Fundamentals of Data Structures - Arrays, Linked Lists, Binary Trees

Fundamentals of Data Structure - Niraj Agarwal

Data Structures ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Data Structure (Cont.) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Data Structures (Cont.) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Collections ,[object Object],[object Object],create Create a new collection add Add an item to a collection delete Delete an item from a collection find Find an item matching some criterion in the collection destroy Destroy the collection

Analyzing an Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Arrays ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],A[0] 1 A[1] 2 A[2] 3 A[n-2] N-1 A[n-1] N

Arrays (Cont.) Multi-dimensional Array A multi-dimensional array of dimension n (i.e., an n -dimensional array or simply n -D array) is a collection of items which is accessed via n subscript expressions. For example, in a language that supports it, the (i,j) th element of the two-dimensional array x is accessed by writing x[i,j]. m x i : : : : : : : : : : : : : : : 2 1 0 n j 10 9 8 7 6 5 4 3 2 1 0 C o l u m n R O W

Array : Limitations ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Linked Lists ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Data Next object

Linked Lists (Cont.) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Head Collection node Tail The variable (or handle) which represents the list is simply a pointer to the node at the head of the list. Data Next object

Linked Lists (Cont.) ,[object Object],[object Object],[object Object],[object Object],[object Object],Head Collection node node Data Next object Data Next object2

Linked Lists - Add implementation ,[object Object],struct t_node { void *item; struct t_node *next; } node; typedef struct t_node *Node; struct collection { Node head; …… }; int AddToCollection( Collection c, void *item ) { Node new = malloc( sizeof( struct t_node ) ); new->item = item; new->next = c->head; c->head = new; return TRUE; } Recursive type definition - C allows it! Error checking, asserts omitted for clarity!

Linked Lists - Find implementation ,[object Object],void *FindinCollection( Collection c, void *key ) { Node n = c->head; while ( n != NULL ) { if ( KeyCmp( ItemKey( n->item ), key ) == 0 ) { return n->item; n = n->next; } return NULL; } Add time Constant - independent of n Search time Worst case - n ,[object Object]

Linked Lists - Delete implementation ,[object Object],void *DeleteFromCollection( Collection c, void *key ) { Node n, prev; n = prev = c->head; while ( n != NULL ) { if ( KeyCmp( ItemKey( n->item ), key ) == 0 ) { prev->next = n->next; return n; } prev = n; n = n->next; } return NULL; } head

Linked Lists - Variations ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],struct t_node { void *item; struct t_node *next; } node; typedef struct t_node *Node; struct collection { Node head, tail; }; head tail By ensuring that the tail of the list is always pointing to the head, we can build a circularly linked list head is tail->next LIFO or FIFO using ONE pointer

Linked Lists - Doubly linked ,[object Object],[object Object],struct t_node { void *item; struct t_node *prev, *next; } node; typedef struct t_node *Node; struct collection { Node head, tail; }; head tail prev prev prev Applications requiring both way search Eg. Name search in telephone directory

Binary Tree ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Note the recursive definition! Each sub-tree is itself a binary tree ,[object Object],[object Object],[object Object],[object Object],[object Object]

Binary Tree (Cont.) ,[object Object],A C D E F G ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Binary Tree - Implementation struct t_node { void *item; struct t_node *left; struct t_node *right; }; typedef struct t_node *Node; struct t_collection { Node root; …… };

Binary Tree - Implementation ,[object Object],extern int KeyCmp( void *a, void *b ); /* Returns -1, 0, 1 for a < b, a == b, a > b */ void *FindInTree( Node t, void *key ) { if ( t == (Node)0 ) return NULL; switch( KeyCmp( key, ItemKey(t->item) ) ) { case -1 : return FindInTree( t->left, key ); case 0: return t->item; case +1 : return FindInTree( t->right, key ); } } void *FindInCollection( collection c, void *key ) { return FindInTree( c->root, key ); } Less, search left Greater, search right

Binary Tree - Performance ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Binary Tree - Traversing ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Binary Tree - Applications ,[object Object],[object Object],[object Object],[object Object],[object Object]

General Tree ,[object Object],[object Object],A Hierarchical Tree

Heaps ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Heaps (Cont.) ,[object Object],[object Object],[object Object],[object Object],[object Object]

Heaps (Cont.) ,[object Object],To add an item to a heap, we follow the reverse procedure. Place it in the next leaf position and move it up. Again, we require O( h ) or O(log n ) exchanges .

Comparisons Arrays Simple, fast Inflexible O(1) O(n) inc sort O(n) O(n) O(logn) binary search Add Delete Find Linked List Simple Flexible O(1) sort -> no adv O(1) - any O(n) - specific O(n) (no bin search) Trees Still Simple Flexible O(log n) O(log n) O(log n)

Queues ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Stacks ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Stack (Cont.) ,[object Object],[object Object],function f( int x, int y) { int a; if ( term_cond ) return …; a = ….; return g( a ); } function g( int z ) { int p, q; p = …. ; q = …. ; return f(p,q); } Context for execution of f

Searching ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Binary Search ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Time complexity O( log n)

Binary Search Implementation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Binary Search vs Sequential Search ,[object Object],[object Object],[object Object],[object Object],[object Object],Logs Base 2 is by far the most common in this course. Assume base 2 unless otherwise noted! Small problems - we’re not interested! Large problems - we’re interested in this gap! n log 2 n ,[object Object],[object Object],[object Object]

Sorting ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Insertion Sort ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],9 A K 10 J 4 5  9   Q 2

Bubble Sort ,[object Object],[object Object],[object Object],[object Object],[object Object],/* Bubble sort for integers */ #define SWAP(a,b) { int t; t=a; a=b; b=t; } void bubble( int a[], int n ) { int i, j; for(i=0;i<n;i++) { /* n passes thru the array */ /* From start to the end of unsorted part */ for(j=1;j<(n-i);j++) { /* If adjacent items out of order, swap */ if( a[j-1]>a[j] ) SWAP(a[j-1],a[j]); } } } Overall O(n 2 ) O( 1 ) statement Inner loop n -1, n -2, n -3, … , 1 iterations Outer loop n iterations

Partition Exchange or Quicksort ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],< pivot > pivot pivot < pivot > pivot pivot < p’ p’ > p’ < p” p” > p”

Heap Sort ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Comparisons of Sorting ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Hashing ,[object Object],[object Object],[object Object],[object Object]

Bucket Arrays ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Direct Access Table ,[object Object],[object Object],[object Object],[object Object]

Analysis of Bucket Arrays ,[object Object],[object Object],[object Object]

Hash Functions ,[object Object],[object Object],[object Object],[object Object],[object Object]

Handling the collisions ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Chaining ,[object Object],[object Object]

Rehashing ,[object Object],[object Object]

Overflow ,[object Object],[object Object],[object Object],[object Object]

Graph ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Labeled Graphs: We may give edges and vertices labels. Graphing applications often require the labeling of vertices Edges might also be numerically labeled. For instance if the vertices represent cities, the edges might be labeled to represent distances.

Graph Terminology ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Graph Terminology ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Graph Terminology ,[object Object],[object Object],[object Object]

Graph Terminology A E D C B F a c b d e f g h i j

Graph Terminology A E D C B F a c b d e f g h i j Vertices A and B are endpoints of edge a

Graph Terminology A E D C B F a c b d e f g h i j Vertex A is the origin of edge a

Graph Terminology A E D C B F a c b d e f g h i j Vertex B is the destination of edge a

Graph Terminology A E D C B F a c b d e f g h i j Vertices A and B are adjacent as they are endpoints of edge a

Graph Terminology U Y X W V Z a c b d e f g h i j Edge 'a' is incident on vertex V Edge 'h' is incident on vertex Z Edge 'g' is incident on vertex Y

Graph Terminology U Y X W V Z a c b d e f g h i j The outgoing edges of vertex W are the edges with vertex W as origin {d, e, f}

Graph Terminology U Y X W V Z a c b d e f g h i j The incoming edges of vertex X are the edges with vertex X as destination {b, e, g, i}

Graph Terminology U Y X W V Z a c b d e f g h i j The degree of vertex X is the number of incident edges on X. deg(X) = ?

Graph Terminology U Y X W V Z a c b d e f g h i j The degree of vertex X is the number of incident edges on X. deg(X) = 5

Graph Terminology U Y X W V Z a c b d e f g h i j The in-degree of vertex X is the number of edges that have vertex X as a destination. indeg(X) = ?

Graph Terminology U Y X W V Z a c b d e f g h i j The in-degree of vertex X is the number of edges that have vertex X as a destination. indeg(X) = 4

Graph Terminology U Y X W V Z a c b d e f g h i j The out-degree of vertex X is the number of edges that have vertex X as an origin. outdeg(X) = ?

Graph Terminology U Y X W V Z a c b d e f g h i j The out-degree of vertex X is the number of edges that have vertex X as an origin. outdeg(X) = 1

Graph Terminology ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Graph Terminology U Y X W V Z a c b d e f g h i j We can see that P1 is a simple path. P1 = {U, a, V, b, X, h, Z} P1

Graph Terminology U Y X W V Z a c b d e f g h i j P2 is not a simple path as not all its edges and vertices are distinct. P2 = {U, c, W, e, X, g, Y, f, W, d, V}

Graph Terminology ,[object Object],[object Object],[object Object],[object Object],[object Object]

Graph Terminology Simple cycle {U, a, V, b, X, g, Y, f, W, c} U Y X W V Z a c b d e f g h i j

Graph Terminology U Y X W V Z a c b d e f g h i j Non-Simple Cycle {U, c, W, e, X, g, Y, f, W, d, V, a}

Graph Representation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Graph Applications ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Reachability ,[object Object],[object Object],[object Object]

Graphs ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Depth First Search Algorithim DFS() Input graph G Output labeling of the edges of G as discovery edges and back edges for all u in G.vertices() setLabel(u, Unexplored) for all e in G.incidentEdges() setLabel(e, Unexplored) for all v in G.vertices() if getLabel(v) = Unexplored DFS(G, v).

Algorithm DFS(G, v) Input graph G and a start vertex v of G Output labeling of the edges of G as discovery edges and back edges setLabel(v, Visited) for all e in G.incidentEdges(v) if getLabel(e) = Unexplored w <--- opposite(v, e) if getLabel(w) = Unexplored setLabel(e, Discovery) DFS(G, w) else setLabel(e, BackEdge)

Depth First Search A A Unexplored Vertex Visited Vertex Unexplored Edge Discovery Edge Back Edge

P J I M L F E N H G K O D C B A

Breadth First Search Algorithm BFS(G) Input graph G Output labeling of the edges and a partitioning of the vertices of G for all u in G.vertices() setLabel(u, Unexplored) for all e in G.edges() setLabel(e, Unexplored) for all v in G.vertices() if getLabel(v) = Unexplored BFS(G, v)

Algorithm BFS(G, v) L 0 <-- new empty list L0.insertLast (v) setLabel(v, Visited) i <-- 0 while( ¬L i.isEmpty()) L i+1 <-- new empty list for all v in G.vertices(v) for all e in G.incidentEdges(v) if getLabel(e) = Unexplored w <-- opposite(v) if getLabel(w) = Unexplored setLabel(e, Discovery) setLabel(w, Visited) Li+1.insertLast (w) else setLabel(e, Cross) i <-- i + 1

A E F B C D Start Vertex A Create a sequence L 0 insert(A) into L 0

A E F B C D Start Vertex A while L 0 is not empty create a new empty list L 1 L 0

A E F B C D Start Vertex A for each v in L 0 do get incident edges of v L 0

A E F B C D Start Vertex A if first incident edge is unexplored get opposite of v, say w if w is unexplored set edge as discovery L 0

A E F B C D Start Vertex A set vertex w as visited and insert(w) into L 1 L 0 L 1

A E F B C D Start Vertex A get next incident edge L 0 L 1

A E F B C D Start Vertex A if edge is unexplored we get vertex opposite v say w, if w is unexplored L 0 L 1

A E F B C D Start Vertex A if w is unexplored set edge as discovery L 0 L 1

A E F B C D Start Vertex A set w as visited and add w to L 1 L 0 L 1

A E F B C D Start Vertex A continue in this fashion until we have visited all incident edge of v L 0 L 1

A E F B C D Start Vertex A as L 0 is now empty we continue with list L 1 L 0 L 1

A E F B C D Start Vertex A as L 0 is now empty we continue with list L 1 L 0 L 1 L 2

A E F B C D Start Vertex A L 0 L 1 L 2

Weighted Graphs ,[object Object],[object Object],[object Object]

Shortest Paths ,[object Object],[object Object]

Dijkstra's Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Dijkstra's Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object]

Edge Relaxation ,[object Object],[object Object],[object Object],[object Object],[object Object]

A(0) D C B F E 8 4 2 1 7 5 9 3 2 Add starting vertex to cloud.

A(0) D C B F E 8 4 2 1 7 5 9 3 2 We store with each vertex v a label d(v) representing the distance of v from s in the subgraph consisting of the cloud and its adjacent vertices.

A(0) D(4) C(2) B(8) F E 8 4 2 1 7 5 9 3 2 We store with each vertex v a label d(v) representing the distance of v from s in the subgraph consisting of the cloud and its adjacent vertices.

A(0) D(4) C(2) B(8) F E 8 4 2 1 7 5 9 3 2 At each step we add to the cloud the vertex outside the cloud with the smallest distance label d(v).

A(0) D(3) C(2) B(8) F(11) E(5) 8 4 2 1 7 5 9 3 2 We update the vertices adjacent to v. d(v) = min{d(z), d(v) + weight(e)}

A(0) D(3) C(2) B(8) F(11) E(5) 8 4 2 1 7 5 9 3 2 At each step we add to the cloud the vertex outside the cloud with the smallest distance label d(v).

A(0) D(3) C(2) B(8) F(8) E(5) 8 4 2 1 7 5 9 3 2 We update the vertices adjacent to v. d(v) = min{d(z), d(v) + weight(e)}.

A(0) D(3) C(2) B(7) F(8) E(5) 8 4 2 1 7 5 9 3 2 We update the vertices adjacent to v. d(v) = min{d(z), d(v) + weight(e)}.

2 2 1 6 7 7 4 2 3 3 2 2 E G B A D H C F

2 2 1 6 7 7 4 2 3 3 2 2 Insert E G B A(0) D H C F

2 2 1 6 7 7 4 2 3 3 2 2 Update E G(6) B(2) A(0) D H C F

2 2 1 6 7 7 4 2 3 3 2 2 Insert E G(6) B(2) A(0) D H C F

2 2 1 6 7 7 4 2 3 3 2 2 Update E(4) G(6) B(2) A(0) D H C(9) F

2 2 1 6 7 7 4 2 3 3 2 2 Insert E(4) G(6) B(2) A(0) D H C(9) F

2 2 1 6 7 7 4 2 3 3 2 2 Update E(4) G(5) B(2) A(0) D H C(9) F(6)

2 2 1 6 7 7 4 2 3 3 2 2 Insert E(4) G(5) B(2) A(0) D H C(9) F(6)

2 2 1 6 7 7 4 2 3 3 2 2 Update E(4) G(5) B(2) A(0) D H(9) C(9) F(6)

2 2 1 6 7 7 4 2 3 3 2 2 Insert E(4) G(5) B(2) A(0) D H(9) C(9) F(6)

2 2 1 6 7 7 4 2 3 3 2 2 Update E(4) G(5) B(2) A(0) D H(8) C(9) F(6)

2 2 1 6 7 7 4 2 3 3 2 2 Insert E(4) G(5) B(2) A(0) D H(8) C(9) F(6)

2 2 1 6 7 7 4 2 3 3 2 2 Update E(4) G(5) B(2) A(0) D(10) H(8) C(9) F(6)

2 2 1 6 7 7 4 2 3 3 2 2 Insert E(4) G(5) B(2) A(0) D(10) H(8) C(9) F(6)

Fundamentals of Data Structures - Arrays, Linked Lists, Binary Trees

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Fundamentals of Data Structures - Arrays, Linked Lists, Binary Trees