Graph ADT and Algorithmic Paradigms Generic Search Breadth First Search Dijkstra's Shortest Paths Algorithm Depth First Search Linear Order Jeff Edmonds York University Lecture 8 COSC 2011

Graphs A graph is a pair (V, E), where Example: PVD ORD SFO LGA HNL V is a set of nodes, called vertices E is a collection of pairs of vertices, called edges Vertices and edges are positions and store elements Example: A vertex represents an airport and stores the three-letter airport code An edge represents a flight route between two airports and stores the mileage of the route ORD PVD MIA DFW SFO LAX LGA HNL 849 802 1387 1743 1843 1099 1120 1233 337 2555 142 Last Update: Dec 4, 2014 Andy Mirzaian

Edge Types Directed edge Undirected edge Directed graph ordered pair of vertices (u,v) first vertex u is the origin second vertex v is the destination e.g., a flight Undirected edge unordered pair of vertices (u,v) e.g., a flight route Directed graph all the edges are directed e.g., route network Undirected graph all the edges are undirected e.g., flight network flight AA 1206 ORD PVD 849 miles ORD PVD Last Update: Dec 4, 2014 Andy Mirzaian

Applications Electronic circuits Transportation networks Printed circuit board Integrated circuit Transportation networks Highway network Flight network Computer networks Local area network Internet Web Databases Entity-relationship diagram Last Update: Dec 4, 2014 Andy Mirzaian

Terminology End vertices (or endpoints) of an edge U and V are the endpoints of a Edges incident on a vertex a, d, and b are incident on V Adjacent vertices U and V are adjacent Degree of a vertex X has degree 5 Parallel edges h and i are parallel edges Self-loop j is a self-loop X U V W Z Y a c b e d f g h i j Last Update: Dec 4, 2014 Andy Mirzaian

Terminology (cont.) Path Simple path Examples: P1 X U V W Z Y a c b e sequence of alternating vertices and edges begins with a vertex ends with a vertex each edge is preceded and followed by its endpoints Simple path path such that all its vertices and edges are distinct Examples: P1 = (V,b,X,h,Z) is a simple path P2 = (U,c,W,e,X,g,Y,f,W,d,V) is a path that is not simple P1 X U V W Z Y a c b e d f g h P2 Last Update: Dec 4, 2014 Andy Mirzaian

Terminology (cont.) Cycle Simple cycle Examples: C1 X U V W Z Y a c b circular sequence of alternating vertices and edges each edge is preceded and followed by its endpoints Simple cycle cycle such that all its vertices and edges are distinct Examples: C1 = (V,b,X,g,Y,f,W,c,U,a,) is a simple cycle C2 = (U,c,W,e,X,g,Y,f,W,d,V,a,) is a cycle that is not simple C1 X U V W Z Y a c b e d f g h C2 Last Update: Dec 4, 2014 Andy Mirzaian

Properties v deg(v) = 2m Notation Property 1 Property 2 n number of vertices m number of edges deg(v) degree of vertex v Property 1 v deg(v) = 2m Proof: each edge is counted twice Property 2 In an undirected graph with no self-loops and no multiple edges m  n (n - 1)/2 Proof: each vertex has degree at most (n - 1) What is the bound for a directed graph? Last Update: Dec 4, 2014 Andy Mirzaian

Vertices and Edges A graph is a collection of vertices and edges. We model the abstraction as a combination of three data types: Vertex, Edge, and Graph. A Vertex is a lightweight object that stores an arbitrary element provided by the user (e.g., an airport code) We assume it supports a method, element(), to retrieve the stored element. An Edge stores an associated object (e.g., a flight number, travel distance, cost), retrieved with the element( ) method. Last Update: Dec 4, 2014 Andy Mirzaian

Graph ADT: part 1 Last Update: Dec 4, 2014 Andy Mirzaian

Graph ADT: part 2 Last Update: Dec 4, 2014 Andy Mirzaian

Edge List Structure Vertex object Edge object Vertex sequence element reference to position in vertex sequence Edge object origin vertex object destination vertex object reference to position in edge sequence Vertex sequence sequence of vertex objects Edge sequence sequence of edge objects Last Update: Dec 4, 2014 Andy Mirzaian

Adjacency List Structure Incidence sequence for each vertex sequence of references to edge objects of incident edges Augmented edge objects references to associated positions in incidence sequences of end vertices Last Update: Dec 4, 2014 Andy Mirzaian

Adjacency Map Structure Incidence sequence for each vertex sequence of references to adjacent vertices, each mapped to edge object of the incident edge Augmented edge objects references to associated positions in incidence sequences of end vertices Last Update: Dec 4, 2014 Andy Mirzaian

Adjacency Matrix Structure Edge list structure Augmented vertex objects Integer key (index) associated with vertex 2D-array adjacency array Reference to edge object for adjacent vertices Null for non-adjacent vertices The “old fashioned” version just has 0 for no edge and 1 for edge Last Update: Dec 4, 2014 Andy Mirzaian

Performance Edge List Adjacency List Adjacency Matrix Space n + m n2 n vertices, m edges no parallel edges no self-loops Edge List Adjacency List Adjacency Matrix Space n + m n2 incidentEdges(v) m deg(v) n areAdjacent (v, w) min(deg(v), deg(w)) 1 insertVertex(o) insertEdge(v, w, o) removeVertex(v) removeEdge(e) max(deg(v), deg(w)) Last Update: Dec 4, 2014 Andy Mirzaian

Subgraphs A subgraph S of a graph G is a graph such that The vertices of S are a subset of the vertices of G The edges of S are a subset of the edges of G A spanning subgraph of G is a subgraph that contains all the vertices of G Subgraph Spanning subgraph Last Update: Dec 4, 2014 Andy Mirzaian

Non connected graph with two connected components Connectivity A graph is connected if there is a path between every pair of vertices A connected component of a graph G is a maximal connected subgraph of G Non connected graph with two connected components Connected graph Last Update: Dec 4, 2014 Andy Mirzaian

Trees and Forests A (free) tree is an undirected graph T such that T is connected T has no cycles This definition of tree is different from the one of a rooted tree A forest is an undirected graph without cycles The connected components of a forest are trees Forest Tree Last Update: Dec 4, 2014 Andy Mirzaian

Spanning Trees and Forests A spanning tree of a connected graph is a spanning subgraph that is a tree A spanning tree is not unique unless the graph is a tree Spanning trees have applications to the design of communication networks A spanning forest of a graph is a spanning subgraph that is a forest Spanning tree Graph Last Update: Dec 4, 2014 Andy Mirzaian

Graph Search

Graph Search Specification: Reachability-from-single-source s <preCond>: The input is a graph G (either directed or undirected) and a source node s. <postCond>: Output all the nodes u that are reachable by a path in G from s.

Graph Search Basic Steps: s v & there is an edge from u to v u Suppose you know that u is reachable from s You know that v is reachable from s Build up a set of reachable nodes.

Graph Search reachable b t b t reachable e h d e h d reachable d v w d f u j s How do we keep track of all of this information? How do we avoid cycling?

Graph Search s b a d e g c f j i h m k l

Graph Search s b a d e g c f j i h m k l

Graph Search s b a d e g c f j i h m k l

Graph Search s We know found nodes are reachable from s because we have traced out a path. b a d e g c f If a node has been handled, then all of its neighbors have been found. j i h m k l l

Graph Search s We know found nodes are reachable from s because we have traced out a path. b a d e g Handle some foundNotHandled node c f If a node has been handled, then all of its neighbors have been found. j i h m k l i.e. find its neighbors Don’t re-find a node.

Graph Search s We know found nodes are reachable from s because we have traced out a path. b a d e g c f j If a node has been handled, then all of its neighbors have been found. i h m k l Handle some foundNotHandled node i.e. find its neighbors

Graph Search s We know found nodes are reachable from s because we have traced out a path. b a d e g c f If a node has been handled, then all of its neighbors have been found. j i h m k l measure progress # of found nodes. Might not increase. # of handled nodes.

Graph Search s We know found nodes are reachable from s because we have traced out a path. b a d e g c f If a node has been handled, then all of its neighbors have been found. j i h m k l Node s is foundNotHandled Other nodes notFound

Graph Search s We know found nodes are reachable from s because we have traced out a path. b a d e g c f If a node has been handled, then all of its neighbors have been found. j i h m k l Exit All nodes found. No. Might not find all. When can’t make any more progress. Handle some foundNotHandled node When all found nodes are have been handled.

Graph Search We know found nodes are reachable from s because we have traced out a path. If a node has been handled, then all of its neighbors have been found. b a d e g c f j i h m k l Exit All found nodes are handled. <postCond>: Output all the nodes u that are reachable by a path in G from s. Exit Output Found nodes

Graph Search We know found nodes are reachable from s because we have traced out a path. b a d e g c f j If a node has been handled, then all of its neighbors have been found. i h m k l Exit All found nodes are handled. <postCond>: Exit Found nodes are reachable from s. Reachable nodes have been found.

Graph Search Exit Found = handled We know found nodes are reachable from s because we have traced out a path. b a d e g c f j If a node has been handled, then all of its neighbors have been found. i h handled notfound m k l Exit All found nodes are handled. <postCond>: Exit Reachable nodes have been Found. [A  B] = [B  A] notFound nodes not reachable.

Graph Search Specification of Reachability-from-single-source s <preCond>: The input is a graph G (either directed or undirected) and a source node s. <postCond>: Output all the nodes u that are reachable by a path in G from s. Ending Initial Conditions Make Progress Maintain Loop Inv Define Exit Condition Define Step Define Measure of Progress Define Loop Invariants Define Problem 79 km to school Exit 75 km 0 km

Graph Search # of handled nodes. Handle some foundNotHandled node f u j O(n) iterations, but iteration takes more than O(1) Time = O(n) O(n) neighbors = O(n2) Could be fewer? Each edge visited, times. 2 = O(E) Linear time. Size = O(E)

Graph Search Stack: Which foundNotHandled node do we handle? Queue: Handle node Found longest ago Likely closest to s (in # of edges). Breadth-First Search Priority Queue: Handle node that seems to be closest to s (weighted). Dijkstra's Shortest-Weighted Paths Stack: Found most recently Likely farthest from s. Depth-First Search

Graph Search Which foundNotHandled node do we handle? Queue: Handle node Found longest ago Likely closest to s (in # of edges). Breadth-First Search So far, the nodes have been found in order of length from s.

Found Not Handled Queue BFS Found Not Handled Queue s b a e d g c f j i h m k l

Found Not Handled Queue BFS Found Not Handled Queue d=0 s b a s d=0 e d g c f j i h m k l

Found Not Handled Queue BFS Found Not Handled Queue d=0 s d=1 b a d=0 e a d=1 d d g g c f b j i h m k l

Found Not Handled Queue BFS Found Not Handled Queue d=0 s d=1 b a a d=1 d e d g g b c f j i h m k l

Found Not Handled Queue BFS Found Not Handled Queue d=0 s d=1 b a d=1 d e d g g b c f c d=2 j f d=2 i h m k l

Found Not Handled Queue BFS Found Not Handled Queue d=0 s d=1 b a d=1 e d g g b c f c d=2 j f d=2 m i e h m k l

Found Not Handled Queue BFS Found Not Handled Queue d=0 s d=1 b a d=1 e d g b c f c d=2 j f d=2 m i e h j m k l

Found Not Handled Queue BFS Found Not Handled Queue d=0 s d=1 b a d=1 e d g c f c d=2 j f d=2 m i e h j m k l

Found Not Handled Queue BFS Found Not Handled Queue d=0 s d=1 b a c d=2 e d f m g e c f j j d=2 i h m k l

Found Not Handled Queue BFS Found Not Handled Queue d=0 s d=1 b a d=2 e d f m g e c f j j h d=3 d=2 i i h m d=3 k l

Found Not Handled Queue BFS Found Not Handled Queue d=0 s d=1 b a d=2 e d m g e c f j j h d=3 d=2 i i h m d=3 k l

Found Not Handled Queue BFS Found Not Handled Queue d=0 s d=1 b a d=2 e d g e c f j j h d=3 d=2 i i h l m d=3 k l

Found Not Handled Queue BFS Found Not Handled Queue d=0 s d=1 b a d=2 e d g c f j j h d=3 d=2 i i h l m d=3 k l

Found Not Handled Queue BFS Found Not Handled Queue d=0 s d=1 b a d=2 e d g c f j h d=3 d=2 i i h l m d=3 k l

Found Not Handled Queue BFS Found Not Handled Queue d=0 s d=1 b a h d=3 i e d l g c f j d=2 i h m d=3 k l

Found Not Handled Queue BFS Found Not Handled Queue d=0 s d=1 b a d=3 i e d l g k d=4 c f j d=2 i h m d=3 k d=4 l

Found Not Handled Queue BFS Found Not Handled Queue d=0 s d=1 b a d=3 e d l g k d=4 c f j d=2 i h m d=3 k d=4 l

Found Not Handled Queue BFS Found Not Handled Queue d=0 s d=1 b a d=3 e d g k d=4 c f j d=2 i h m d=3 k d=4 l

Found Not Handled Queue BFS Found Not Handled Queue d=0 s d=1 b a k d=4 e d g c f j d=2 i h m d=3 k d=4 l

Found Not Handled Queue BFS Found Not Handled Queue d=0 s d=1 b a d=4 e d=5 d g c f j d=2 i h m d=3 k d=4 l

BFS So far, the nodes have been found in order of length from s.

BFS So far, the nodes have been found in order of length from s. Finds a shortest path from s to each node v and its length. <postCond>: When we find v, we know there isn't a shorter path to it because ? Otherwise, we would have found it already.

BFS Data structure for storing a path to each node: For each node v, store (v) to be parent of v.

BFS Basic Steps: s u The shortest path to u has length d v & there is an edge from u to v There is a path to v with length d+1. Parent of v is (v) = u.

Graph Search Stack: Which foundNotHandled node do we handle? Queue: Handle node Found longest ago Likely closest to s (in # of edges). Breadth-First Search Priority Queue: Handle node that seems to be closest to s (weighted). Dijkstra's Shortest-Weighted Paths Stack: Found most recently Likely farthest from s. Depth-First Search

Dijkstra's Shortest-Weighted Paths Specification: Dijkstra's Shortest-Weighted Paths Reachability-from-single-source s <preCond>: The input is a graph G (either directed or undirected) with positive edge weights and a source node s. <postCond>: Finds a shortest weighted path from s to each node v and its length.

Dijkstra's Shortest-Weighted Paths The king wanted to know the shortest path from his castle s to each node (home) v. So that people could come and go quickly and so that he could guard these home and paths. 5 3 ( ) = previous node in path D( ) = length = 8.

Dijkstra's Shortest-Weighted Paths Rome was not built in a day. He decided to handle the nodes in terms of their distance from s. He put a wall around these handled nodes.

Dijkstra's Shortest-Weighted Paths We have the answer for handled nodes (i.e. closest) i.e. D(u) and (u) give shortest path from s to u. Exit When all the nodes are handled, done!

Dijkstra's Shortest-Weighted Paths When a node has been handled: He put a knight on each handled node to guard all edges (roads) leading out of the node. An edge is handled if it comes out of a handled node. i.e. all edges in and leaving city.

Dijkstra's Shortest-Weighted Paths Unhandled edges are unguarded, so we don’t want to travel along them, even if they make a shorter path.

Dijkstra's Shortest-Weighted Paths A path is handled if has handled edges. i.e. path travels from s between the handled nodes within the walls and then one last edge out at gate. For unhandled nodes, (u) and d(u) values give the shortest handled path from s. d( )= d( )=8.  d( )=20.

Dijkstra's Shortest-Weighted Paths This red path to has length d( ) + w( , ) = 8 + 5 = 13 This is better but unhandled and hence unguarded. 5 d( )=8. d( )=20.

Dijkstra's Shortest-Weighted Paths Theorem: If has the smallest d value, then its shortest path from s is handled. Proof: Any unhandled path must leave at a different gate, and hence already has gone further than d( ). 5 d( )=8. D d( )=20.

Dijkstra's Shortest-Weighted Paths Consider the unhandled node with the smallest d( ). Priority Queue (FoundNotHandled): Handle node that seems to be closest to s. 5 D( )=8. d( )=20.

Dijkstra's Shortest-Weighted Paths Consider the unhandled node with the smallest d( ). Handled it. Handle out going edges. Update its neighbor’s d values. 5 D( )=8. d( )=20.

Dijkstra's Shortest-Weighted Paths This red path to has length d( ) + w( , ) = 8 + 5 = 13 This is better but was unhandled . But with handled, this edge is now handled so we are good. 5 D( )=8. d( )=20.

Dijkstra's Shortest-Weighted Paths This red path to has length d( ) + w( , ) = 8 + 5 = 13 This is better but was unhandled . But with handled, this edge is now handled so we are good. 5 D( )=8. ( ) d( )=20. 13

Dijkstra's Shortest-Weighted Paths We have the answer for handled nodes (i.e. closest) i.e. D(u) and (u) give shortest path from s to u. Done! 5 D( )=8. ( ) d( )=13.

Dijkstra's Shortest-Weighted Paths For unhandled nodes, (u) and d(u) values give the shortest handled path from s. Done! 5 D( )=8. ( ) d( )=13.

Dijkstra's Shortest-Weighted Paths Ending Initial Conditions Make Progress Maintain Loop Inv Define Exit Condition Define Step Define Measure of Progress Define Loop Invariants Define Problem 79 km to school Exit 75 km 0 km Done! 5 D( )=8. ( ) d( )=13.

Definition of handled paths and d(v) Dijkstra's Definition of handled paths and d(v) Handled nodes Found nodes s v u Handled Edges?

Definition of handled paths and d(v) Dijkstra's Definition of handled paths and d(v) Handled nodes Found nodes s v u Handled Edges Handled Paths?

Definition of handled paths and d(v) Dijkstra's Definition of handled paths and d(v) Handled nodes Found nodes s v u Handled Edges Handled Paths?

Dijkstra's Definition of handled paths and d(v) Handled nodes s v NotFound d(v’)= u  For handled u, d(u) is the length of the shortest paths to u. d(v) is the length of the shortest handled paths to v.

Dijkstra's Basic Steps: Handle node that “seems” to be closest to s. v The shortest of handled paths to v has length d(v) u The shortest of handled paths to u has length d(u) & there is an edge from u to v w<u,v> The shortest handled path to v has length min( d(v), d(u)+w<u,v> ).

Dijkstra's s b a d e c f h g i j d=0 10 1 d= d= 30 2 30 40 3 3 1 2 8 15 1 2 f h d= 1 d= g 6 d= i 3 j d= d=

Dijkstra's s b a d e c f h g i j d=0 10 1 d=10 d=1 30 2 30 40 3 3 1 2 8 15 1 2 f h d= 1 d= g 6 d= i 3 j d= d=

Dijkstra's s b a d e c f h g i j d=0 10 1 d=10 d=1 30 2 30 40 3 3 1 2 8 15 1 2 f h d=3 1 d= g 6 d= i 3 j d= d=

Dijkstra's s b a d e c f h g i j d=0 10 1 d=10 d=1 30 2 30 40 3 3 1 2 8 15 1 2 f h d=3 1 d= g 6 d=4 i 3 j d= d=

Dijkstra's s b a d e c f h g i j d=0 10 1 d=10 d=1 30 2 30 40 3 3 1 2 8 15 1 1 f h d=3 1 d=10 g 6 d=4 i 3 j d= d=

Dijkstra's s b a d e c f h g i j d=0 10 1 d=10 d=1 30 2 30 40 3 3 1 2 8 15 1 1 f h d=3 1 d=7 g 6 d=4 i 3 j d= d=

Dijkstra's s b a d e c f h g i j d=0 10 1 d=9 d=1 30 2 30 40 3 3 1 2 8 15 1 1 f h d=3 1 d=7 g 6 d=4 i 3 j d= d=

Dijkstra's s b a d e c f h g i j d=0 10 1 d=8 d=1 30 2 30 40 3 3 1 2 15 1 1 f h d=3 1 d=7 g 6 d=4 i 3 j d= d=

Dijkstra's s b a d e c f h g i j d=0 10 1 d=8 d=1 30 2 30 40 3 3 1 2 15 1 1 f h d=3 1 d=7 g 6 d=4 i 3 j d= d=

Dijkstra's s b a d e c f h g i j d=0 10 1 d=8 d=1 30 2 30 40 3 3 1 2 15 1 1 f h d=3 1 d=7 g 6 d=4 i 3 j d= d=

Dijkstra's s b a d e c f h g i j d=0 10 1 d=8 d=1 30 2 30 40 3 3 1 2 15 1 1 f h d=3 1 d=7 g 6 d=4 i 3 j d= d=

Dijkstra's s b a d e c f h g i j d=0 10 1 d=8 d=1 30 2 30 40 3 3 1 2 15 1 1 f h d=3 1 d=7 g 6 d=4 DONE i 3 j d= d=

Dijkstra's 13 16 19 37 20 ∞ 17 Π(e) =c Π(d) =c Handle c

Dijkstra's Each edge visited, times. 2 Time = O(E) ? This number of times following an edge. For each update, d(v) = min( d(v), d(u)+w<u,v> ). Must update Priority Queue. Takes time O(log(n)) Heap Time = O(E log(n))

Adaptable Heap Pop/Push/Changes But now it is not a heap! The 39 “bubbles” down or up until it finds its spot. Suppose some outside user knows about some data item c and remembers where it is in the heap. And changes its priority from 21 to 39 27 39 21 39 21 c

Adaptable Heap Pop/Push/Changes But now it is not a heap! The 39 “bubbles” down or up until it finds its spot. Suppose some outside user also knows about data item f and its location in the heap just changed. The Heap must be able to find this outside user and tell him it moved. 39 27 39 27 f 21 c Time = O(log n)

Heap Implementation A location-aware heap entry is an object storing 4 a 2 d 6 b 8 g 5 e 9 c A location-aware heap entry is an object storing key value position of the entry in the underlying heap In turn, each heap position stores an entry Back pointers are updated during entry swaps Last Update: Oct 23, 2014 Andy

Dijkstra's

Dijkstra's

Graph Search Stack: Which foundNotHandled node do we handle? Queue: Handle node Found longest ago Likely closest to s (in # of edges). Breadth-First Search Priority Queue: Handle node that seems to be closest to s (weighted). Dijkstra's Shortest-Weighted Paths Stack: Found most recently Likely farthest from s. Depth-First Search

DFS Breadth first search makes a lot of sense for dating in general actually. It suggests dating a bunch of people casually before getting serious rather than having a series of five year relationships.

Found Not Handled Stack DFS Found Not Handled Stack s b a e d g c f j i h m k l

Found Not Handled Stack DFS Found Not Handled Stack s b a e d g c f j i s h m k l

Found Not Handled Stack DFS Found Not Handled Stack s b a e d g c f b j g i d a h m k l

Found Not Handled Stack DFS Found Not Handled Stack s b a e d g c e f m j g i d a h m k l

DFS

DFS The nodes in the stack form a path starting at s.

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f j i h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f j i s,0 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f j i a,0 s,1 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f j c,0 i a,1 s,1 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f h,0 j c,1 i a,1 s,1 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c k,0 f h,1 j c,1 i a,1 s,1 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f h,1 j c,1 Path on Stack i a,1 s,1 h m Tree Edge k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f j c,1 i a,1 s,1 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f i,0 j c,2 i a,1 s,1 h m k l

DFS Found Not Handled Stack <node,# edges> s b a e d g c f i,1 j m Cross Edge to handled node k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f i,2 j c,2 i a,1 s,1 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c l,0 f i,3 j c,2 i a,1 s,1 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c l,1 f i,3 j c,2 i a,1 s,1 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f i,3 j c,2 i a,1 s,1 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c g,0 f i,4 j c,2 i a,1 s,1 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g j,0 c g,1 f i,4 j c,2 i a,1 s,1 h m k l

DFS Found Not Handled Stack <node,# edges> s b a e d g j,1 c g,1 i,4 j c,2 i a,1 s,1 h Back Edge to node on Stack m k l Forms a cycle

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d m,0 g j,2 c g,1 f i,4 j c,2 i a,1 s,1 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d m,1 g j,2 c g,1 f i,4 j c,2 i a,1 s,1 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g j,2 c g,1 f i,4 j c,2 i a,1 s,1 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c g,1 f i,4 j c,2 i a,1 s,1 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f i,4 j c,2 i a,1 s,1 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f,0 f i,5 j c,2 i a,1 s,1 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f,1 f i,5 j c,2 i a,1 s,1 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f i,5 j c,2 i a,1 s,1 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f j c,2 i a,1 s,1 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f j c,3 i a,1 s,1 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f j i a,1 s,1 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f j i a,2 s,1 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f j i s,1 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f j i d,1 s,2 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f j i d,2 s,2 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f j e,0 i d,3 s,2 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f j e,1 i d,3 s,2 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f j i d,3 s,2 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f j i s,2 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f j i s,3 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f j i b,0 s,4 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f j i b,1 s,4 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f j i b,2 s,4 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f j i b,3 s,4 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f j i s,4 h m k l

Found Not Handled Stack <node,# edges> DFS Found Not Handled Stack <node,# edges> s b a e d g c f j i h m k done l

Recursive Depth First Search Get help from friends

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Linear Order a b h c i d j e k g f l ….. l <preCond>: A Directed Acyclic Graph (DAG) <postCond>: Find one valid linear order a b h c i d j e k Algorithm: Find a sink. Put it last in order. Delete & Repeat g ? f l ….. l

Linear Order a b h c i d j e k g f l ….. l <preCond>: A Directed Acyclic Graph (DAG) <postCond>: Find one valid linear order a b h c i d j e k Algorithm: Find a sink. Put it last in order. Delete & Repeat g f l (n) (n2) ….. l

Found Not Handled Stack Linear Order Found Not Handled Stack Alg: DFS a b h c i d j e k g f g l e f d ….. f

Linear Order Found Not Handled Stack a b h c i d j e k g l g e f l d Alg: DFS a b h c i d j e k g l g e f l d When take off stack add to backwards order ….. f

Linear Order Found Not Handled Stack a b h c i d j e k g g e f l d l,f Alg: DFS a b h c i d j e k g g l e f d When take off stack add to backwards order l,f

Linear Order Found Not Handled Stack a b h c i d j e k g e f l d g,l,f Alg: DFS a b h c i d j e k g f l e d When take off stack add to backwards order g,l,f

Linear Order Found Not Handled Stack a b h c i d j e k g f l d e,g,l,f Alg: DFS a b h c i d j e k g f l d When take off stack add to backwards order e,g,l,f

Linear Order Found Not Handled Stack a b h c i d j e k g f l d,e,g,l,f Alg: DFS a b h c i d j e k g f l When take off stack add to backwards order d,e,g,l,f

Linear Order Found Not Handled Stack a b h c i d j e k g k j f l i Alg: DFS a b h c i d j e k g k j f l i When take off stack add to backwards order d,e,g,l,f

Linear Order Found Not Handled Stack a b h c i d j e k g j f l i Alg: DFS a b h c i d j e k g j f l i When take off stack add to backwards order k,d,e,g,l,f

Linear Order Found Not Handled Stack a b h c i d j e k g f l i Alg: DFS a b h c i d j e k g f l i When take off stack add to backwards order j,k,d,e,g,l,f

Linear Order Found Not Handled Stack a b h c i d j e k g f l Alg: DFS a b h c i d j e k g f l When take off stack add to backwards order i,j,k,d,e,g,l,f

Linear Order Found Not Handled Stack a b h c i d j e k g c f l b Alg: DFS a b h c i d j e k g c f l b When take off stack add to backwards order i,j,k,d,e,g,l,f

Linear Order Found Not Handled Stack a b h c i d j e k g f l b Alg: DFS a b h c i d j e k g f l b When take off stack add to backwards order c,i,j,k,d,e,g,l,f

Linear Order Found Not Handled Stack a b h c i d j e k g f l Alg: DFS a b h c i d j e k g f l When take off stack add to backwards order b,c,i,j,k,d,e,g,l,f

Linear Order Found Not Handled Stack a b h c i d j e k g h f l a Alg: DFS a b h c i d j e k g h f l a When take off stack add to backwards order b,c,i,j,k,d,e,g,l,f

Linear Order Found Not Handled Stack a b h c i d j e k g f l a Alg: DFS a b h c i d j e k g f l a When take off stack add to backwards order h,b,c,i,j,k,d,e,g,l,f

Linear Order Found Not Handled Stack a b h c i d j e k g f l Alg: DFS a b h c i d j e k g f l When take off stack add to backwards order a,h,b,c,i,j,k,d,e,g,l,f done

Found Not Handled Stack Linear Order Found Not Handled Stack Proof: Case 1: u goes on stack first before v. Because of edge, v goes on before u comes off v comes off before u comes off v goes after u in order.  Consider each edge v … u … u v u… v…

Found Not Handled Stack Linear Order Found Not Handled Stack Proof: Case 1: u goes on stack first before v. Case 2: v goes on stack first before u. v comes off before u goes on. v goes after u in order.  Consider each edge u … v … u v u… v…

Found Not Handled Stack Linear Order Found Not Handled Stack Proof: Case 1: u goes on stack first before v. Case 2: v goes on stack first before u. v comes off before u goes on. Case 3: v goes on stack first before u. u goes on before v comes off. Panic: u goes after v in order.  Cycle means linear order is impossible  Consider each edge u … The nodes in the stack form a path starting at s. v … u v u… v… done

Optimization Problems Ingredients: Instances: The possible inputs to the problem. Solutions for Instance: Each instance has an exponentially large set of solutions. Cost of Solution: Each solution has an easy to compute cost or value. Specification <preCond>: The input is one instance. <postCond>: A valid solution with optimal cost. (minimum or maximum)

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