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Graphs and Sets Dr. Andrew Wallace PhD BEng(hons) EurIng andrew.wallace@cs.umu.se Overview • Sets • Implementation • Complexity • Graphs • Constructing Graphs • Graph examples Sets • Collection of items • No specified ordered • Unique values • Implementation of mathematical concept of finite set • Static of dynamic Sets • Items in a set are members of the set • 𝒙∈𝑿 • No sets of sets but … • Subsets • 𝑨⊆𝑨 • Union of sets • 𝑨 ∪𝑩 • Intersections • 𝑨 ∩𝑩 Sets • Examples: • • • • int, float, char Arrays Functions Objects (struct) Set operations • Create • Insert • Remove • Is member of • Is empty • Select • Size of • Enumerate Implementation • Simple • Array • List • Efficient • Trees • Radix trees • Hash tables Implementation • Insert • Check for duplicates • Union • If list has duplicates • Check when doing • • • • Equal Remove Intersection Difference Implementation • Bit vector • 0 or 1 • Set the bit if the item is in the queue • Faster operations • Bit operations in hardware (bitwise AND or OR) • Masking • 0010110101 AND 0010000000 • Minimise memory 𝒏 • 𝒘 Implementation • Bit field struct bField { int int int int } m_nVal1 m_nVal2 m_nVal3 : 6; : 3; : 4; : 6; Implementation • Limitations • Can’t use bit field variables in an array • Can’t take the memory address of a bit field variable • Can’t overlap integer boundaries Implementation • Priority Queues • Linux kernel • Caching algorithms / memory pages Complexity • Depends on implementation • Improve set operations such as union or intersection • Improve insert, search, remove • O(n) or O(logn) • Some set operations can take O(m*n) Graphs • Set of nodes or vertices + pairs of nodes • G = (V, A) a • Directed or undirected b • Undirected a to b is the same as b to a • Node - undirected • Vertices - directed • Edge • Arcs (directed) • Connection between nodes • Weighted or unweighted c d Graphs a b • V = {a, b, c, d} • A = {(a, b), (a, c), (b, d), (c, b)} • Adjacency • 2 edges are adjacent if the share a common vertex • (a, c) and (a, b) • 2 vertices are adjacent if they share a common edge • a and c • Join • Incident • An edge and vertex on the edge c d Graphs struct Node { int Node* int }; nNodeID; pOut; nOut; Graphs struct Edge { int int int }; nEdgeID; nStart; nEnd; Graphs • Trivial graph • One vertex • Edgeless graph • Vertices and no edges • Null graph • Empty set Graphs a b • Paths • Sequence of nodes • {a, b, d} • Simple path • No repetition of nodes • Cyclic path • Around and around in circles! • Walk • Open walk = path • Closed walk = cyclic path • Trail = walk with unique edges c d Graphs d a b • Connected graph • All nodes have a path to all other nodes • Sub graphs • Vertices are a sub set of G • Adjacency relationship are a subset of G’s and restricted to the subgraph • Complete graph • All nodes connected to all other nodes • Undirected graph A = n(n-1)/2 • If A < n-1, then the graph is not connected c Graphs • Weighted graphs • • • • Maps Places as node Roads as arcs Distances as weights on the edges a 10 Graphs • Weights can represent “cost” • Some algorithms require restrictions on weights • Rational numbers or integers • All positive integers • Weight of a path • Sum of the weights for a given path b 20 13 23 c d Constructing Graphs a b • Adjacency lists • An array of arrays of adjacent vertices a c b c b d c b d d Constructing Graphs int** pNodeArray; pNodeArray = (int**)malloc(sizeof(int) * 10); for(i=0;i<10;i++) { pNodeArray[i] = malloc(sizeof(int) * NumNodesOut[i]); } pNodeArray[i][j] = nNodeOut; Constructing Graphs a b • Adjacency matrix • Node x node matrix (n x n) c 1 1 1 1 d Constructing Graphs a b • Undirected graph • symmetrical c 1 1 1 1 1 1 1 1 d Constructing Graphs int** pNodeArray; pNodeArray = (int**)malloc(sizeof(int) * 10); for(i=0;i<10;i++) { pNodeArray[i] = malloc(sizeof(int) * 10); } pNodeArray[i][j] = 1; Graph examples • Robot navigation • AGV (automatic Guided Vehicles) • Free space paths • Pick up and drop off points • Map as a graph Graph example Graph example • Computer Networks Questions?