CS1022 Computer Programming & Principles

1. CS1022Computer Programming & Principles Lecture 7.2 Graphs (2)

2. Plan of lecture • Hamiltonian graphs • Trees • Sorting and searching CS1022

3. Hamiltonian graphs (1) • Euler looked into the problem of using all edges once (visiting vertices as many times as needed) • Interesting related problem: • Find a cycle which passes through all vertices once • A Hamiltonian cycle includes all vertices in a graph • Hamiltonian graphs have Hamiltonian cycles • Useful for planning train timetables • Useful for studying telecommunications W. R. Hamilton CS1022

4. Hamiltonian graphs (2) • Unlike the Eulerian problem, there is no simple rule for detecting Hamiltonian cycles • One of the major unsolved problems in graph theory • Many graphs are Hamiltonian • If each vertex is adjacent (has an edge) to every other vertex, then there is always a Hamiltonian cycle • These are called complete graphs • A complete graph with n vertices is denoted Kn CS1022

5. Hamiltonian graphs (3) • Example: complete graph K5 • A Hamiltonian cycle is a b c d e a • There are several others • Since each vertex is adjacent to every other vertex • We have 4 options to move to the 2nd vertex, then • We have 3 options to move to the 3rd vertex, then • We have 2 options to move to the 4th vertex, then • We have 1 option to move to the 5th vertex • That is, 4  3  2  1  4!  24 cycles b c a e d CS1022

6. Hamiltonian graphs (4) • Finding Hamiltonian cycles in an arbitrary graph is not straightforward • Deciding if a graph is Hamiltonian is can be quite demanding • Problem: • Input a graph G (V, E) • Analyse G • If it is Hamiltonian output YES, otherwise output NO • The test “if it is Hamiltonian” is not straightforward CS1022

7. Travelling salesperson problem (1) • Hamiltonian graphs model many practical problems • Classic problem: travelling salesperson • A salesperson wishes to visit a number of towns connected by roads • Find a route visiting each town exactly once, and keeping travelling costs to a minimum • The graph modelling the problem is Hamiltonian • Vertices are town and edges are roads • Additionally, edges have a weight to represent cost of travel along road (e.g., petrol, time/distance it takes) • Search a Hamiltonian cycle of minimal total weight CS1022

8. Travelling salesperson problem (2) • No efficient algorithm to solve problem • Complex graphs have too many Hamiltonian cycles • They all have to be considered, in order to find the one with minimal total weight • There are algorithms for sub-optimal solutions • Sub-optimal: not minimal, but considerably better than an arbitrary choice of Hamiltonian cycle CS1022

9. Travelling salesperson problem (3) • Nearest neighbour (sub-optimal) algorithm CS1022

10. Travelling salesperson problem (4) • Trace nearest neighbour (sub-optimal) algorithm 5 8 7 6 10 3 B D A C CS1022

11. Travelling salesperson problem (5) • “Nearest”  “with lower weight on edge” • Exhaustive search finds 2 other solutions: • ABCDA (total weight 23) • ACBDA (total weight 31) • It is not the best solution, but it’s better than 31 • A complete graph with 20 vertices has 6  1016 Hamiltonian cycles • Enumerating all would take too much time and memory 5 8 7 6 10 3 B D A C CS1022

12. Trees (1) • Special type/class of graphs called trees • Very popular in computing applications/solutions • A tree is a connected and acyclic graph G (V, E) • In the literature, trees are drawn upside down D B C A E F CS1022

13. Trees (2) • Let G (V, E) be a tree, |V| n, |E| m • We can state (all are equivalent) • There is exactly one path between any vertices of G • G is connected and mn– 1 • G is connected and the removal of one single edge disconnects it • G is acyclic and adding a new edge creates a cycle CS1022

14. Spanning trees (1) • Any connected graph G contains trees as sub-graphs • A sub-graph of G which is a tree and includes all vertices is a spanning tree • It is straightforward to build a spanning tree: • Select an edge of G • Add further edges of G without creating cycles • Do 2 until no more edges can be added (w/o creating cycle) CS1022

15. Spanning trees (2) • Find two spanning trees for the graph f • Solution 1 • Solution 2 g a b b c b a c d e d e f g CS1022

16. Minimal spanning tree • Process adapted for minimum connector problem: • A railway network connecting many towns is to be built • Given the costs of linking 2 towns, find a network of minimal total cost • Spanning tree for a graph with weighted edges, with minimal total weight • This is called minimal spanning tree (MST) • Unlike the travelling salesperson, we have efficient algorithms to solve this problem • We can find the optimal solution! CS1022

17. Minimal spanning tree algorithm (1) • G (V, E) is a connected graph with weighted edges • Algorithm finds MST for Gby successively selecting edges of least possible weight to build an MST • MST is stored as a set T of edges CS1022

18. Rooted trees (1) • We often need to represent information which is naturally hierarchical • Example: family trees • We make use of rooted trees • A special vertex is called the root of the tree • The root of the tree has unique features • Oldest, youngest, smallest, highest, etc. D B C A E F CS1022

19. Rooted trees (2) Rooted trees defined recursively: • A single vertex is a tree (with that vertex as root) • If T1, T2, , Tk are disjoint trees with roots v1, v2, , vk we can “attach” a new vertex v to each vi to form a new tree T with root v Each vertex in a rooted tree T forms the root of another rooted tree which we call a subtree of T v v1 v2 vk ... T1 T2 Tk CS1022

20. Rooted trees (3) • Top vertex is the root and vertices at bottom of tree (those with no children) are called leaves • Vertices other than root or leaves are called internal vertices CS1022

21. Binary (rooted) trees • Rooted trees used as models in many areas • Computer science, biology, management • Very important in computing: binary rooted trees • Each vertex has at most two children • Subtrees: left-and rightsubtrees of the vertex • A missing subtree is called a null tree v Left Right CS1022

22. Sorting and searching (1) • Binary rooted trees are useful to support decisions, especially those requiring sorting/searching data • Ordered numbers, strings ordered lexicographically • Ordered data stored as vertices of binary tree • Data in left-subtree less than data item stored in v • Data in right-subtree greater than data item stored in v • These are called binary search trees v < > Left Right CS1022

23. Sorting and searching (2) • Example of binary search tree with words MYCOMPUTERHASACHIPONITSSHOULDER MY COMPUTER ON SHOULDER A HAS ITS CHIP CS1022

24. Sorting and searching (3) • Binary search trees allow efficient algorithms for • Searching for data items • Inserting new data items • Printing all data in an ordered fashion CS1022

25. Binary search (1) • Algorithm to find (or not) item in binary tree CS1022

26. Binary search (2) • Algorithm to find (or not) item in binary tree K K C M C C T T T T V M M M M V V V V V search(R, ) left_sub= root= right_sub = K search(R, ) left_sub= root= right_sub = T search(R, ) left_sub= null root= right_sub = null V search(R, null) false CS1022

27. Further reading • R. Haggarty. “Discrete Mathematics for Computing”. Pearson Education Ltd. 2002. (Chapter 7) • Wikipedia’s entry on graph theory • Wikibooks entry on graph theory CS1022