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Data Structures and Algorithms(CS210/ESO207/ESO211)

Lecture 27 Miscellaneous application of binary trees

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Outline of the lecture

Elegant solution for two interesting problem

An important lesson :

Lack of proper understanding of a problem is a big hurdle to solve the problem

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Two interesting problems on sequences

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What is a sequence ?

A sequence S = , , Can be viewed as a mapping from [ 0, ]. Order does matter.

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Problem 1

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Multi-increment

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Problem 1Given an initial sequence S = , ,

of numbers, maintain a compact data

structure to perform the following operations efficiently for any 0 < < . ReportElement ( ):

Report the current value of . Multi-Increment ( , , ):

Add to each for each

Trivial solution :

Store S in an array A[0.. -1] such that A[ ] stores the current value of .Multi-Increment ( , , )

{For ( ) A[ ] A[ ]+ ;

}

ReportElement ( ){ return A[ ] }

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O( ) = O( )

O(1)

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Problem 1Given an initial sequence S = , ,

of numbers, maintain a compact data

structure to perform the following operations efficiently for any 0 < < . ReportElement ( ):

Report the current value of . Multi-Increment ( , , ):

Add to each for each

Trivial solution :

Store S in an array A[0.. -1] such that A[ ] stores the current value of .

Question: the source of difficulty in breaking the O (n) barrier for Multi-Increment () ?Answer: we need to explicitly maintain in S.

Question: who asked/inspired us to maintain S explicitly.Answer: 1. incomplete understanding of the problem

2. conditioning based on incomplete understanding 8

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Towards efficient solution of Problem 1

Assumption: without loss of generality assume is power of 2.

Explore ways to maintain sequence S implicitly such that Multi-Increment ( , , ) is efficient Report ( ) is efficient too.

Main hurdle: To perform Multi-Increment ( , , ) efficiently

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How to perform Multi-Increment ( , , ) efficiently ?

A simple idea: Any positive integer can be expressed as a sum of powers of 2.Inference: an O(log n) size set A such that any arbitrary integer upto n can beexpressed as a sum of numbers from A.

Key idea for our problem:

a small set of intervals I such that every interval can be expressed as aunion of very small subset of I.

You are encouraged to explore how this idea might lead to efficient implementation ofMulti-Increment ( , , ).

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A small set of intervals

Expressing an interval [ , ] as union of a small set of intervals:For intervals [ , ]with = 0.EXAMPLE: [0 , 12 ]

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0,3 [4,7] 8,11 [12,15]

0,1 2,3 4,5 6,7 8,9 10,11 12,13 [14,15]

[8,15] [0,7]

[0,15]

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A small set of intervals

Expressing an interval [ , ] as union of a small set of intervals:For intervals [ , ]with = 0.EXAMPLE: [0 , 12 ]

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0,3 [4,7] 8,11 [12,15]

0,1 2,3 4,5 6,7 8,9 10,11 12,13 [14,15]

[8,15] [0,7]

[0,15]

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A small set of intervals

Expressing an interval [ , ] as union of a small set of intervals:For intervals [ , ]with = 0.EXAMPLE: [0 , 12 ]

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0,3 [4,7] 8,11 [12,15]

0,1 2,3 4,5 6,7 8,9 10,11 12,13 [14,15]

[8,15] [0,7]

[0,15]

c

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A small set of intervals

Expressing an interval [ , ] as union of a small set of intervals:For general intervals [ , ].EXAMPLE: [3 , 11 ]

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0,3 [4,7] 8,11 [12,15]

0,1 2,3 4,5 6,7 8,9 10,11 12,13 [14,15]

[8,15] [0,7]

[0,15]

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Multi-Increment ( , , ) efficiently

Observation: Any interval [ , ] can be expressed as union of O(log n ) basic intervals.Question: What data structure to use ?

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0,3 [4,7] 8,11 [12,15]

0,1 2,3 4,5 6,7 8,9 10,11 12,13 [14,15]

[8,15] [0,7]

[0,15]

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Data Structure for efficient Multi-Increment ( , , )

How to do Multi-Increment (3,11 ,10 ) ?

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0,1 2,3 4,5 6,7 8,9 10,11 12,13 [14,15]

0,3 [4,7] 8,11 [12,15]

[0,7] [8,15]

[0,15]

0 0 0 0 0 0 0 0

0 0 0 0

0 0

0

What value should you keepin leaf nodes initially ?

What value should you keepin internal nodes initially ?

4 14 9 17 23 21 29 91 37 25 8 33 2 67 11 44Initial

sequence

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Data Structure for efficient Multi-Increment ( , , )

How to do Multi-Increment (3,11 ,10 ) ?

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0,1 2,3 4,5 6,7 8,9 10,11 12,13 [14,15]

0,3 [4,7] 8,11 [12,15]

[0,7] [8,15]

[0,15]

4 14 9 27 23 21 29 91 37 25 8 43 2 67 11 44

0 0 0 0 0 0 0 0

0 0 0 0

0 0

0

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Data Structure for efficient Multi-Increment ( , , )

How to do Multi-Increment (3,11 ,10 ) ?

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0,1 2,3 4,5 6,7 8,9 10,11 12,13 [14,15]

0,3 [4,7] 8,11 [12,15]

[0,7] [8,15]

[0,15]

4 14 9 27 23 21 29 91 37 25 18 43 2 67 11 44

0 0 0 0 0 0 0 0

0 0 0 0

0 0

0

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Data Structure for efficient Multi-Increment ( , , )

How to do Multi-Increment (3,11 ,10 ) ?

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0,1 2,3 4,5 6,7 8,9 10,11 12,13 [14,15]

0,3 [4,7] 8,11 [12,15]

[0,7] [8,15]

[0,15]

4 14 9 27 23 21 29 91 37 25 18 43 2 67 11 44

0 0 0 0 10 0 0 0

0 0 0 0

0 0

0

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Data Structure for efficient Multi-Increment ( , , )

How to do Multi-Increment (3,11 ,10 ) ?

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0,1 2,3 4,5 6,7 8,9 10,11 12,13 [14,15]

0,3 [4,7] 8,11 [12,15]

[0,7] [8,15]

[0,15]

4 14 9 27 23 21 29 91 37 25 18 43 2 67 11 44

0 0 0 0 10 0 0 0

0 10 0 0

0 0

0

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Data Structure for efficient Multi-Increment ( , , )

How to do Multi-Increment (3,11 ,10 ) ?

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0,1 2,3 4,5 6,7 8,9 10,11 12,13 [14,15]

0,3 [4,7] 8,11 [12,15]

[0,7] [8,15]

[0,15]

4 14 9 27 23 21 29 91 37 25 18 43 2 67 11 44

0 0 0 0 10 0 0 0

0 10 0 0

0 0

0

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Multi-Increment ( , , ) efficiently

Sketch:

1. Let u and v be the leaf nodes corresponding to and .

2. Increment the value stored at u and v.

3. Keep repeating the following step as long as parent (u ) parent (v)Move up by one step simultaneously from u and v - If u is left child of its parent, increment value stored in sibling of u .

- If v is right child of its parent, increment value stored in sibling of v.

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Executing Report ( ) efficiently

Sketch:

1. Let u be the leaf nodes corresponding to .2. val 0;3. Keep moving up from u and keep adding the value of all the nodes on the

path to the root to val .4. Return val .

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What is an efficient implementation of the treedata structure for these two algorithms?

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Exploiting complete binary tree structure

Data structure: An array A of size 2n -1.Copy the sequence S = , , into ??Leaf node corresponding to = ??How to check if a node is left child or right child of its parent ?

(if index of the node is odd, then the node is left child, else the node is right child) 24

0

1 2

3 4 5 6

7 8 9 10 11 12 13 14

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

A[( 1 ) + ]A[ 1 ]A[2 2 ]

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MultiIncrement( , , ) MultiIncrement( , , )

( ) + ; ( ) + ;A( ) A( ) + ;If ( > )

{ A( ) A( ) + ;While( ?? ){

If ( %2=1) A( + ) A( + ) + ;If ( %2=0) A( ) A( ) + ; ( 1)/ 2 ;

( 1)/ 2 ;}

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( 1)/ 2 ( 1)/ 2

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Report( )

Report( ) ( ) + ;

val 0;While( ?? ){

val val + A[ ]; ( 1)/ 2 ;

}

return val ;

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> 0

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The solution of Multi-Increment Problem

Theorem: There exists a data structure of size O( ) for maintaining asequence S = , , such that each Multi-Increment () and Report ()operation takes O(log ) time.

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Problem 2

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Dynamic Range-minima

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Problem 2

Given an initial sequence S = , , of numbers, maintain a compact datastructure to perform the following operations efficiently for any 0 < < . ReportMin ( , ):

Report the minimum element from { for each } Update ( , a ):

a becomes the new value of .

AIM: O( ) size data structure. ReportMin ( , ) in O(log ) time. Update ( , a ) in O(log ) time.

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Efficient dynamic range minima

What to store at internal nodes ?How to perform ReportMin ( , ) ?How to perform Update ( , a ) ?

Make sincere attempts to solve the problem. We shall discuss it in next class 30