UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Operations on Data Structures Notes for Ch.9 of Bratko For CSCE 580 Sp03 Marco Valtorta

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Quicksort Delete some element X from L and split the rest of the list into two lists, Small and Big, as follows: Big contains all elements of L that are bigger than X; Small contains all other elements Sort Small obtaining SortedSmall Sort Big obtaining SortedBig The whole sorted list is the concatenation of SortedSmall, [X], and SortedBig fig9_2.pl

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Quicksort with Difference Lists quicksort1 in fig9_3.pl (with split in fig9_2.pl) concat( A1-Z1,Z1-Z2,A1-Z2) Example: unify concat( A1-Z1,Z1-Z2,A1-Z2) concat( [a,b,c|T1]-T1,[d,e|T2]-T2,L) –Obtain: A1 = [a,b,c|T1] T1 = Z1 = [d,e|T2] Z2 = T2 L = [a,b,c,d,e|T2]-T2

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Mergesort ch9_1.pl mergesort is faster than quicksort on large arrays, on average, at least in some Prolog implementations mergesort has much better worst-case behavior than quicksort

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Binary Trees Binary trees Binary search trees –All nodes in the left subtree are less than the root –All nodes in the right subtree are greater than the root –Good implementation of dictionaries (if balanced)

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Binary Tree Representation Use an atom to represent the empty tree –nil Use a special functor to indicate a non-empty tree –t( L,X,R) –Where X is the root, L is the left subtree and R is the right subtree –E.g., t( t( nil,b,nil),a,t( t( nil,d,nil),c,nil)) a b c d

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Binary Search Trees Search in BST: in( X,S): fig9_7.pl Inserting a leaf in the BST: addleaf( Tree,X,NewTree): fig9_10.pl Deleting from the BST del( Tree,X,NewTree): fig9_13.pl Inserting anywhere in the BST add( Tree,X,NewTree): fig9_15.pl Reversible: can be used to delete! Use the program of fig9_17.pl to display

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Adding a Leaf addleaf( D, X, D1)—adding X as a leaf to D gives D1 Adding X to the empty tree is t( nil,X,nil) If X is the root of D then D1 = D (no duplicates) If the root of D is greater than X than add X into the left subtree of D; if the root of D is less than X, then insert into the right subtree of D fig9_10.pl

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Deleting from a BST Reversing addleaf does not work addleaf only adds at the leaf level addleaf does not correctly remove an internal node We take the minimum item in the right subtree of the node to be deleted and replace the deleted node with it (cf. Fig 9.12) delmin( Tree,Y,Tree1) if Y is the minimal (i.e., leftmost) node in Tree and Tree1 is Tree with Y deleted del( Tree,X,NewTree): fig9_13.pl

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering General add and delete To add X to a binary dictionary D, either –(1) add X at the root of D, or –(2) if the root of D is greater than X then insert X into the left subtree, else insert X into the right subtree of D The difficult part is (1) –addroot( D,X,D1)--- add X to D obtaining D1 –See Fig. 9.14

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Figure 9.14 L1 and L2 must be BSTs The union of L1 and L2 is L All nodes in L1 are less than X, and all nodes in L2 are greater than X addroot imposes these constraints: If X were added as the root into L, then the subtrees of the resulting tree would be L1 and L2 Fig9_15.pl

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Representing Graphs By edges, e.g. connected( a,b). connected( b,c). arc( s,t,3). arc( t,v,1). arc( u,t,2). –(weighted digraph) –Analogous to representation for FSAs As a pair of sets, each represented by a list: –G1 = graph([a,b,c,d],[e(a,b),e(b,d),e(b,c),e(c,d)]). –G2 = digraph([s,t,u,v], [a(s,t,3),a(t,v,1),a(t,u,5),a(u,t,2),a(v,u,2)]).

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Representing Graphs II Using adjacency lists –G1 = [a->[b],b->[a,c,d],c->[b,d],d->[b,c]] –G2 = [s->[t/3],t->[u/5,v/1],u->[t/2],v->[u/2]]

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Finding a (Simple) Path path( A,Z,G,P) if –P is a (simple) path from A to Z in G –A path is defined here as a list of nodes (not a list of edges) If A = Z then P [A], otherwise Find a path P1 from some node Y to Z, and find a path from A to Y avoiding the nodes in P1

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Simple Paths in Graphs Program to find a simple path: fig9_20.pl –Program includes code for Hamiltonian paths Extra parentheses are needed for not (in SWI- Prolog, as usual) Spanning trees (fig9_22.pl, fig9_23.pl)

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Paths in Weighted Graphs Extra arguments for costs: path( A,Z,G,P,C) path1( A,P1,C1,G,P,C) Program 9_21 Extra parentheses needed for not (as usual) Test1: any path Test2: shortest path (very inefficient: will see better ways soon) Test3: longest path (very inefficient)

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Spanning Trees A spanning tree of a graph G = (V,E) is a graph T = (V,E’) such that –T is connected –There is no cycle in T –E’ is a subset of E

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Spanning Trees: a Procedural Solution stree(G,T) if T is a spanning tree of G Spread( Tree1,Tree,Graph) if –Tree is a spanning tree of Graph obtaining by adding zero or mode edges to Tree1 Program fig9_22.pl Program uses “edge list” representation of graphs –Program can easily be modified to implement the Jarník-Prim-Dijkstra algorithm for minimum spanning trees

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Spanning Trees: a Declarative Solution G and T are represented as lists of edges T is a spanning tree of G if T is a subset of G T is a tree T covers G, i.e. each node of G is also in T A set of edges T is a tree if T is connected T has no cycle Program fig9_23.pl Is inefficient Modification can compute minimum spanning trees