Set4. 8 February 2019 OSU CSE 1

Transcription

1 Set4 8 February 2019 OSU CSE 1

2 Documenting Set4 (and Map4) By now you understand hashing and its benefits. But the algorithms we are using are not trivial, and make a lot of assumptions about the representation. How can we document those assumptions in our code? 8 February 2019 OSU CSE 2

3 The correspondence of this = union i: integer, s: finite set of T where (0 <= i and (s) 8 February 2019 OSU CSE 3

4 The correspondence of this = union i: integer, s: finite set of T where (0 <= i and The prefixed union (s) introduces a universal quantifier. This is the union over all possible values for i and s 8 February 2019 OSU CSE 4

5 The correspondence of Set4 What does the where clause tell us about the values of i and this = union i: integer, s: finite set of that Tare of interest to us? where (0 <= i and (s) 8 February 2019 OSU CSE 5

6 The correspondence of Set4 From the first two parts of the where clause we can see that i has to be a this = union i: integer, s: finite set of index T of our hash table where (0 <= i and (s) 8 February 2019 OSU CSE 6

7 The correspondence of Set4 The third part of the where clause tells is that s has to be the set at position i this = union i: integer, s: finite set of T our hash table. where (0 <= i and (s) 8 February 2019 OSU CSE 7

8 The correspondence of this = union i: integer, s: finite set of T where (0 <= i and (s) And the abstract value of this is the union of all such sets s. 8 February 2019 OSU CSE 8

9 The correspondence of Set4 Thus the value of this is the union of all the sets in our hash this = union i: integer, s: finite set of T where (0 <= i and (s) 8 February 2019 OSU CSE 9

10 The convention of $this.hashtable.entries > 0 and for all i: integer, s: finite set of T, x: T where (0 <= i and <s> = $this.hashtable.entries[i, i+1)] and x is in s) ([computed result of x.hashcode()] mod $this.hashtable.entries = i)) and $this.hashtable.examinableindices = $this.hashtable.entries and $this.size = sum i: integer, s: finite set of T where (0 <= i and ( s ) 8 February 2019 OSU CSE 10

11 The convention of Set4 Perhaps it is better if we break this into $this.hashtable.entries > 0 and for all i: integer, s: finite set of T, x: T where (0 <= i and <s> = $this.hashtable.entries[i, i+1)] and x is in s) ([computed result of x.hashcode()] mod $this.hashtable.entries = i)) and $this.hashtable.examinableindices = $this.hashtable.entries and $this.size = sum i: integer, s: finite set of T where (0 <= i and ( s ) 8 February 2019 OSU CSE 11

12 Set4 s convention (1 of 4) $this.hashtable.entries > 0 and 8 February 2019 OSU CSE 12

13 Set4 s convention (1 of 4) $this.hashtable.entries > 0 and Surprisingly Array (just like Java arrays) can have zero elements, but we want to have buckets in our hash table 8 February 2019 OSU CSE 13

14 Set4 s convention (2 of 4) and for all i: integer, s: finite set of T, x: T where (0 <= i and <s> = $this.hashtable.entries[i, i+1)] and x is in s) ([computed result of x.hashcode()] mod $this.hashtable.entries = i)) and 8 February 2019 OSU CSE 14

15 Set4 s convention (2 of 4) The first three parts of the where clause look and for all i: integer, familiar What values of s: finite set of T, i and s are we x: T interested in? where (0 <= i and <s> = $this.hashtable.entries[i, i+1)] and x is in s) ([computed result of x.hashcode()] mod $this.hashtable.entries = i)) and 8 February 2019 OSU CSE 15

16 Set4 s convention (2 of 4) and for all i: integer, s: finite set of T, x: T where (0 <= i and <s> = $this.hashtable.entries[i, i+1)] and x is in s) What about the values of x? ([computed result of x.hashcode()] mod $this.hashtable.entries = i)) and 8 February 2019 OSU CSE 16

17 Set4 s convention (2 of 4) and for all i: integer, s: finite set of T, This tells us that if x x: T is in a set in our where (0 <= i and hash table, then that i < $this.hashtable.entries set is at a specific and <s> = $this.hashtable.entries[i, position in our hash i+1)] table... and x is in s) ([computed result of x.hashcode()] mod $this.hashtable.entries = i)) and 8 February 2019 OSU CSE 17

18 Set4 s convention (2 of 4) In other words, this tells us that we are hashing and for all i: integer, the entries in our set! s: finite set of T, x: T where (0 <= i and <s> = $this.hashtable.entries[i, i+1)] and x is in s) ([computed result of x.hashcode()] mod $this.hashtable.entries = i)) and 8 February 2019 OSU CSE 18

19 Set4 s convention (3 of 4) and $this.hashtable.examinableindices = $this.hashtable.entries and 8 February 2019 OSU CSE 19

20 Set4 s convention (3 of 4) and $this.hashtable.examinableindices = $this.hashtable.entries and If this is true, can there be a non-examinable index in our hash table? 8 February 2019 OSU CSE 20

21 Set4 s convention (4 of 4) and $this.size = sum i: integer, s: finite set of T where (0 <= i and ( s ) 8 February 2019 OSU CSE 21

22 Set4 s convention (4 of 4) and $this.size = sum i: integer, s: finite set of T where (0 <= i ( s ) and The prefixed sum introduces a universal quantifier. This is the sum over all possible values for i and s 8 February 2019 OSU CSE 22

23 Set4 s convention (4 of 4) and $this.size = sum i: integer, s: finite set of T where (0 <= i ( s ) and The where clause looks familiar What values of i and s are we interested in? 8 February 2019 OSU CSE 23

24 Set4 s convention (4 of 4) and $this.size = sum i: integer, s: finite set of T where (0 <= i ( s ) and What does this tell us about the value of $this.size with respect to the value of this? 8 February 2019 OSU CSE 24