Symbol Tables

COS 265 - Data Structures & Algorithms

Data Structures

“

Smart data structures and dumb code works a lot better than the other way around.
—Eric S. Raymond

”

Symbol Tables

API

symbol tables

Key-value pair abstraction

• Insert a value with specified key
• Given a key, search for the corresponding value

Ex: DNS Lookup

• Insert domain name (key) with specified IP address (value)
• Given domain name, find corresponding IP address

symbol table applications

symbol tables: context

Also known as: maps, dictionaries, associative arrays

Generalizes arrays: Keys need not be between \(0\) and \(N-1\)

Language support

• External libraries: C, VisualBasic, Standard ML, bash, ...
• Built-in libraries: Java, C#, C++, Scala, ...
• Built-in to language: Awk, Perl, PHP, Tcl, JavaScript, Python, Ruby, Lua, ...

PHP: every array is an associative array
JavaScript: every object is an associative array
Lua: table is the only primitive data structure

hasNiceSyntaxForAssociativeArrays['Python'] = True
hasNiceSyntaxForAssociativeArrays['Java']   = False
# legal Python code

Basic symbol table API

Associative array abstraction: associate one value with each key

conventions

• Values are not null (Java allows null value)
• Method get() returns null if key not present
• Method put() overwrites old value with new value

Intended consequences

• Easy to implement contains()

public boolean contains(Key key) {
    return get(key) != null;
}

• Can implement lazy version of delete()

public void delete(Key key) {
    put(key, null);
}

keys and values

Value type: any generic type

Key type: several natural assumptions

• Assume keys are Comparable, use compareTo() (specify Comparable in API)
• Assume keys are any generic type, use equals() to test equality
  • use hashCode() to scramble key
  • built-in to Java (stay tuned)

Best practices: Use immutable types for symbol table keys

• Immutable in Java: Integer, Double, String, ...
• Mutable in Java: StringBuilder, arrays, ...

equality test

All java classes inherit a method equals()

Java requirements: for any references x, y, and z:

• Reflexive: x.equals(x) is true
• Symmetric: x.equals(y) iff y.equals(x)
• Transitive: if x.equals(y) and y.equals(z), then x.equals(z)
• Non-null: x.equals(null) is false

Equivalence relation: reflexive, symmetric, transitive

Default implementation: (x==y: do x and y refer to same object?)

Customized implementations: Integer, Double, String, ...

User-defined implementations: some care needed

implementing equals for user-defined types

Seems easy, but...

public class Date implements Comparable<Date> {
    private final int month;
    private final int day;
    private final int year;

    /* ... */

    public boolean equals(Date that) {
        // check that all significant fields are the same
        if(this.day   != that.day  ) return false;
        if(this.month != that.month) return false;
        if(this.year  != that.year ) return false;
        return true;
    }
}

implementing equals for user-defined types

Seems easy, but requires some care

// typically unsafe to use equals() with inheritance
//    (would violate symmetry)
public final class Date implements Comparable<Date> {
    private final int month, day, year;
    // ...
    // must be Object (why? experts still debate)
    public boolean equals(Object y) {
        if(y == this) return true;   // optimize for true object equality
        if(y == null) return false;  // check for null

        // objects must be in the same class (religion: getClass() vs instanceof)
        if(y.getClass() != this.getClass()) return false;

        Date that = (Date) y;  // cast is guaranteed to succeed

        // check that all significant fields are the same
        if(this.day   != that.day  ) return false;
        if(this.month != that.month) return false;
        if(this.year  != that.year ) return false;
        return true;
    }
}

equals design

"Standard" recipe for user-defined types

• Optimization for reference equality (i.e., this == that)

• Check against null

• Check that two objects are of the same type; cast

• Compare each significant field:

  • if field is a primitive type, use == (but use Double.compare() with double to deal with -0.0 and NaN)

  • if field is an object, use equals() (apply rule recursively)

  • if field is an array, apply to each entry (can use Arrays.deepEquals(a,b) but not a.equals(b))

equals design

Best practices

• No need to use calculated fields that depend on other fields

• Compare fields mostly likely to differ first

• Make compareTo() consistent with equals() (x.equals(y) iff x.compareTo(y) == 0)

ST test client for traces

Build ST by associating value i with ith string from standard input

public static void main(String[] args) {
    ST<String, Integer> st = new ST<String, Integer>();
    for(int i = 0; !StdIn.isEmpty(); i++) {
        String key = StdIn.readString();
        st.put(key, i);
        StdOut.print(s + " " + i + ", ");
    }
    StdOut.println();
    for(String s : st.keys())
        StdOut.print(s + " " + st.get(s) + ", ");
    StdOut.println();
}

$ java STTestClient < searchexample.txt
S 0, E 1, A 2, R 3, C 4, H 5, E 6, X 7, A 8, M 9, P 10, L 11, E 12,
A 8, C 4, E 12, H 5, L 11, M 9, P 10, R 3, S 0, X 7,

st test client for analysis

Frequency counter: Read a sequence of strings from standard input and print out one that occurs with highest frequency

$ cat tinyTale.txt
it was the best of times
it was the worst of times
it was the age of wisdom
it was the age of foolishness
it was the epoch of belief
it was the epoch of incredulity
it was the season of light
it was the season of darkness
it was the spring of hope
it was the winter of despair

$ # tiny example (60 words, 20 distinct)
$ java FrequencyCounter 1 < tinyTale.txt
it 10

$ # real example (135,635 words, 10,769 distinct)
$ java FrequencyCounter 8 < tale.txt
business 122

$ # real example (21,191,455 words, 534,580 distinct)
$ java FrequencyCounter 10 < leipzip1M.txt
government 24763

Argument: minimum length of words to consider

Sources: tale.txt, leipzig1m.txt

frequency counter implementation

public class FrequencyCounter {
    public static void main(String[] args) {
        int minlen = Integer.parseInt(args[0]);
        ST<String, Integer> st = new ST<>();    // create ST
        while(!StdIn.isEmpty()) {
            // read string and update frequency
            String word = StdIn.readString();
            if(word.length() < minlen) continue; // ignore short str
            if(!st.contains(word)) st.put(word, 1);
            else st.put(word, st.get(word) + 1);
        }
        // print a string with max freq
        String max = "";
        st.put(max, 0);
        for(String word : st.keys())
            if(st.get(word) > st.get(max)) max = word;
        StdOut.println(max + " " + st.get(max));
    }
}

symbol tables

elementary implementations

sequential search in a linked list

Data structure: Maintain an (unordered) linked list of key-value pairs

Search: Scan through all keys until find a match

Insert: Scan through all keys until find a match; if no match add to front

k  v    first
- --    -----
S  0 -> S, 0
E  1 -> E, 1 > S, 0
A  2 -> A, 2 > E, 1 > S,0
R  3 -> R, 3 > A, 2 > E,1 > S,0
C  4 -> C, 4 > R, 3 > A,2 > E,1 > S,0
H  5 -> H, 5 > C, 4 > R,3 > A,2 > E,1 > S,0
E  6 -> H, 5 > C, 4 > R,3 > A,2 > E,6 > ...
X  7 -> X, 7 > H, 5 > C,4 > R,3 > A,2 > E,1 > S,0
A  8 -> X, 7 > H, 5 > C,4 > R,3 > A,8 > ...
M  9 -> M, 9 > X, 7 > H,5 > C,4 > R,3 > A,2 > E,1 > S,0
P 10 -> P,10 > M, 9 > X,7 > H,5 > C,4 > R,3 > A,2 > E,1 > S,0
L 11 -> L,11 > P,10 > M,9 > X,7 > H,5 > C,4 > R,3 > A,2 > E,1 > S,0
E 12 -> L,11 > P,10 > M,9 > X,7 > H,5 > C,4 > R,3 > A,2 > E,12 > ...

elementary ST implementations: summary

\(^*\)guarantee, \(^\dagger\)average

Challenge: Efficient implementations of both search and insert

binary search in an ordered array

Data structure: Maintain an ordered array of key-value pairs

Helper function rank: How many keys < key?

           0 1 2 3 4 5 6 7 8 9
keys[] =   A C E H L M P R S X

lo hi  m   0 1 2 3 4 5 6 7 8 9  successful search for P
 0  9  4   A C E H L>M P R S X
 5  9  7   . . . . . M P<R S X
 5  6  5   . . . . . M>P . . .
 6  6  6   . . . . . . P . . .  loop exist with keys[m] = P: return 6

lo hi  m   0 1 2 3 4 5 6 7 8 9  unsuccessful search for Q
 0  9  4   A C E H L>M P R S X
 5  9  7   . . . . . M P<R S X
 5  6  5   . . . . . M>P . . .
 6  6  6   . . . . . . P>. . .
 7  6  -   . . . . . . .*. . .  loop exits with lo > hi: return 7

binary search: java implementation

public Value get(Key key) {
    if(isEmpty()) return null;
    int i = rank(key);
    if(i < N && keys[i].compareTo(key) == 0) return vals[i];
    else return null;
}

// find number of keys < key
private int rank(Key key) {
    int lo = 0; int hi = N-1;
    while(lo <= hi) {
        int mid = lo + (hi - lo) / 2;
        int cmp = key.compareTo(keys[mid]);
        if     (cmp < 0) hi = mid - 1;
        else if(cmp > 0) lo = mid + 1;
        else             return mid;
    }
    return lo;
}

binary search: trace of indexing client

Repeat the ST client example from before.

k  v      keys[]            N              vals[]
- --  -------------------  ---  -----------------------------
S  0  S                     1    0
E  1  E S                   2    1  0
A  2  A E S                 3    2  1  0
R  3  . . R S               4    .  .  3  0
C  4  . C E R S             5    .  4  1  3  0
H  5  . . . H R S           6    .  .  .  5  3  0
E  6  . . . . . .           6    .  .  6  .  .  .
X  7  . . . . . . X         7    .  .  .  .  .  .  7
A  8  . . . . . . .         7    8  .  .  .  .  .  .
M  9  . . . . M R S X       8    .  .  .  .  9  3  0  7
P 10  . . . . . P R S X     9    .  .  .  .  . 10  3  0  7
L 11  . . . . L M P R S X  10    .  .  .  . 11  9 10  3  0  7
E 12  . . . . . . . . . .  11    .  . 12  .  .  .  .  .  .  .
      A C E H L M P R S X        8  4 12  5 11  9 10  3  0  7

Problem: To insert, need to shift all greater keys over

elementary ST implementations: summary

\(^*\)guarantee, \(^\dagger\)average

Challenge: Efficient implementations of both search and insert

Symbol Tables

Ordered operations

examples of ordered symbol table API

min()                     // 09:00:00
max()                     // 09:37:44

select(7)                 // 09:10:25
rank(09:10:25)            // 7
get(09:10:25)             // Seattle

floor(09:05:00)           // 09:03:13
ceiling(09:05:00)         // 09:10:11

size(09:15:00, 09:25:00)  // 5
keys(09:15:00, 09:25:00)  // [ 09:19:32, 09:19:46, 09:21:05, 09:22:43,
                          //   09:22:54 ]

ordered symbol table api

st implementation summary