Last time Today Next (Midterm) Hash tables Unbounded arrays - - PowerPoint PPT Presentation

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Last time Today Next (Midterm) Hash tables Unbounded arrays - - PowerPoint PPT Presentation

Apr 15, 2023 15 likes •409 views

Last time Today Next (Midterm) Hash tables Unbounded arrays Implementation Amortized analysis Stacks Queues Linked Lists Toward unbounded arrays Unsorted Arrays Linked Lists O(1) access Self-resizing PROS Built-in support

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SLIDE 1

Last time

(Midterm) Implementation § Stacks § Queues Linked Lists Unbounded arrays Amortized analysis

Today Next

Hash tables

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Unsorted Arrays Linked Lists PROS CONS O(1) access Built-in support Self-resizing O(1) insertion (given right pointers) Fixed size O(n) insertion O(n) access No built-in support

Toward unbounded arrays

A data structure that combines the best properties of arrays and linked lists

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Amortized analysis

§ Average frequent cheap operations with infrequent expensive operations

Example: 100 operations at cost 1, followed by 1 operation at cost 100 The amortized cost per operation is: 200/101

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Binary Counter

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Binary Counter

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Binary Counter

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Binary Counter

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Binary Counter

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n-bit counter

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 Per press Cost $ 1 $ 2 $ 1 $ 3 $ 1 $ 2 $ 1 $ 4 Press # 1 2 3 4 5 6 7 8

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n-bit counter

1 0 1 0 1 1 _ _ _ _ _ _ Per press Cost $ ?

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n-bit counter

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 Per press Cost 1 2 1 3 1 2 1 4 Press # 1 2 3 4 5 6 7 8 Total Cost 1 3 4 7 8 10 11 15 $1 / press Total Revenue 1 2 3 4 5 6 7 8 Bank Account

  • 1
  • 1
  • 3
  • 3
  • 4
  • 4
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n-bit counter

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 Per press Cost 1 2 1 3 1 2 1 4 Press # 1 2 3 4 5 6 7 8 Total Cost 1 3 4 7 8 10 11 15 $2 / press Total Revenue 2 4 6 8 10 12 14 16 Bank Account 1 1 2 1 2 2 3 1

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n-bit counter

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 Per press Cost 1 2 1 3 1 2 1 4 Press # 1 2 3 4 5 6 7 8 Total Cost 1 3 4 7 8 10 11 15 $2 / press Total Revenue 2 4 6 8 10 12 14 16 Bank Account 1 1 2 1 2 2 3 1 Average cost $2, O(1) Will this keep working? Can we prove it?

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Invariant?

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 Per press Cost 1 2 1 3 1 2 1 4 Press # 1 2 3 4 5 6 7 8 Total Cost 1 3 4 7 8 10 11 15 $2 / press Total Revenue 2 4 6 8 10 12 14 16 Bank Account 1 1 2 1 2 2 3 1

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000000 000001 000010 000011 000100 000101 000110 000111 001000 charge 2 tokens Pay 1 1 in reserve charge 2 tokens Pay 2 1 in reserve charge 2 tokens Pay 1 2 in reserve charge 2 tokens Pay 3 1 in reserve charge 2 tokens Pay 1 2 in reserve charge 2 tokens Pay 2 2 in reserve charge 2 tokens Pay 1 3 in reserve charge 2 tokens Pay 4 1 in reserve

n-bit counter

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000000 000001 000010 000011 000100 000101 000110 000111 001000 charge 2 tokens Pay 1 1 in reserve charge 2 tokens Pay 2 1 in reserve charge 2 tokens Pay 1 2 in reserve charge 2 tokens Pay 3 1 in reserve charge 2 tokens Pay 1 2 in reserve charge 2 tokens Pay 2 2 in reserve charge 2 tokens Pay 1 3 in reserve charge 2 tokens Pay 4 1 in reserve

n-bit counter

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Invariant: Preservation

0 1 0 1 0 1 1 _ _ _ _ _ _ _

  • 1. Assume invariant

k tokens for k ones

  • 2. Prove preservation

k’ tokens for k’ ones

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Amortized Analysis

# of times the counter has been incremented # of total bit flips Budgeted by amortized analysis (2 flips per increment) Actual number (will never be bigger than budget)

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Unbounded arrays

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Unsorted Arrays Linked Lists PROS CONS O(1) access Built-in support Self-resizing O(1) insertion (given right pointers) Fixed size O(n) insertion O(n) access No built-in support

Recall: Toward unbounded arrays

A data structure that combines the best properties of arrays and linked lists

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Unbounded arrays interface

int uba_len(uba_t A) /*@requires A != NULL; @*/ /*@ensures \result >= 0; @*/ ; uba_t uba_new(int size) /*@requires 0 <= size; @*/ /*@ensures \result != NULL; @*/ /*@ensures uba_len(\result) == size; @*/ ; string uba_get(uba_t A, int i) /*@requires A != NULL; @*/ /*@requires 0 <= i && i < uba_len(A); @*/; void uba_set(uba_t A, int i, string x) /*@requires A != NULL; @*/ /*@requires 0 <= i && i < uba_len(A); @*/; // typedef ______* uba_t;

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string uba_rem(uba_t A) /*@requires A != NULL; @*/ /*@requires 0 < uba_len(A); @*/ ; void uba_add(uba_t A, string x) /*@requires A != NULL; @*/ ;

new functionality Increase and decrease the array’s size by 1 Goal: All operations in the interface are O(1)

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Unbounded Array

Action Start with [4, 7] § add(5) § rem() § add(9) User View Implementation View

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Data structure

typedef struct uba_header uba; struct uba_header { int size; // 0 <= size && size < limit int limit; // 0 < limit string[] data; // \length(data) == limit }; reported to user actual size

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Data structure invariant

bool is_uba(uba* A) { return A != NULL && is_array_expected_length(A->data, A->limit) && 0 <= A->size && A->size < A->limit; }

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Cost assumptions

Each array write costs 1 New array allocation does not cost anything

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Resizing the array

  • Make uba_add have constant amortized cost
  • Arrange so that resizing happens rarely
  • making new array of length limit+1 won’t work. Why?
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“a” “b” “c” “d” “e” “f” “g” “h”

size limit data 7 8 size limit data 8 16 size limit data 9 16

“a” “b” “c” “d” “e” “f” “g” “h” “I” “a” “b” “c” “d” “e” “f” “g” “h” “a” “b” “c” “d” “e” “f” “g”

size limit data 6 8

“a” “b” “c” “d” “e” “f”

size limit data 5 8

“a” “b” “c” “d” “e” “a” “b” “c” “d”

size limit data 4 8

“a” “b” “c” “d”

size limit data 3 4

“a” “b” “c” “a” “b”

size limit data 2 4

“a” “b”

size limit data 1 2

1 3 3 2 4 1 3 9 5 4 10 1 5 11 1 6 12 1 7 21 9 8 22 1

“a”

cost of this add

Resize 2x

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“a” “b” “c” “d” “e” “f” “g” “h”

size limit data 7 8 size limit data 8 16 size limit data 9 16

“a” “b” “c” “d” “e” “f” “g” “h” “I” “a” “b” “c” “d” “e” “f” “g” “h” “a” “b” “c” “d” “e” “f” “g”

size limit data 6 8

“a” “b” “c” “d” “e” “f”

size limit data 5 8

“a” “b” “c” “d” “e” “a” “b” “c” “d”

size limit data 4 8

“a” “b” “c” “d”

size limit data 3 4

“a” “b” “c” “a” “b”

size limit data 2 4

“a” “b”

size limit data 1 2

3 3 4 1 9 5 10 1 11 1 12 1 21 9 22 1

“a”

t

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a l c

  • s

t c

  • s

t

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t h i s a d d

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“a” “b” “c” “d” “e” “f” “g” “h”

size limit data 7 8 size limit data 8 16 size limit data 9 16

“a” “b” “c” “d” “e” “f” “g” “h” “I” “a” “b” “c” “d” “e” “f” “g” “h” “a” “b” “c” “d” “e” “f” “g”

size limit data 6 8

“a” “b” “c” “d” “e” “f”

size limit data 5 8

“a” “b” “c” “d” “e” “a” “b” “c” “d”

size limit data 4 8

“a” “b” “c” “d”

size limit data 3 4

“a” “b” “c” “a” “b”

size limit data 2 4

“a” “b”

size limit data 1 2

1 3 3 2 4 1 3 9 5 4 10 1 5 11 1 6 12 1 7 21 9 8 22 1

“a”

#

  • f

a d d s t

  • t

a l c

  • s

t c

  • s

t

  • f

t h i s a d d

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“a” “b” “c” “d” “e” “f” “g” “h”

size limit data 7 8 size limit data 8 16 size limit data 9 16

“a” “b” “c” “d” “e” “f” “g” “h” “I” “a” “b” “c” “d” “e” “f” “g”

size limit data 6 8

“a” “b” “c” “d” “e” “f”

size limit data 5 8

“a” “b” “c” “d” “e” “a” “b” “c” “d”

size limit data 4 8 size limit data 3 4

“a” “b” “c” “a” “b”

size limit data 2 4 size limit data 1 2

3 3 3 6 4 1 9 9 5 12 10 1 15 11 1 18 12 1 21 21 9 24 22 1

“a”

t

  • t

a l t

  • k

e n s t

  • t

a l c

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t c

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t

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t h i s a d d b a n k a c c

  • u

n t 2 2 4 6 2

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“r” “o” “b” size limit data 3 6 “r” “o” “b” “e” size limit data 4 6 “r” “o” “b” “e” “r” size limit data 5 6 “r” “o” “b” “e” “r” “t” size limit data 6 12 size limit data 7 12 “r” “o” “b” “e” “r” “t” “r” “o” “b” “e” “r” “t” “s” size limit data 8 12 “r” “o” “b” “e” “r” “t” “s” “i"

Unit of cost: array add 3 tokens

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“r” “o” “b” size limit data 3 6 “r” “o” “b” “e” size limit data 4 6 “r” “o” “b” “e” “r” size limit data 5 6 “r” “o” “b” “e” “r” “t” size limit data 6 12 size limit data 7 12 “r” “o” “b” “e” “r” “t” “r” “o” “b” “e” “r” “t” “s” size limit data 8 12 “r” “o” “b” “e” “r” “t” “s” “i"

Unit of cost: array add 3 tokens

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“r” “o” “b” size limit data 3 6 “r” “o” “b” “e” size limit data 4 6 “r” “o” “b” “e” “r” size limit data 5 6 “r” “o” “b” “e” “r” “t” size limit data 6 12 size limit data 7 12 “r” “o” “b” “e” “r” “t” “r” “o” “b” “e” “r” “t” “s” size limit data 8 12 “r” “o” “b” “e” “r” “t” “s” “i"

Unit of cost: array add 3 tokens

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“r” “o” “b” size limit data 3 6 “r” “o” “b” “e” size limit data 4 6 “r” “o” “b” “e” “r” size limit data 5 6 “r” “o” “b” “e” “r” “t” size limit data 6 12 size limit data 7 12 “r” “o” “b” “e” “r” “t” “r” “o” “b” “e” “r” “t” “s” size limit data 8 12 “r” “o” “b” “e” “r” “t” “s” “i"

Unit of cost: array add 3 tokens

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“r” “o” “b” size limit data 3 6 “r” “o” “b” “e” size limit data 4 6 “r” “o” “b” “e” “r” size limit data 5 6 “r” “o” “b” “e” “r” “t” size limit data 6 12 size limit data 7 12 “r” “o” “b” “e” “r” “t” “r” “o” “b” “e” “r” “t” “s” size limit data 8 12 “r” “o” “b” “e” “r” “t” “s” “i"

Unit of cost: array add 3 tokens

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“r” “o” “b” size limit data 3 6 “r” “o” “b” “e” size limit data 4 6 “r” “o” “b” “e” “r” size limit data 5 6 “r” “o” “b” “e” “r” “t” size limit data 6 12 size limit data 7 12 “r” “o” “b” “e” “r” “t” “r” “o” “b” “e” “r” “t” “s” size limit data 8 12 “r” “o” “b” “e” “r” “t” “s” “i’’

Unit of cost: array add 3 tokens

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Amortized analysis guarantees the average cost of each operation in the worst case

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  • 1. If you access Q->front, what Boolean

expression must be true to ensure safety?

  • 2. What Boolean expression indicates

whether this queue is empty?

  • 3. Write the statements in the correct
  • rder that will make node y the front
  • f this non-empty queue.
  • 4. TRUE or FALSE: In C0, setting y->next = y is allowed.
  • 5. If y is of type list*, the type of *y is ___________
  • 6. Q->back->data contains ______________________

y Q

front back

Go to

  • r

cs.cmu.edu/~1 5122/quiz

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