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Chapter 20: Graphs CS 302 - Data Structures Mehmet H Gunes Modified from authors’ slides

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What is a graph? A data structure that consists of a set of nodes (vertices) and a set of edges between the vertices. The set of edges describes relationships among the vertices. 1 2 34

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Terminology Definition: – A set of points that are joined by lines Graphs also represent the relationships among data items G = { V, E } – a graph is a set of vertices and edges A subgraph consists of a subset of a graph’s vertices and a subset of its edges Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Formally a graph G is defined as follows: G = (V,E) where – V(G) is a finite, nonempty set of vertices – E(G) is a set of edges written as pairs of vertices

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An undirected graph The order of vertices in E is not important for undirected graphs!! A graph in which the edges have no direction

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A directed graph A graph in which each edge is directed from one vertex to another (or the same) vertex The order of vertices in E is important for directed graphs!!

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A directed graph Trees are special cases of graphs!

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Terminology Undirected graphs: edges do not indicate a direction Directed graph, or digraph: each edge has a direction Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Terminology (a) A campus map as a graph; (b) a subgraph Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Terminology Graphs that are (a) connected; (b) disconnected; and (c) complete Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Terminology (a) A multigraph is not a simple graph; (b) a self edge is not allowed in a simple graph Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Path: A sequence of vertices that connects two nodes in a graph The length of a path is the number of edges in the path. e.g., a path from 1 to 4 1 2 34 Graph terminology

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Terminology Simple path: passes through vertex only once Cycle: a path that begins and ends at same vertex Simple cycle: cycle that does not pass through other vertices more than once Connected graph: each pair of distinct vertices has a path between them Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Graph terminology Complete graph: A graph in which every vertex is directly connected to every other vertex

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Terminology Complete graph: each pair of distinct vertices has an edge between them Graph cannot have duplicate edges between vertices – Multigraph: does allow multiple edges When labels represent numeric values, graph is called a weighted graph Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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What is the number of edges E in a complete undirected graph with V vertices? E=V* (V-1) / 2 Graph terminology (cont.) or O(V 2 )

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What is the number of edges E in a complete directed graph with V vertices? E=V * (V-1) Graph terminology (cont.) or O(V 2 )

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A weighted graph Weighted graph: A graph in which each edge carries a value 18

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Terminology (a) a weighted graph; (b) a directed graph Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Graphs as ADTs ADT graph operations – Test whether graph is empty. – Get number of vertices in a graph. – Get number of edges in a graph. – See whether edge exists between two given vertices. – Insert vertex in graph whose vertices have distinct values that differ from new vertex’s value. – Insert edge between two given vertices in graph. Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Graphs as ADTs ADT graph operations, ctd. – Remove specified vertex from graph and any edges between the vertex and other vertices. – Remove edge between two vertices in graph. – Retrieve from graph vertex that contains given value. View interface for undirected, connected graphs, Listing 20-1Listing 20-1 Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Array-Based Implementation Use a 1D array to represent the vertices Use a 2D array (i.e., adjacency matrix) to represent the edges Adjacency Matrix: – for a graph with N nodes, an N by N table that shows the existence (and weights) of all edges in the graph

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to node x ? from node x ? Adjacency Matrix for Flight Connections

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Array-Based Implementation (cont.) Memory required – O(V+V 2 )=O(V 2 ) Preferred when – The graph is dense: E = O(V 2 ) Advantage – Can quickly determine if there is an edge between two vertices Disadvantage – Consumes significant memory for sparse large graphs

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Linked Implementation Use a 1D array to represent the vertices Use a list for each vertex v which contains the vertices which are adjacent from v (i.e., adjacency list) Adjacency List: – A linked list that identifies all the vertices to which a particular vertex is connected; each vertex has its own adjacency list

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Adjacency List Representation of Graphs to node x ? from node x ?

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Link-List-based Implementation (cont.) Memory required – O(V + E) Preferred when – for sparse graphs: E = O(V) Disadvantage – No quick way to determine the vertices adjacent to a given vertex Advantage – Can quickly determine the vertices adjacent from a given vertex O(V) for sparse graphs since E=O(V) O(V 2 ) for dense graphs since E=O(V 2 )

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Implementing Graphs (a) A directed graph and (b) its adjacency matrix Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Implementing Graphs (a) A weighted undirected graph and (b) its adjacency matrix Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Implementing Graphs (a) A directed graph and (b) its adjacency list Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Implementing Graphs (a) A weighted undirected graph and (b) its adjacency list Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Graph Traversals Visits all of the vertices that it can reach – Happens if graph is connected Connected component is subset of vertices visited during traversal that begins at given vertex Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Graph searching Problem: find if there is a path between two vertices of the graph – e.g., Austin and Washington Methods: Depth-First-Search (DFS) or Breadth-First-Search (BFS)

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Depth-First-Search (DFS) Main idea: – Travel as far as you can down a path – Back up as little as possible when you reach a "dead end“ i.e., next vertex has been "marked" or there is no next vertex startVertex endVertex

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Depth First Search: Follow Down DFS uses Stack ! 2 1 3

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startVertex endVertex (initialization) 36

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endVertex 38

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Depth-First Search Goes as far as possible from a vertex before backing up Recursive algorithm Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Depth-First Search Iterative algorithm, using a stack Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Depth-First Search Iterative algorithm, using a stack, ctd. Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Depth-First Search Visitation order for (a) a depth-first search; (b) a breadth-first search Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Depth-First Search A connected graph with cycles Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Depth-First Search The results of a depth-first traversal, beginning at vertex a, of the graph Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Breadth-First-Searching (BFS) Main idea: – Look at all possible paths at the same depth before you go at a deeper level – Back up as far as possible when you reach a "dead end“ i.e., next vertex has been "marked" or there is no next vertex startVertex endVertex

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Breadth First: Follow Across BFS uses Queue ! 1 2 3 4 5 6 7 8

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Breadth First Uses Queue

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startVertexendVertex (initialization) 48

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Breadth-First Search Visits all vertices adjacent to vertex before going forward Breadth-first search uses a queue Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Breadth-First Search The results of a breadth-fi rst traversal, beginning at vertex a, of the graph Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Applications of Graphs A directed graph without cycles Topological Sorting Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Applications of Graphs The graph arranged according to the topological orders (a) a, g, d, b, e, c, f and (b) a, b, g, d, e, f, c Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Applications of Graphs Topological sorting algorithm Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Applications of Graphs A trace of topSort1 for the graph Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Applications of Graphs A trace of topSort1 for the graph Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Applications of Graphs A trace of topSort1 for the graph Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Applications of Graphs A trace of topSort1 for the graph Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Applications of Graphs A trace of topSort2 for the graph Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Spanning Trees Tree: an undirected connected graph without cycles Observations about undirected graphs 1.Connected undirected graph with n vertices must have at least n – 1 edges. 2.Connected undirected graph with n vertices and exactly n – 1 edges cannot contain a cycle 3.A connected undirected graph with n vertices and more than n – 1 edges must contain at least one cycle Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Spanning Trees The DFS spanning tree rooted at vertex a for the graph Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Spanning Trees DFS spanning tree algorithm Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Spanning Trees BFS spanning tree algorithm Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Spanning Trees The BFS spanning tree rooted at vertex a for the graph Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Minimum Spanning Trees A weighted, connected, undirected graph Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Minimum Spanning Trees Minimum spanning tree algorithm Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Minimum Spanning Trees A trace of primsAlgorithm for the graph, beginning at vertex a Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Minimum Spanning Trees A trace of primsAlgorithm for the graph, beginning at vertex a Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Minimum Spanning Trees A trace of primsAlgorithm for the graph, beginning at vertex a Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Graph Algorithms Depth-first search – Visit all the nodes in a branch to its deepest point before moving up Breadth-first search – Visit all the nodes on one level before going to the next level Single-source shortest-path – Determines the shortest path from a designated starting node to every other node in the graph 71

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Single Source Shortest Path 72

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Single Source Shortest Path What does “shortest” mean? What data structure should you use? 73

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Shortest-path problem There might be multiple paths from a source vertex to a destination vertex Shortest path: the path whose total weight (i.e., sum of edge weights) is minimum Austin Houston Atlanta Washington: 1560 miles Austin Dallas Denver Atlanta Washington: 2980 miles 74

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Variants of Shortest Path Single-pair shortest path – Find a shortest path from u to v for given vertices u and v Single-source shortest paths – G = (V, E) find a shortest path from a given source vertex s to each vertex v V 75

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Variants of Shortest Paths (cont’d) Single-destination shortest paths – Find a shortest path to a given destination vertex t from each vertex v – Reversing the direction of each edge single-source All-pairs shortest paths – Find a shortest path from u to v for every pair of vertices u and v 76

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Shortest Paths Shortest path between two vertices in a weighted graph has smallest edge-weight sum (a) A weighted directed graph and (b) its adjacency matrix Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Weight of path p = v 0, v 1,..., v k Shortest-path weight from s to v : min w(p) : s v if there exists a path from s to v ∞ otherwise Notation 78 0 39 5 11 3 6 5 7 6 s tx yz 2 2 1 4 3 p δ(v) =

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Negative Weights and Negative Cycles Negative-weight edges may form negative-weight cycles. If negative cycles are reachable from the source, the shortest path is not well defined. – i.e., keep going around the cycle, and get w(s, v) = - for all v on the cycle 0 3 -4 2 8 -6 s ab ef -3 3 5 6 4 7 c d g 79

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Could shortest path solutions contain cycles? Negative-weight cycles – Shortest path is not well defined Positive-weight cycles: – By removing the cycle, we can get a shorter path Zero-weight cycles – No reason to use them; can remove them to obtain a path with same weight 80

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Shortest-path algorithms Solving the shortest path problem in a brute-force manner requires enumerating all possible paths. – There are O(V!) paths between a pair of vertices in an acyclic graph containing V nodes. We will discuss two algorithms – Dijkstra’s algorithm – Bellman-Ford’s algorithm 81

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Shortest-path algorithms Dijkstra’s and Bellman-Ford’s algorithms are “greedy” algorithms! – Find a “globally” optimal solution by making “locally” optimum decisions. Dijkstra’s algorithm – Does not handle negative weights. Bellman-Ford’s algorithm – Handles negative weights but not negative cycles reachable from the source. 82

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Shortest-path algorithms (cont’d) Both Dijkstra’s and Bellman-Ford’s algorithms are iterative: – Start with a shortest path estimate for every vertex: d[v] – Estimates are updated iteratively until convergence: d[v] δ(v) 83

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Shortest-path algorithms (cont’d) Two common steps: (1) Initialization (2) Relaxation (i.e., update step) 84

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0 6 5 7 7 9 s tx 8 -3 2 -4 -2 Initialization Step Set d[s]=0 – i.e., source vertex Set d[v]=∞ for – i.e., large value 85

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Relaxing an edge (u, v) implies testing whether we can improve the shortest path to v found so far by going through u: If d[v] > d[u] + w(u, v) we can improve the shortest path to v d[v]=d[u]+w(u,v) Relaxation Step 59 2 uv 57 2 uv RELAX(u, v, w) 56 2 uv 56 2 uv ss no change 86

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Bellman-Ford Algorithm Can handle negative weights. Detects negative cycles reachable from the source. Returns FALSE if negative-weight cycles are reachable from the source s no solution 87

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Bellman-Ford Algorithm (cont’d) Each edge is relaxed |V–1| times by making |V-1| passes over the whole edge set – to make sure that each edge is relaxed exactly |V – 1| times it puts the edges in an unordered list and goes over the list |V – 1| times 0 6 5 7 7 9 s tx yz 8 -3 2 -4 -2 (t, x), (t, y), (t, z), (x, t), (y, x), (y, z), (z, x), (z, s), (s, t), (s, y) 88

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Example 0 6 5 7 7 9 s tx yz 8 -3 2 -4 -2 0 6 5 7 7 9 s tx yz 8 -3 2 -4 -2 E: (t, x), (t, y), (t, z), (x, t), (y, x), (y, z), (z, x), (z, s), (s, t), (s, y) 6 7 Pass 1 89

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Example 0 6 7 6 5 7 7 9 s tx yz 8 -3 2 -4 -2 (t, x), (t, y), (t, z), (x, t), (y, x), (y, z), (z, x), (z, s), (s, t), (s, y) 0 6 7 6 5 7 7 9 s tx yz 8 -3 2 -4 -2 11 2 4 0 6 7 6 5 7 7 9 s tx yz 8 -3 2 -4 -2 11 2 4 2 0 6 7 6 5 7 7 9 s tx yz 8 -3 2 -4 -2 11 2 4 2 -2 Pass 1 (from previous slide) Pass 2 Pass 3 Pass 4 90

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Detecting Negative Cycles: needs an extra iteration for each edge (u, v) E do if d[v] > d[u] + w(u, v) then return FALSE return TRUE 0 c s b 2 3 -8 0 c s b 2 3 2 5 -3 2 5 c s b 2 3 -8 2 -6 Consider edge (s, b): d[b] = -1 d[s] + w(s, b) = -4 d[b] > d[s] + w(s, b) d[b]=-4 (d[b] keeps changing!) 1 st pass 2 nd pass (s,b) (b,c) (c,s) 91

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BELLMAN-FORD Algorithm 1. INITIALIZE-SINGLE-SOURCE(V, s) 2. for i ← 1 to |V| - 1 3. for each edge (u, v) E 4. RELAX(u, v, w) 5. for each edge (u, v) E 6. if d[v] > d[u] + w(u, v) 7. return FALSE 8. return TRUE Time: O(V+VE+E)=O(VE) O(V) O(E) O(VE) 92

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Dijkstra’s Algorithm Cannot handle negative-weights! – w(u, v) > 0, (u, v) E Each edge is relaxed only once! 93

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Dijkstra’s Algorithm (cont’d) At each iteration, it maintains two sets of vertices: d[v]=δ (v)d[v]≥δ (v) V SV-S estimates have converged to the shortest path solution estimates have not converged yet Initially, S is empty 94

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Dijkstra’s Algorithm (cont.) Vertices in V–S reside in a min-priority queue Q – Priority of u determined by d[u] – The “highest” priority vertex will be the one having the smallest d[u] value. 95

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Dijkstra (G, w, s) 0 10 1 5 2 s tx yz 2 3 9 7 4 6 0 1 5 2 s tx yz 2 3 9 7 4 6 5 S=<> Q= S= Q= Initialization 96

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Example (cont.) 0 10 5 1 5 2 s tx yz 2 3 9 7 4 6 8 14 7 0 8 5 7 10 1 5 2 s tx yz 2 3 9 7 4 6 13 S= Q= S= Q= 97

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Example (cont.) 0 8 13 57 10 1 5 2 s tx yz 2 3 9 7 4 6 9 0 8 9 57 1 5 2 s tx yz 2 3 9 7 4 6 S= Q= S= Q=<> Note: use back-pointers to recover the shortest path solutions! 98

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Dijkstra (G, w, s) INITIALIZE-SINGLE-SOURCE( V, s ) S ← Q ← V[G] while Q u ← EXTRACT-MIN(Q) S ← S { u } for each vertex v Adj[u] RELAX( u, v, w ) Update Q (DECREASE_KEY) Overall: O(V+2VlogV+(E v1 +E v2 +...)logV) =O(VlogV+ElogV)=O(ElogV) build priority heap O(V logV) O(V) times O(logV) O(E vi ) O(logV) O(V) O(E vi logV) 99

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Improving Dijkstra’s efficiency Suppose the shortest path from s to w is the following: If u is the i-th vertex in this path, it can be shown that d[u] δ (u) at the i-th iteration: – move u from V-S to S – d[u] never changes again w sx u … … 100

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Add a flag for efficiency! INITIALIZE-SINGLE-SOURCE( V, s ) S ← Q ← V[G] while Q u ← EXTRACT-MIN(Q) S ← S { u }; for each vertex v Adj[u] RELAX( u, v, w ) Update Q (DECREASE_KEY) If v not marked mark u 101

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Dijkstra vs Bellman-Ford Bellman-Ford O(VE) Dijkstra O(ElogV) V2V2 V3V3 if G is sparse: E=O(V) if G is dense: E=O(V 2 ) VlogV V 2 logV if G is sparse: E=O(V) if G is dense: E=O(V 2 ) 102

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Shortest Paths Dijkstra’s shortest-path algorithm Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Shortest Paths Dijkstra’s shortest-path algorithm, ctd. Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Shortest Paths A trace of the shortest-path algorithm applied to the graph Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Shortest Paths Checking weight [u] by examining the graph: (a) weight [2] in step 2; (b) weight [1] in step 3; Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Shortest Paths Checking weight [u] by examining the graph: (c) weight [3] in step 3; (d) weight [3] in step 4 Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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weightless BFS can be used to solve the shortest path problem when the graph is weightless or when all the weights are equal. – Path with lowest number of edges i.e., connections Need to “mark” vertices before Enqueue! – i.e., do not allow duplicates Revisiting BFS 108

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Circuits Another name for a type of cycle common in statement of certain problems Circuits either visit every vertex once or visit every edge once An Euler circuit begins at a vertex v, passes through every edge exactly once, and terminates at v Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Circuits (a) Euler’s bridge problem and (b) its multigraph representation Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Circuits Pencil and paper drawings Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Circuits Connected undirected graphs based on the drawings Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Circuits The steps to determine an Euler circuit for the graph Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Circuits The steps to determine an Euler circuit for the graph Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Some Difficult Problems Hamilton circuit – Path that begins at a vertex v, passes through every vertex in the graph exactly once, and terminates at v The traveling salesperson problem – Variation of Hamilton circuit – Involves a weighted graph that represents a road map – Circuit traveled must be the least expensive Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Some Difficult Problems The three utilities problem Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Some Difficult Problems Planar graph – Can draw it in a plane in at least one way so that no two edges cross The four-color problem – Given a planar graph, can you color the vertices so that no adjacent vertices have the same color, if you use at most four colors? Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Some Difficult Problems 1.Describe the graphs in Figure 20-32. For example, are they directed? Connected? Complete? Weighted? 2.Use the depth-first strategy and the breadth- first strategy to traverse the graph in Figure 20-32 a, beginning with vertex 0. List the vertices in the order in which each traversal visits them. Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Some Difficult Problems 3.Write the adjacency matrix for the graph in Figure 20-32 a. 4.Add an edge to the directed graph in Figure 20-14 that runs from vertex d to vertex b. Write all possible topological orders for the vertices in this new graph. 5.Is it possible for a connected undirected graph with fi ve vertices and four edges to contain a simple cycle? Explain. Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Some Difficult Problems 6.Draw the DFS spanning tree whose root is vertex 0 for the graph in Figure 20-33. 7.Draw the minimum spanning tree whose root is vertex 0 for the graph in Figure 20-33. 8.What are the shortest paths from vertex 0 to each vertex of the graph in Figure 20-24 a? (Note the weights of these paths in Figure 20-25.) Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Some Difficult Problems FIGURE 20-32 Graphs for Checkpoint Questions 1, 2, and 3 Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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Some Difficult Problems FIGURE 20-33 A graph for Checkpoint Questions 6 and 7 and for Exercises 1 and 4 Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

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