cse 326: data structures lecture #6 (end of lists, then) trees alon halevy spring quarter 2001
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CSE 326: Data StructuresLecture #6
(end of Lists, then) Trees
Alon Halevy
Spring Quarter 2001
Application: Polynomial ADT
Ai is the coefficient of the xn-i term:
3x2 + 2x + 5 ( 3 2 5 )
8x + 7 ( 8 7 )
x2 + 3 ( 1 0 3 )
Problem?
3x2001 + 4 ( 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 )
Sparse List Data Structure:3x2001 + 4
40
32001
(<4 0> <2001 3>)
Addition of Two Polynomials• Similar to merging two sorted lists – O(n+m)
1050
31200
15+10x50+3x1200
150
p
3050
4100
5+30x50+4x100
50
q
4100
31200
4050
200
r
Multiple Linked Lists
• Many ADTS such as graphs, relations, sparse matrices, multivariate polynomials use multiple linked lists
• Several options– array of lists– lists of lists– multi lists
• General principle throughout the course: use one ADT to implement a more complicated one.
Array of Linked Lists:Adjacency List for Graphs
4 3 5
1 4
1 3
4
2
5
G12345
5 3
5 2
•Array G of unordered linked lists•Each list entry corresponds to an edge in the graph
Reachability by Marking• Suppose we want to mark all the nodes in the
graph which are reachable from a given node k.– Let G[1..n] be the adjacency list rep. of the graph
– Let M[1..n] be the mark array, initially all falses.
mark(int i){ M[i] = true; x = G[i] while (x != NULL) { if (M[x->node] == false) mark(G[x->node]) x = x->next }}
Multi-Lists
• Suppose we have a set of movies and cinemas, and we want a structure that stores which movies are playing where.
More on Multi-Lists
• What if we also want to store the playing times of movies?
Trees• Family Trees• Organization Charts• Classification trees
– is this mushroom poisonous?
• File directory structure• Parse Trees (x+y*z)
• Search Trees – often better than lists for sorted data
Definition of a TreeRecursive definition:
– empty tree has no root
– given trees T1,…,Tk and a node r, there is a tree T where
• r is the root of T
• the children of r are the roots of T1, T2, …, Tk
r
T1 T2 T3
Tree Terminology
rootchildparentsibling
path
descendentancestor
a
i
d
h j
b
f
k l
ec
g
More Tree Terminology
a
i
d
h j
b
f
k l
subtreeleaf
depthheight
branching factorn-arycomplete
ec
g
Basic Tree Data Structure
data
next_siblingfirst_child
a
b c d e
Logical View of Tree
a
i
d
h j
b
f
k l
ec
g
Actual Data Structurea
b c d e
f gh i j
k l
Combined View of Tree
a
i
d
h j
b
f
k l
ec
g
Traversals
• Many algorithms involve walking through a tree, and performing some computation at each node
• Walking through a tree is called a traversal• Common kinds of traversal
– Pre-order
– Post-order
– Level-order
Pre-Order Traversal
• Perform computation at the node, then recursively perform computation on each child
preorder(node * n){
node * c;
if (! n==NULL){
DO SOMETHING;
c = n->first_child;
while (! c==NULL){
preorder(c);
c = c->next_sibling; } }
Pre-Order Traversal Example
a
i
d
h j
b
f
k l
ec
g
Start with a -
Pre-Order Applications
• Use when computation at node depends upon values calculated higher in the tree (closer to root)
• Example: computing depthdepth(node) = 1 + depth( parent of node )
• Another example: printing out a directory structure.
Computing Depth of All Nodes• Add a field “depth” to all nodes
Depth(node * n, int d){node * c;if (! n==NULL){
n->depth = d; d = d+1; c = n->first_child;while (! c==NULL){
Depth(c, d);c = c->next_sibling; } }
• Call Depth(root,0) to set depth field correctly
Depth Calculation
a
i
d
h j
b
f
k l
ec
g
Post-Order Traversal• Recursively perform computation on each child,
and then perform computation at node
postorder(node * n){node * c;if (! n==NULL){
c = n->first_child;while (! c==NULL){
postorder(c);c = c->next_sibling;
} DO SOMETHING;}
Post-Order Applications
• Use when computation at node depends upon values calculated lower in the tree (closer to leafs)
• Example: computing heightheight(node) = 1 +
MAX( height(child1), height(child2), … height(childk))
• Example: size of tree rooted at node size(node) = 1 +
size(child1) + size(child2) + … + size(childk))
Computing Size of TreeSize(node * n){
node * c;
if (! n==NULL) return 0;
else { int m=1;
c = n->first_child;
while (! c==NULL){
m = m + Size(c);
c = c->next_sibling; }
return m; }
• Call Size(root) to compute number of nodes in tree
Depth-First Search
• Both Pre-Order and Post-Order traversals are examples of depth-first search– nodes are visited deeply on the left-most branches
before any nodes are visited on the right-most branches• visiting the right branches deeply before the left would still be
depth-first! Crucial idea is “go deep first!”
• In DFS the nodes “being worked on” are kept on a stack (where?)
Level-Order/Breadth-first Traversal
• Consider task of traversing tree level by level from top to bottom (alphabetic order)
• What data structure to use to keep track of nodes??
a
i
d
h j
b
f
k l
ec
g
Level-Order (Breadth First) Traversal
• Put root in a Queue
• Repeat until Queue is empty:– Dequeue a node– Process it– Add it’s children to queue
Example: Printing the Treeprint(node * root){node * n, c; queue Q;Q.enqueue(root);while (! Q.empty()){
n = Q.dequeue();print n->data;c = n->first_child;while (! c==NULL){
Q.enqueue(c);c = c->next_sibling; } } }
QUEUE
a
b c d e
c d e f g
d e f g
e f g h i j
f g h i j
g h i j
h i j k
i j k
j k l
k l
l
a
i
d
h j
b
f
k l
ec
g
Applications of BFS
• Find the shortest path from the root to a given node N– if N is at depth k, BFS will never visit a node at
depth>k
– important for really deep trees
• Generalizes to finding shortest paths in graphs• Spidering the world wide web
– From a root URL, fetch pages that are further and further away
Coming Up
• Binary Trees & Binary Search Trees (finally!)– Weiss 4.2 – 4.3
Section 4.3 is quite long. It will probably take us two lectures to get through it.