Graph Algorithms. Textbook reading. Chapter 3 Chapter 4. CSci 3110 Graph Algorithms 1/41
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1 CSci 3110 Graph Algorithms 1/41 Graph Algorithms Textbook reading Chapter 3 Chapter 4
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3 CSci 3110 Graph Algorithms 2/41 Overview Design principle: Learn the structure of a graph by systematic exploration Proof techniques: Proof by contradiction Problems: Bipartiteness Connectivity Strong connectivity Topological sorting
4 CSci 3110 Graph Algorithms 3/41 Graphs, Vertices, and Edges A graph is an ordered pair G = (V,E). V is the set of vertices of G. E is the set of edges of G. The elements of E are pairs (v,w) of vertices.
5 CSci 3110 Graph Algorithms 3/41 Graphs, Vertices, and Edges A graph is an ordered pair G = (V,E). V is the set of vertices of G. E is the set of edges of G. The elements of E are pairs (v,w) of vertices. For an edge (v,w) E, we call vertices v and w adjacent. Edge (v,w) is incident to v and w. Vertices v and w are the endpoints of edge (v,w). The degree of a vertex v is the number of edges incident to v.
6 CSci 3110 Graph Algorithms 4/41 Undirected and Directed Graphs A graph is undirected if its edges are unordered pairs (v, w), that is, (v,w) = (w,v). A graph is directed if its edges are ordered pairs (v,w), that is, (v,w) (w,v). Edge (v,w) is an out-edge of v and an in-edge of w. The in-degree and out-degree of a vertex v are the numbers of in-edges and out-edges incident to v.
7 CSci 3110 Graph Algorithms 5/41 Paths and Cycles A path P = (x = v 0,v 1,...,v k = y) from a vertex x to a vertex y is a sequence of vertices such that (v i 1,v i ) is an edge, for all 1 i k. A cycle is a path from a vertex x back to itself. A path or cycle is simple if it contains every vertex of G at most once.
8 CSci 3110 Graph Algorithms 6/41 Connectivity and Connected Components A graph is connected if there exists a path from x to y, for any two vertices x and y of G. connected not connected
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10 CSci 3110 Graph Algorithms 7/41 Adjacency-List Representation of Graphs Doubly-linked vertex list Doubly-linked edge list e a b One doubly-linked adjacency list per vertex Pointers from adjacency list entries to vertices c f d Cross-pointers between edges and adjacency list entries a b c d e f (b,d) (e,f) (c,f) (c,e) (a,e)
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12 CSci 3110 Graph Algorithms 8/41 Modelling Real-World Problems (1) Many problems are quite naturally expressed as graph problems: Example: Stable matching is a special case of bipartite perfect matching w 1 w 2 w 3 w 4 m 1 m 2 m 3 m 4 A graph is bipartite if its vertices can be divided into sets X and Y so that every edge has one endpoint in X and the other in Y. A matching of a graph is a subset of edges so that no two edges in the set share an endpoint. w 5 m 5 A matching is perfect if every vertex is the endpoint of an edge.
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14 CSci 3110 Graph Algorithms 9/41 Modelling Real-World Problems (2) Example: Airline scheduling There are n lucrative flight segments to be serviced Flight segment = (source, destination, departure time, arrival time) Question: Can we service all n segments using the k planes in our fleet? Rules: Same plane can service two segments (s 1,t 1,d 1,a 1 ) and (s 2,t 2,d 2,a 2 ) if t 1 = s 2 There is enough time for maintenance between arrival time a 1 and departure time d 2 We can add other flight segments to get a plane that arrived at destination t 1 to service segment from destination s 2 by adding extra flight segments that get us from t 1 to s 2. The same rules about maintenance periods between arrivals and departures apply.
15 CSci 3110 Graph Algorithms 10/41 Graph-theoretic formulation: BOS 6:00 DCA 7:00 DCA 8:00 LAX 11:00 LAS 5:00 SEA 6:00 PHL 7:00 PIT 8:00 PHL 11:00 SFO 2:00 SFO 2:15 SEA 3:15
16 CSci 3110 Graph Algorithms 10/41 Graph-theoretic formulation: BOS 6:00 DCA 7:00 DCA 8:00 LAX 11:00 LAS 5:00 SEA 6:00 PHL 7:00 PIT 8:00 PHL 11:00 SFO 2:00 SFO 2:15 SEA 3:15
17 CSci 3110 Graph Algorithms 10/41 Graph-theoretic formulation: BOS 6:00 DCA 7:00 DCA 8:00 LAX 11:00 LAS 5:00 SEA 6:00 PHL 7:00 PIT 8:00 PHL 11:00 SFO 2:00 SFO 2:15 SEA 3:15 Are there k paths in this network whose union includes all solid edges?
18 CSci 3110 Graph Algorithms 10/41 Graph-theoretic formulation: BOS 6:00 DCA 7:00 DCA 8:00 LAX 11:00 LAS 5:00 SEA 6:00 PHL 7:00 PIT 8:00 PHL 11:00 SFO 2:00 SFO 2:15 SEA 3:15 Are there k paths in this network whose union includes all solid edges?
19 CSci 3110 Graph Algorithms 10/41 Graph-theoretic formulation: BOS 6:00 DCA 7:00 DCA 8:00 LAX 11:00 LAS 5:00 SEA 6:00 PHL 7:00 PIT 8:00 PHL 11:00 SFO 2:00 SFO 2:15 SEA 3:15 Are there k paths in this network whose union includes all solid edges? This reduces to a flow problem. See textbook, Chapter 7.
20 CSci 3110 Graph Algorithms 11/41 Modelling Real-World Problems (3) Example: Ordering tasks under constraints Building a shack Buy boards 1 Buy nails 2 Buy hammer 3 Buy hinges 6 4 Erect sides of shack Assemble door 7 Add roof 5 8Insert door in door frame
21 CSci 3110 Graph Algorithms 11/41 Modelling Real-World Problems (3) Example: Ordering tasks under constraints Building a shack Buy boards 1 Buy nails 2 Buy hammer 3 Buy hinges 6 4 Erect sides of shack Assemble door 7 Add roof 5 8Insert door in door frame Topological sorting: Number the vertices so that, for every edge (v,w), v < w.
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23 CSci 3110 Graph Algorithms 12/41 Modelling Real-World Problems (4) More examples: Communication networks Transportation networks Data structures...
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25 CSci 3110 Graph Algorithms 13/41 Graph Exploration (1) EXPLORE-GRAPH(G) 1 Mark every vertex and every edge of G as unexplored 2 for every vertex v of G 3 do if v is unexplored 4 then Explore-from-Vertex(G, v) EXPLORE-FROM-VERTEX(G, v) 1 Mark v as explored 2 S Adj(v) 3 while S is not empty 4 do Remove an edge (x,y) from S 5 if (x,y) is not explored 6 then if y is not explored 7 then Mark (x,y) as a tree edge 8 Mark y as explored 9 S S Adj(y) 10 else Mark (x,y) as a non-tree edge
26 CSci 3110 Graph Algorithms 14/41 Graph Exploration (2) "Explorable" Explored Source Unexplored
27 CSci 3110 Graph Algorithms 15/41 Graph Exploration Variants (1): Depth-First Search DFS-FROM-VERTEX(G, v) 1 Mark v as explored 2 S is a stack 3 for every edge (v, w) incident to v 4 do Push(S,(v, w)) 5 while S is not empty 6 do (x,y) Pop(S) 7 if (x,y) is not explored 8 then if y is not explored 9 then Mark (x,y) as a tree edge 10 Mark y as explored 11 for every edge (y,z) incident to y 12 do Push(S,(y, z)) 13 else Mark (x,y) as a non-tree edge
28 CSci 3110 Graph Algorithms 15/41 Graph Exploration Variants (1): Depth-First Search DFS-FROM-VERTEX(G, v) 1 Mark v as explored 2 S is a stack 3 for every edge (v, w) incident to v 4 do Push(S,(v, w)) 5 while S is not empty 6 do (x,y) Pop(S) 7 if (x,y) is not explored 8 then if y is not explored 9 then Mark (x,y) as a tree edge 10 Mark y as explored 11 for every edge (y,z) incident to y 12 do Push(S,(y, z)) 13 else Mark (x,y) as a non-tree edge Lemma: Depth-first search takes O(n + m) time.
29 CSci 3110 Graph Algorithms 16/41 Graph Exploration Variants (2): Breadth-First Search BFS-FROM-VERTEX(G, v) 1 Mark v as explored 2 S is a queue 3 for every edge (v, w) incident to v 4 do Enqueue(S,(v, w)) 5 while S is not empty 6 do (x,y) Dequeue(S) 7 if (x,y) is not explored 8 then if y is not explored 9 then Mark (x,y) as a tree edge 10 Mark y as explored 11 for every edge (y,z) incident to y 12 do Enqueue(S,(y, z)) 13 else Mark (x,y) as a non-tree edge
30 CSci 3110 Graph Algorithms 16/41 Graph Exploration Variants (2): Breadth-First Search BFS-FROM-VERTEX(G, v) 1 Mark v as explored 2 S is a queue 3 for every edge (v, w) incident to v 4 do Enqueue(S,(v, w)) 5 while S is not empty 6 do (x,y) Dequeue(S) 7 if (x,y) is not explored 8 then if y is not explored 9 then Mark (x,y) as a tree edge 10 Mark y as explored 11 for every edge (y,z) incident to y 12 do Enqueue(S,(y, z)) 13 else Mark (x,y) as a non-tree edge Lemma: Breadth-first search takes O(n + m) time.
31 CSci 3110 Graph Algorithms 17/41 Graph Exploration Variants (3): Dijkstra s Algorithm DIJKSTRA-FROM-VERTEX(G, v) 1 Mark v as explored 2 S is a priority queue 3 for every edge (v,w) incident to v 4 do Insert(S,(v,w),w(v,w)) 5 while S is not empty 6 do (x, y) Delete-Min(S) 7 if (x,y) is not explored 8 then if y is not explored 9 then Mark (x,y) as a tree edge 10 Mark y as explored 11 Let dist(v,y) be the priority of edge (x,y) 12 for every edge (y,z) incident to y 13 do Insert(S,(y,z),dist(s,y)+w(y,z)) 14 else Mark (x,y) as a non-tree edge
32 CSci 3110 Graph Algorithms 18/41 Graph Exploration Variants (4): Prim s Algorithm PRIM-FROM-VERTEX(G, v) 1 Mark v as explored 2 S is a priority queue 3 for every edge (v,w) incident to v 4 do Insert(S,(v,w),w(v,w)) 5 while S is not empty 6 do (x, y) Delete-Min(S) 7 if (x,y) is not explored 8 then if y is not explored 9 then Mark (x,y) as a tree edge 10 Mark y as explored 11 for every edge (y,z) incident to y 12 do Insert(S,(y, z), w(y, z)) 13 else Mark (x,y) as a non-tree edge
33 CSci 3110 Graph Algorithms 19/41 Graph Exploration Summary Depth-first search, breadth-first search, Dijkstra s algorithm, and Prim s algorithm are variants of the same graph exploration procedure: Maintain a set of explored vertices. Grow this set by exploring edges incident to these vertices. What differs is the order in which edges are explored: DFS: Choose the edge whose source vertex has been discovered most recently. BFS: Choose the edge whose source vertex has been discovered first. Dijkstra: Choose the edge whose target is unexplored and has the minimum tentative distance among all unexplored vertices. Prim: Choose the edge whose target is unexplored and which has minimum weight.
34 CSci 3110 Graph Algorithms 20/41 Breadth-First Search in Undirected Graphs Lemma: Let s be a vertex of G, let T be a BFS-tree rooted at s, and let (u,v) be an edge of G. Then dist T (s,u) dist T (s,v) 1. In other words, u and v are on the same level or on adjacent levels of G.
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36 CSci 3110 Graph Algorithms 21/41 Testing Bipartiteness (1) Problem: Given a graph G, decide whether it is bipartite.
37 CSci 3110 Graph Algorithms 21/41 Testing Bipartiteness (1) Problem: Given a graph G, decide whether it is bipartite.
38 CSci 3110 Graph Algorithms 21/41 Testing Bipartiteness (1) Problem: Given a graph G, decide whether it is bipartite. bipartite non-bipartite
39 CSci 3110 Graph Algorithms 21/41 Testing Bipartiteness (1) Problem: Given a graph G, decide whether it is bipartite. bipartite non-bipartite Lemma: A graph is bipartite if and only if it does not contain an odd cycle.
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41 CSci 3110 Graph Algorithms 22/41 Testing Bipartiteness (2) Lemma: A graph G is bipartite if and only if there are no two adjacent vertices that are on the same level in a BFS-tree of G.
42 CSci 3110 Graph Algorithms 22/41 Testing Bipartiteness (2) Lemma: A graph G is bipartite if and only if there are no two adjacent vertices that are on the same level in a BFS-tree of G. If:
43 CSci 3110 Graph Algorithms 22/41 Testing Bipartiteness (2) Lemma: A graph G is bipartite if and only if there are no two adjacent vertices that are on the same level in a BFS-tree of G. If: Only if:
44 CSci 3110 Graph Algorithms 23/41 Testing Bipartiteness (3) TEST-BIPARTITENESS(G) 1 Run BFS on G to label all vertices with their distance d(v) from some vertex s 2 for every edge (v,w) of G 3 do if d(v) = d(w) 4 then Report that G is not bipartite and exit 5 X {v G d(v) is odd} 6 Y {v G d(v) is even}
45 CSci 3110 Graph Algorithms 23/41 Testing Bipartiteness (3) TEST-BIPARTITENESS(G) 1 Run BFS on G to label all vertices with their distance d(v) from some vertex s 2 for every edge (v,w) of G 3 do if d(v) = d(w) 4 then Report that G is not bipartite and exit 5 X {v G d(v) is odd} 6 Y {v G d(v) is even} Lemma: Given a graph G, one can decide in O(n+m) time whether G is bipartite.
46 CSci 3110 Graph Algorithms 24/41 A Recursive Depth-First Search Procedure DFS(G) 1 Mark every vertex and every edge of G as unexplored 2 for every vertex v of G 3 do if v is unexplored 4 then DFS-from-Vertex(G, v) DFS-FROM-VERTEX(G, v) 1 Mark v as explored 2 for every out-edge (v,w) of v 3 do if w is unexplored 4 then Mark (v,w) as a tree edge 5 DFS-from-Vertex(G, w) 6 else Mark (v,w) as a non-tree edge
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48 CSci 3110 Graph Algorithms 25/41 Depth-First Search in Undirected Graphs Lemma: For every non-tree edge (u,v) in an undirected graph G w.r.t. a DFS-tree T of G, either u is an ancestor of v or vice versa; that is, there are no cross edges.
49 CSci 3110 Graph Algorithms 26/41 Connected Components and Depth-First Search Lemma: Every call to DFS-from-Vertex in Line 4 of Procedure DFS completely explores a connected component of G. v x y w v x y w We do not explore more. We do not explore less.
50 CSci 3110 Graph Algorithms 27/41 Computing Connected Components CONNECTED-COMPONENTS(G) 1 c 0 2 Mark every vertex and every edge of G as unexplored 3 for every vertex v of G 4 do if v is unexplored 5 then c c+1 6 Label-Component-from-Vertex(G, v, c) LABEL-COMPONENT-FROM-VERTEX(G, v, c) 1 Mark v as explored 2 component(v) c 3 for every out-edge (v,w) of v 4 do if w is unexplored 5 then Mark (v,w) as a tree edge 6 Label-Component-from-Vertex(G, w, c) 7 else Mark (v,w) as a non-tree edge
51 CSci 3110 Graph Algorithms 27/41 Computing Connected Components CONNECTED-COMPONENTS(G) 1 c 0 2 Mark every vertex and every edge of G as unexplored 3 for every vertex v of G 4 do if v is unexplored 5 then c c+1 6 Label-Component-from-Vertex(G, v, c) LABEL-COMPONENT-FROM-VERTEX(G, v, c) 1 Mark v as explored 2 component(v) c 3 for every out-edge (v,w) of v 4 do if w is unexplored 5 then Mark (v,w) as a tree edge 6 Label-Component-from-Vertex(G, w, c) 7 else Mark (v,w) as a non-tree edge Lemma: The connected components of a graph with n vertices and m edges can be computed in O(n+m) time.
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53 CSci 3110 Graph Algorithms 28/41 Topological Sorting (1) Lemma: When depth-first search backtracks from a vertex v, all outneighbours of v are explored. explored source current vertex unexplored
54 CSci 3110 Graph Algorithms 29/41 Topological Sorting (2) TOP-SORT(G) 1 c n 2 Mark every vertex and every edge of G as unexplored 3 for every vertex v of G 4 do if v is unexplored 5 then c Label-Vertex(G, v, c) LABEL-VERTEX(G, v, c) 1 Mark v as explored 2 for every out-edge (v,w) of v 3 do if w is unexplored 4 then c Label-Vertex(G, w, c) 5 number(v) c 6 c c 1 7 return c
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56 CSci 3110 Graph Algorithms 29/41 Topological Sorting (2) TOP-SORT(G) 1 c n 2 Mark every vertex and every edge of G as unexplored 3 for every vertex v of G 4 do if v is unexplored 5 then c Label-Vertex(G, v, c) LABEL-VERTEX(G, v, c) 1 Mark v as explored 2 for every out-edge (v,w) of v 3 do if w is unexplored 4 then c Label-Vertex(G, w, c) 5 number(v) c 6 c c 1 7 return c Lemma: A directed acyclic graph can be topologically sorted in O(n+m) time.
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58 CSci 3110 Graph Algorithms 30/41 Detecting Cycles (1) Lemma: If graph G contains a directed cycle C, the vertices of C are discovered after the first vertex v in C is discovered and before the recursive call DFS-from-Vertex(G, v) returns. s v w
59 CSci 3110 Graph Algorithms 30/41 Detecting Cycles (1) Lemma: If graph G contains a directed cycle C, the vertices of C are discovered after the first vertex v in C is discovered and before the recursive call DFS-from-Vertex(G, v) returns. s v w For procedure Top-Sort, this means that an out-edge of the last vertex w on the cycle leads to a discovered, but unnumbered vertex, namely v. The vertices in C are the vertices on the call stack between v and w.
60 CSci 3110 Graph Algorithms 31/41 Detecting Cycles (2) DETECT-CYCLE-FROM-VERTEX(G, v) 1 Mark v as visited 2 for every out-edge (v,w) of v 3 do if w is unvisited 4 then u Detect-Cycle-from-Vertex(G, w) 5 if u is not nil 6 then Output v 7 if u = v 8 then exit 9 else return u 10 else if w is not finished 11 then Output v 12 return w 13 Mark v as finished 14 return nil
61 CSci 3110 Graph Algorithms 31/41 Detecting Cycles (2) DETECT-CYCLE-FROM-VERTEX(G, v) 1 Mark v as visited 2 for every out-edge (v,w) of v 3 do if w is unvisited 4 then u Detect-Cycle-from-Vertex(G, w) 5 if u is not nil 6 then Output v 7 if u = v 8 then exit 9 else return u 10 else if w is not finished 11 then Output v 12 return w 13 Mark v as finished 14 return nil Lemma: One can test in O(n+m) time whether a given directed graph G contains a directed cycle and, if so, output such a cycle.
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63 CSci 3110 Graph Algorithms 32/41 Strongly Connected Components A directed graph G is strongly connected if, for any two vertices v and w in G, there exists a path from v to w. The strongly connected components of a directed graph G are the maximal strongly connected subgraphs of G. Alternative formulation: For any two vertices v and w in a strongly connected component, there exists a directed cycle that contains v and w.
64 CSci 3110 Graph Algorithms 32/41 Strongly Connected Components A directed graph G is strongly connected if, for any two vertices v and w in G, there exists a path from v to w. The strongly connected components of a directed graph G are the maximal strongly connected subgraphs of G. Alternative formulation: For any two vertices v and w in a strongly connected component, there exists a directed cycle that contains v and w.
65 CSci 3110 Graph Algorithms 33/41 A Strong Connectivity Algorithm (1) Maintain partition of vertices into three sets: Finished Live Unexplored Finished + live components are strongly connected components of the graph defined by explored edges. Finished components are strongly connected components of G. Live components form a path and can merge into larger components as more edges are discovered.
66 CSci 3110 Graph Algorithms 34/41 Updating the Partition (1) Explore edges out of last live component. Three cases, depending on location of target of the edge: Target is finished. Target is unexplored. Target is live. When last live component has no unexplored out-edges, mark it as finished and continue processing the previous live component. Finished Live Unexplored
67 CSci 3110 Graph Algorithms 35/41 Updating the Partition (2) Case 1: Edge to finished vertex Finished Live Unexplored
68 CSci 3110 Graph Algorithms 35/41 Updating the Partition (2) Case 1: Edge to finished vertex Finished Live Unexplored Do nothing.
69 CSci 3110 Graph Algorithms 36/41 Updating the Partition (3) Case 2: Edge to unexplored vertex Finished Live Unexplored
70 CSci 3110 Graph Algorithms 36/41 Updating the Partition (3) Case 2: Edge to unexplored vertex Finished Live Unexplored Finished Live Unexplored
71 CSci 3110 Graph Algorithms 37/41 Case 3: Edge to live vertex Updating the Partition (4) Finished Live Unexplored
72 CSci 3110 Graph Algorithms 37/41 Case 3: Edge to live vertex Updating the Partition (4) Finished Live Unexplored Finished Live Unexplored
73 CSci 3110 Graph Algorithms 38/41 Representation of Live Components Two stacks: Vertex stack S: Contains vertices in order of discovery, numbered in order of discovery Component stack C: Contains one entry per component, the number of the first vertex in this component Finished Live Unexplored S C
74 CSci 3110 Graph Algorithms 39/41 A Strong Connectivity Algorithm (2) LABEL-COMPONENTS-FROM-VERTEX(G, v, c, S, C) 1 c c+1 2 Label v as live 3 number(v) c 4 Push(S, v) 5 Push(C, c) 6 for every out-edge (v,w) of v 7 do if w is unexplored 8 then Label-Components-from-Vertex(G, w, c, S, C) 9 else if w is live 10 then repeat c Pop(C) 11 until c number(w) 12 Push(C,c ) 13 c Pop(C) 14 if number(v) = c 15 then repeat w Pop(S) 16 Mark w as finished 17 number(w) c 18 until w = v 19 else Push(C,c )
75 CSci 3110 Graph Algorithms 40/41 A Strong Connectivity Algorithm (3) Lemma: The strongly connected components of a directed graph G can be computed in O(n+m) time.
76 CSci 3110 Graph Algorithms 41/41 Summary Graphs are fundamental in computer science: Many problems are quite natural to express as graph problems Matching problems Scheduling problems... Data structures are graphs whose nodes store useful information Graph exploration lets us learn the structure of a graph: Connectivity properties Distances between vertices Planarity...
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