closure. or which vertices v and w is there a path from v to w? PageRank. What is the importance of a web page?

7 Digraph API Digraph search DIRECED GRAPHS

8 Digraph API % Digraph tinydg.txt -> -> -> -> -> -> -> -> -> -> -> -

9 Set-of-edges digraph representation Store a list of the edges (linked list or array). 7 7

10 Adjacency-matrix digraph representation Maintain a two-dimensional V-by-V boolean array; for each edge v w in the digraph: adj[v][w] = true. to 7 from 7

11 Adjacency-lists digraph representation Maintain vertex-indexed array of lists.

for the new node { struct node *next; int vertex; q=(node*)malloc(sizeof(node)); }node;

12 Adjacency-list graph representation: implementation //for each edge call addedge twice addedge(v,w); typedef struct node public void addedge(int v, int w){ node *q; //acquire memory for the new node { struct node *next; int vertex; q=(node*)malloc(sizeof(node)); }node; q->vertex=w; q->next=null; node * adj []; //insert the node to beginning of the linked list q->next=adj[v]; adj[v]= q; }

13 Digraph representations In practice. Use adjacency-lists representation. Algorithms based on iterating over vertices pointing from v. Real-world digraphs tend to be sparse. huge number of vertices, small average vertex degree representation space insert edge from v to w edge from v to w? iterate over vertices pointing from v? list of edges E E E adjacency matrix V V adjacency lists E + V outdegree(v) outdegree(v)

14 Digraph API Digraph search DIRECED GRAPHS

15 Reachability Problem. ind all vertices reachable from s along a directed path. s

16 in digraphs Same method as for undirected graphs. Every undirected graph is a digraph (with edges in both directions). DS is a digraph algorithm. DS (to visit a vertex v) Mark v as visited. Recursively visit all unmarked vertices w pointing from v.

17 o visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices pointing from v. a directed graph 7 7 7

18 o visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices pointing from v. v marked[] edgeo[] 7 7 a directed graph

19 o visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices pointing from v. v marked[] edgeo[] 7 7 visit : check and check

20 o visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices pointing from v. v marked[] edgeo[] 7 7 visit : check

21 o visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices pointing from v. v marked[] edgeo[] 7 7 visit : check and check

22 o visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices pointing from v. v marked[] edgeo[] 7 7 visit : check and check

23 o visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices pointing from v. v marked[] edgeo[] 7 7 visit : check and check

24 o visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices pointing from v. v marked[] edgeo[] 7 7 visit : check and check

25 o visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices pointing from v. v marked[] edgeo[] 7 7 visit : check and check

26 o visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices pointing from v. v marked[] edgeo[] 7 7 done

27 o visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices pointing from v. v marked[] edgeo[] 7 7 done 7

28 o visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices pointing from v. v marked[] edgeo[] 7 7 visit : check and check

29 o visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices pointing from v. v marked[] edgeo[] 7 7 done

30 o visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices pointing from v. v marked[] edgeo[] 7 7 done

31 o visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices pointing from v. v marked[] edgeo[] 7 7 visit : check and check

32 o visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices pointing from v. v marked[] edgeo[] 7 7 visit

33 o visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices pointing from v. v marked[] edgeo[] 7 7 done

34 o visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices pointing from v. v marked[] edgeo[] 7 7 done

35 o visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices pointing from v. v marked[] edgeo[] 7 7 done

36 o visit a vertex v : Mark vertex v as visited. Recursively visit all unmarked vertices pointing from v. v marked[] edgeo[] 7 reachable from vertex 7 reachable from

37 void DS(int i) { node *p = adj[i]; visited[i]=; while(p!= NULL) { i = p->vertex; if(!visited[i]) DS(i); p = p->next; } } typedef struct node { struct node *next; int vertex; }node; //GLOBAL PARAMEERS node * adj []; int visited[]; Same as undirected graph 7

38 Reachability application: program control-flow analysis Every program is a digraph. Vertex = basic block of instructions (straight-line program). Edge = jump. Dead-code elimination. ind (and remove) unreachable code. Infinite-loop detection. Determine whether exit is unreachable.

39 Reachability application: mark-sweep garbage collector Every data structure is a digraph. Vertex = object. Edge = reference. Roots. Objects known to be directly accessible by program (e.g., stack). Reachable objects. Objects indirectly accessible by program (starting at a root and following a chain of pointers). roots

40 Reachability application: mark-sweep garbage collector Mark-sweep algorithm. [McCarthy, ] Mark: mark all reachable objects. Sweep: if object is unmarked, it is garbage (so add to free list). Memory cost. Uses extra mark bit per object (plus DS stack). roots

Breadth-first search in digraphs Same method as for undirected graphs. Every undirected graph is a digraph (with edges in both directions). BS is a digraph algorithm.

41 Breadth-first search in digraphs Same method as for undirected graphs. Every undirected graph is a digraph (with edges in both directions). BS is a digraph algorithm. BS (from source vertex s) Put s onto a IO queue, and mark s as visited. Repeat until the queue is empty: - remove the least recently added vertex v - for each unmarked vertex pointing from v: add to queue and mark as visited. s

CMSC 132: Object-Oriented Programming II

CMSC 132: Object-Oriented Programming II DIRECTED GRAPHS Graphs slides are modified from COS 126 slides of Dr. Robert Sedgewick. 1 Directed graphs Digraph Set of vertices connected pairwise by directed