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Graphs, Trees, Paths and Their Representations
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Graphs can be weighted/unweighted, directed, undirected. Lecture 2 Slides - Advanced Data Structures - GWU

Graphs can be weighted/unweighted, directed, undirected. Lecture 2 Slides - Advanced Data Structures - GWU

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Graphs, Trees, Paths and Their Representations Graphs, Trees, Paths and Their Representations Presentation Transcript

  • CS 6213 – Advanced Data Structures Lecture 2 GRAPHS AND THEIR REPRESENTATION TREES, PATHS, CYCLES TOPOLOGICAL SORTING
  •  Instructor Prof. Amrinder Arora amrinder@gwu.edu Please copy TA on emails Please feel free to call as well  TA Iswarya Parupudi iswarya2291@gwmail.gwu.edu CS 6213 – Arora – L2 Advanced Data Structures - Graphs 2 LOGISTICS
  • L4 - BTrees CS 6213 - Advanced Data Structures - Arora 3 CS 6213 Basics Record / Struct Arrays / Linked Lists / Stacks / Queues Graphs / Trees / BSTs Advanced Trie, B-Tree Splay Trees R-Trees Heaps and PQs Union Find
  •  Graphs – Basics  Degrees, Number of Edges, Min/Max Degree  Kinds of Graphs  How to Represent in Data Structures  Trees, Paths, Cycles  Journeys  Topological Sorting, DAGs CS 6213 – Arora – L2 Advanced Data Structures - Graphs 4 AGENDA
  •  A graph G=(V,E) consists of a finite set V, which is the set of vertices, and set E, which is the set of edges. Each edge in E connects two vertices v1 and v2, which are in V.  Can be directed or undirected Not to be confused with a bar graph!!!!  CS 6213 – Arora – L2 Advanced Data Structures - Graphs 5 GRAPH
  • A core data structure that shows up in many circumstances:  Transportation and Logistics (paths, etc.)  Circuit Design  Social Networking – Model connections between people  Sociology – Model influence  Zoology and Wildlife  Project Task Management  Job Scheduling and Resource Assignment (matching)  Time table scheduling  Task parallelization (graph coloring) CS 6213 – Arora – L2 Advanced Data Structures - Graphs 6 GRAPH – APPLICATIONS
  •  (Undirected) Degree of a node  (Directed) Indegree / Outdegree  Min degree in a graph:   Max degree in a graph:   Basic observations  (Undirected) Sum of degrees = 2 x number of edges  (Directed) Sum of indegree = Sum of outdegree = number of edges CS 6213 – Arora – L2 Advanced Data Structures - Graphs 7 DEGREE
  •  If (x,y) is an edge, then x is said to be adjacent to y, and y is adjacent from x.  In the case of undirected graphs, if (x,y) is an edge, we just say that x and y are adjacent (or x is adjacent to y, or y is adjacent to x). Also, we say that x is the neighbor of y.  The indegree of a node x is the number of nodes adjacent to x  The outdegree of a node x is the number of nodes adjacent from x  The degree of a node x in an undirected graph is the number of neighbors of x  A path from a node x to a node y in a graph is a sequence of node x, x1,x2,...,xn,y, such that x is adjacent to x1, x1 is adjacent to x2, ..., and xn is adjacent to y.  The length of a path is the number of its edges.  A cycle is a path that begins and ends at the same node  The distance from node x to node y is the length of the shortest path from x to y. CS 6213 – Arora – L2 Advanced Data Structures - Graphs 8 GRAPH DEFINITIONS
  •  Using a matrix A[1..n,1..n] where A[i,j] = 1 if (i,j) is an edge, and is 0 otherwise. This representation is called the adjacency matrix representation. If the graph is undirected, then the adjacency matrix is symmetric about the main diagonal.  Using an array Adj[1..n] of pointers, which Adj[i] is a linked list of nodes which are adjacent to i.  The matrix representation requires more memory, since it has a matrix cell for each possible edge, whether that edge exists or not. In adjacency list representation, the space used is directly proportional to the number of edges.  If the graph is sparse (very few edges), then adjacency list may be a more efficient choice. CS 6213 – Arora – L2 Advanced Data Structures - Graphs 9 GRAPH REPRESENTATIONS
  •  A very practical choice is to use graphing libraries, such as:  JGraphT (Java)  Boost (C++)  GraphStream (Java)  JUNG (Java) CS 6213 – Arora – L2 Advanced Data Structures - Graphs 10 GRAPH REPRESENTATION (CONT.)
  •  Graphs can be characterized in many ways. Two important ones being:  Directed or Undirected  Weighted or Unweighted  Both Adjacency Matrix (AM) and Adjacency List (AL) representations can be used for graphs – weighted or unweighted, directed or undirected.  A[i,j] = A[j,i] if graph is undirected. So, we could decide to use just the upper triangle.  If graph is weighted, in adjacency list, we can also store the weight. AL[i] = [(j,w(i,j)), (k,w(i,k)), …] CS 6213 – Arora – L2 Advanced Data Structures - Graphs 11 GRAPH CHARACTERIZATIONS
  •  A tree is a connected acyclic graph (i.e., it has no cycles)  Rooted tree: A tree in which one node is designated as a root (the top node) CS 6213 – Arora – L2 Advanced Data Structures - Graphs 12 TREE Example: Node A is root node F and D are child nodes of A. P and Q are child nodes of J. Etc.
  •  Definitions  Leaf is a node that has no children  Ancestors of a node x are all the nodes on the path from x to the root, including x and the root  Subtree rooted at x is the tree consisting of x, its children and their children, and so on and so forth all the way down  Height of a tree is the maximum distance from the root to any node CS 6213 – Arora – L2 Advanced Data Structures - Graphs 13 TREE (CONT.)
  • Option 1 •Trees are graphs, so we can use standard graph representation – Adjacency Matrix or Adjacency List Option 2 •Use parent node and list for child nodes Option 3 •Use parent node, and two pointers – one for first child and the other for nextSibling CS 6213 – Arora – L2 Advanced Data Structures - Graphs 14 REPRESENTING TREES 3 Basic Options
  •  We can use standard graph representation – Adjacency Matrix or Adjacency List  These options are overkill for trees, but in some instances, the graph is not known to be a tree beforehand, so this option works. CS 6213 – Arora – L2 Advanced Data Structures - Graphs 15 TREE REPRESENTATION – OPTION 1
  •  Rather than using Adjacency Matrix or Adjacency List representation, we can use a simpler representation  Each node has a pointer to parent, and a linked list of child nodes  The root node’s parent pointer is null  This representation works for “rooted” trees. If the tree is not rooted, we can designate one node as root. CS 6213 – Arora – L2 Advanced Data Structures - Graphs 16 TREE REPRESENTATION – OPTION 2 Node { Node parent, List<Node> childNodes }
  •  Another representation for Trees: Left Child, Right Sibling representation for a tree. Each node has 3 pointers:  Parent – Points to parent (null if this is the root node)  Left pointer – Points to first child (null if this is a leaf node)  Right pointer – Points to right sibling (null if no more siblings)  For example:  is represented by CS 6213 – Arora – L2 Advanced Data Structures - Graphs 17 TREE REPRESENTATION – OPTION 3 Node { Node parent, Node firstChild, Node nextSibling }
  •  When updating values in a tree, such as the size of the subtree rooted at a node, the weight of the subtree rooted at a node, etc, there are two main methods:  Recompute whenever there is a modify operation (addition of a node, deletion of a node, changing the weight of a node, etc). Recomputations usually only need to propagate from the change node upwards to the root. [Advantage: Values always up to date, Disadvantages: Lot of time spent in Recompute, Method cannot be run concurrently.]  Use a dirty flag and set to true when there is a change. Set dirty flag to true for all ancestors (navigate to parent, until the root). Recompute when there is a need, or as per a schedule. [Advantages: Efficient, Methods (other than the “recomputed” operation can be run concurrently. Disadvantages: Values are out of date at times.] CS 6213 – Arora – L2 Advanced Data Structures - Graphs 18 UPDATING VALUES
  •  Able to hold the entire graph in memory?  If your graph is large and changing fast, like the Facebook interconnection graph, you simply cannot hold it in memory using traditional methods. You need to replicate it across multiple servers and use reliable services to get partial data out from the graph. Your graph may simply be backed by a database with which you interact directly (without ever loading a complete “graph” object.) CS 6213 – Arora – L2 Advanced Data Structures - Graphs 19 LARGE GRAPHS
  •  Given each of the graph representations, how do we find a path?  Shortest path algorithms  Dijkstra  All Pairs Shortest Paths  [Refer to CS 6212 Notes for details] CS 6213 – Arora – L2 Advanced Data Structures - Graphs 20 PATHS
  •  Path, spread over time  Useful concept if the graph changes over time  Specifically, consider this scenario:  Edge e1 existed from x to y at time t1  Edge e2 existed from y to z at time t2  t1 < t2  Then, we say that there exists a journey from x to z CS 6213 – Arora – L2 Advanced Data Structures - Graphs 21 JOURNEY
  •  How can we detect cycles in a graph? CS 6213 – Arora – L2 Advanced Data Structures - Graphs 22 CYCLES
  •  Assuming a graph is a Directed Acyclic Graph, topological ordering can be produced in linear time. O(n + m) CS 6213 – Arora – L2 Advanced Data Structures - Graphs 23 TOPOLOGICAL SORTING
  •  Graphs are important data structures with numerous applications  Graphs can be of different kinds  Many convenient ways to model graphs  Adjacency Matrix  Adjacency List  Special structures for trees  Many commercial and open source libraries exist, such as JGraphT CS 6213 – Arora – L2 Advanced Data Structures - Graphs 24 SUMMARY