Hashing hash functions collision resolution applications

References: Algorithms in Java, Chapter 14 http://www.cs.princeton.edu/introalgsds/42hash 1

Summary of symbol-table implementations

implementation unordered array ordered array unordered list ordered list BST randomized BST red-black tree

guarantee search N lg N N N N 7 lg N 3 lg N insert N N N N N 7 lg N 3 lg N delete N N N N N 7 lg N 3 lg N search N/2 lg N N/2 N/2 1.39 lg N 1.39 lg N lg N

average case insert N/2 N/2 N N/2 1.39 lg N 1.39 lg N lg N delete N/2 N/2 N/2 N/2 ? 1.39 lg N lg N

ordered iteration? no yes no yes yes yes yes

Can we do better?

2

Optimize Judiciously

More computing sins are committed in the name of efficiency (without necessarily achieving it) than for any other single reason including blind stupidity. - William A. Wulf

We should forget about small efficiencies, say about 97% of the time: premature optimization is the root of all evil. - Donald E. Knuth

We follow two rules in the matter of optimization: Rule 1: Don't do it. Rule 2 (for experts only). Don't do it yet - that is, not until you have a perfectly clear and unoptimized solution. - M. A. Jackson

Reference: Effective Java by Joshua Bloch.
3

Hashing: basic plan Save items in a key-indexed table (index is a function of the key). Hash function. Method for computing table index from key. hash(“it”) = 3 ?? hash(“times”) = 3
0 1 2 3 4 5 “it”

Issues. 1. Computing the hash function 2. Collision resolution: Algorithm and data structure to handle two keys that hash to the same index. 3. Equality test: Method for checking whether two keys are equal. Classic space-time tradeoff. No space limitation: trivial hash function with key as address. No time limitation: trivial collision resolution with sequential search. Limitations on both time and space: hashing (the real world).

• • •

4

hash functions collision resolution applications

5

Computing the hash function Idealistic goal: scramble the keys uniformly. Efficiently computable. Each table position equally likely for each key.

• •

thoroughly researched problem, still problematic in practical applications

Practical challenge: need different approach for each type of key Ex: Social Security numbers. Bad: first three digits. Better: last three digits. Ex: date of birth. Bad: birth year. Better: birthday. Ex: phone numbers. Bad: first three digits. Better: last three digits.

• • • • • •

573 = California, 574 = Alaska assigned in chronological order within a given geographic region

6

Hash Codes and Hash Functions Java convention: all classes implement hashCode() hashcode() returns a 32-bit int (between -2147483648 and 2147483647)

Hash function. An int between 0 and M-1 (for use as an array index) First try:
String s = "call"; int code = s.hashCode(); int hash = code % M;
7121 8191 3045982

Bug. Don't use (code % M) as array index

1-in-a billion bug. Don't use (Math.abs(code) % M) as array index.

OK. Safe to use ((code & 0x7fffffff) % M) as array index. hex literal 31-bit mask
7

Java’s hashCode() convention Theoretical advantages Ensures hashing can be used for every type of object Allows expert implementations suited to each type

• • • • • • • • •

Requirements: If x.equals(y) then x and y must have the same hash code. Repeated calls to x.hashCode() must return the same value. x y

Practical realities True randomness is hard to achieve Cost is an important consideration
x.hashCode() y.hashCode()

Available implementations default (inherited from Object): Memory address of x ( ! ! ! ) customized Java implementations: String, URL, Integer, Date. User-defined types: users are on their own that’s you!
8

A typical type Assumption when using hashing in Java: Key type has reasonable implementation of hashCode() and equals() Ex. Phone numbers: (609) 867-5309. exchange extension

public final class PhoneNumber { private final int area, exch, ext; public PhoneNumber(int area, int exch, int ext) { this.area = area; this.exch = exch; this.ext = ext; } public boolean equals(Object y) { // as before } public int hashCode() { return 10007 * (area + 1009 * exch) + ext; } }

sufficiently random?

Fundamental problem: Need a theorem for each data type to ensure reliability.

9

A decent hash code design Java 1.5 string library [see also Program 14.2 in Algs in Java]. public int hashCode() { int hash = 0; for (int i = 0; i < length(); i++) hash = s[i] + (31 * hash); return hash; } ith character of s char … 'a' 'b' 'c' … Unicode … 97 98 99 …

• Equivalent to h = 31 s + … + 31 s + 31 s + s . • Horner's method to hash string of length L: L multiplies/adds
L-1 0 2 L-3 L-2 L-1

Ex.

String s = "call"; int code = s.hashCode();
3045982 = 99 313 + 97 312 + 108 311 + 108 310 = 108 + 31 (108 + 31 (99 + 31 (97)))

Provably random? Well, no.

10

A poor hash code design Java 1.1 string library. For long strings: only examines 8-9 evenly spaced characters. Saves time in performing arithmetic…

• •

public { int int for

int hashCode()

hash = 0; skip = Math.max(1, length() / 8); (int i = 0; i < length(); i += skip) hash = (37 * hash) + s[i]; return hash;

}

but great potential for bad collision patterns. http://www.cs.princeton.edu/introcs/13loop/Hello.java http://www.cs.princeton.edu/introcs/13loop/Hello.class http://www.cs.princeton.edu/introcs/13loop/Hello.html http://www.cs.princeton.edu/introcs/13loop/index.html http://www.cs.princeton.edu/introcs/12type/index.html

Basic rule: need to use the whole key.

11

Digression: using a hash function for data mining Use content to characterize documents. Applications Search documents on the web for documents similar to a given one. Determine whether a new document belongs in one set or another

• • • • •

Approach Fix order k and dimension d • Compute hashCode() % d for all k-grams in the document Result: d-dimensional vector profile of each document To compare documents: Consider angle θ separating vectors cos θ close to 0: not similar cos θ close to 1: similar

a

b

θ cos θ = a ｜a｜｜b｜
12

b/

Digression: using a hash function for data mining k = 10 d = 65536
% more tale.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 ... % more genome.txt CTTTCGGTTTGGAACC GAAGCCGCGCGTCT TGTCTGCTGCAGC ATCGTTC ... tale.txt i 10-grams with hashcode() i freq

genome.txt
10-grams with hashcode() i freq

0 1 2

0 0 0

0 0 0

435

best of ti foolishnes

2

TTTCGGTTTG TGTCTGCTGC

2

8999 ... 12122 ... 34543

it was the

8

0

0

CTTTCGGTTT

3

t was the b

5

ATGCGGTCGA

4

cos θ small: not similar

... 65535 65536 profiles
13

Digression: using a hash function to profile a document for data mining

public class Document { private String name; private double[] profile; public Document(String name, int k, int d) { this.name = name; String doc = (new In(name)).readAll(); int N = doc.length(); profile = new double[d]; for (int i = 0; i < N-k; i++) { int h = doc.substring(i, i+k).hashCode(); profile[Math.abs(h % d)] += 1; } } public double simTo(Document other) { // compute dot product and divide by magnitudes } }

14

Digression: using a hash function to compare documents

public class CompareAll { public static void main(String args[]) { int k = Integer.parseInt(args[0]); int d = Integer.parseInt(args[1]); int N = StdIn.readInt(); Document[] a = new Document[N]; for (int i = 0; i < N; i++) a[i] = new Document(StdIn.readString(), k, d); System.out.print(" "); for (int j = 0; j < N; j++) System.out.printf(" %.4s", a[j].name()); System.out.println(); for (int i = 0; i < N; i++) { System.out.printf("%.4s ", a[i].name()); for (int j = 0; j < N; j++) System.out.printf("%8.2f", a[i].simTo(a[j])); System.out.println(); } } }
15

Digression: using a hash function to compare documents
Cons TomS Huck Prej Pict DJIA Amaz ACTG
US Constitution “Tom Sawyer” “Huckleberry Finn” “Pride and Prejudice” a photograph financial data Amazon.com website .html source genome

% java CompareAll Cons Cons 1.00 TomS 0.89 Huck 0.87 Prej 0.88 Pict 0.35 DJIA 0.70 Amaz 0.63 ACTG 0.58

5 1000 < docs.txt TomS Huck Prej 0.89 0.87 0.88 1.00 0.98 0.96 0.98 1.00 0.94 0.96 0.94 1.00 0.34 0.32 0.34 0.75 0.74 0.76 0.66 0.65 0.67 0.62 0.61 0.63

Pict 0.35 0.34 0.32 0.34 1.00 0.29 0.48 0.24

DJIA 0.70 0.75 0.74 0.76 0.29 1.00 0.62 0.58

Amaz 0.63 0.66 0.65 0.67 0.48 0.62 1.00 0.45

ACTG 0.58 0.62 0.61 0.63 0.24 0.58 0.45 1.00
16

hash functions collision resolution applications

17

Helpful results from probability theory Bins and balls. Throw balls uniformly at random into M bins.

0

1

2

3

4

5

6

7

8

9

10

11

12

13 14

15

Birthday problem. Expect two balls in the same bin after Coupon collector. Expect every bin has
M / 2 tosses.

1 ball after

(M ln M) tosses.

Load balancing. After M tosses, expect most loaded bin has

(log M / log log M) balls.

18

Collisions Collision. Two distinct keys hashing to same index. Conclusion. Birthday problem a ridiculous amount of memory. can't avoid collisions unless you have

Challenge. Deal with collisions efficiently.

Approach 1: accept multiple collisions

Approach 2: minimize collisions

19

Collision resolution: two approaches 1. Separate chaining. [H. P. Luhn, IBM 1953] Put keys that collide in a list associated with index. 2. Open addressing. [Amdahl-Boehme-Rocherster-Samuel, IBM 1953] When a new key collides, find next empty slot, and put it there. st[0] st[1] st[0] st[1] st[2] st[3] jocularly listen null null suburban untravelled considerating seriously jocularly

null listen suburban

st[2] st[3]

st[30001] st[8190] browsing browsing

separate chaining (M = 8191, N = 15000) easy extension of linked list ST implementation

linear probing (M = 30001, N = 15000) easy extension of array ST implementation

20

Collision resolution approach 1: separate chaining good choice: M Use an array of M < N linked lists. Hash: map key to integer i between 0 and M-1. Insert: put at front of ith chain (if not already there). Search: only need to search ith chain.

• • •

N/10

key

hash 7121 3480 5017 0 3 3 .

st[0] st[1] st[2] st[3]

jocularly listen

seriously

call me ishmael

null suburban untravelled considerating

seriously untravelled suburban

st[8190]

browsing

. . .

21

Separate chaining ST implementation (skeleton) public class ListHashST { private int M = 8191; private Node[] st = new Node[M]; private class Node { Object key; Object val; Node next; Node(Key key, Value val, Node next) { this.key = key; this.val = val; this.next = next; } } private int hash(Key key) { return (key.hashcode() & 0x7ffffffff) % M; public void put(Key key, Value val) // see next slide public Val get(Key key) // see next slide }
22

could use doubling

compare with linked lists

no generics in arrays in Java

}

Separate chaining ST implementation (put and get)

public void put(Key key, Value val) { int i = hash(key); for (Node x = st[i]; x != null; x = x.next) if (key.equals(x.key)) { x.val = val; return; } st[i] = new Node(key, value, first); } public Value get(Key key) { int i = hash(key); for (Node x = st[i]; x != null; x = x.next) if (key.equals(x.key)) return (Value) x.val; return null; }

Identical to linked-list code, except hash to pick a list.

23

Analysis of separate chaining Separate chaining performance. Cost is proportional to length of list. Average length = N / M. Worst case: all keys hash to same list.

• • •

Theorem. Let = N / M > 1 be average length of list. For any t > 1, probability that list length > t is exponentially small in t. depends on hash map being random map

Parameters. M too large too many empty chains. M too small chains too long. Typical choice: = N / M 10 constant-time ops.

• • •

24

Collision resolution approach 2: open addressing good choice: M 2N Use an array of size M >> N. Hash: map key to integer i between 0 and M-1. Linear probing: Insert: put in slot i if free; if not try i+1, i+2, etc. Search: search slot i; if occupied but no match, try i+1, i+2, etc.

• • •

0

1

2

S
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H
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5

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A
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C
8

E
9

R
10

11

N
12

0

1

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S
3

H
4

5

6

A
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C
8

E
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R
10

I
11

12

insert I hash(I) = 11

0

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S
3

H
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A
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C
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E
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I
11

N
12

insert N hash(N) = 8

25

Linear probing ST implementation

public class ArrayHashST { standard ugly casts private int M = 30001; private Value[] vals = (Value[]) new Object[maxN]; private Key[] keys = (Key[]) new Object[maxN]; privat int hash(Key key) // as before public void put(Key key, Value val) { int i; for (i = hash(key); keys[i] != null; i = (i+1) % M) if (key.equals(keys[i])) break; vals[i] = val; keys[i] = key; } public Value get(Key key) { for (int i = hash(key); keys[i] != null; i = (i+1) % M) if (key.equals(keys[i])) return vals]i]; return null; } }

compare with elementary unordered array implementation standard array doubling code omitted (double when half full)

26

Clustering Cluster. A contiguous block of items. Observation. New keys likely to hash into middle of big clusters.

-

-

-

S

H

A

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E

-

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M

I

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P

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L

-

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cluster

Knuth's parking problem. Cars arrive at one-way street with M parking spaces. Each desires a random space i: if space i is taken, try i+1, i+2, … What is mean displacement of a car?

Empty. With M/2 cars, mean displacement is about 3/2. Full. Mean displacement for the last car is about
M/2
27

Analysis of linear probing Linear probing performance. Insert and search cost depend on length of cluster. but keys more likely to Average length of cluster = = N / M. hash to big clusters Worst case: all keys hash to same cluster.

• • •

Theorem. [Knuth 1962] Let

= N / M < 1 be the load factor.

Average probes for insert/search miss
1 — 2

(1 + (1 +

1 ( 1 1 ( 1 α ) α )
2

Average probes for search hit
1 — 2

= )2

( 1 + α + 2α2 + 3α3 + 4α4 + . . . ) /

)=

1 + ( α + α2 + α3 + α4 + . . . ) /2

Parameters. Load factor too small Load factor too large Typical choice: M 2N

• • •

too many empty array entries. clusters coalesce. constant-time ops.
28

Hashing: variations on the theme Many improved versions have been studied: Ex: Two-probe hashing hash to two positions, put key in shorter of the two lists reduces average length of the longest list to log log N

• • • • •

Ex: Double hashing use linear probing, but skip a variable amount, not just 1 each time effectively eliminates clustering can allow table to become nearly full

29

Double hashing Idea Avoid clustering by using second hash to compute skip for search. Hash. Map key to integer i between 0 and M-1. Second hash. Map key to nonzero skip value k. Ex: k = 1 + (v mod 97). hashCode() Effect. Skip values give different search paths for keys that collide.

Best practices. Make k and M relatively prime.

30

Double Hashing Performance

Theorem. [Guibas-Szemerédi] Let
Average probes for insert/search miss
1 ( 1 α ) 1 ( 1 α )

= N / M < 1 be average length of list.

=

1 + α + α2 + α3 + α4 + . . .

Average probes for search hit
1 — α

ln

= 1 + α/2 + α2 /3 + α3 /4 + α4 /5 +...

Parameters. Typical choice: α

1.2

constant-time ops.

Disadvantage. Delete cumbersome to implement.

31

Hashing Tradeoffs Separate chaining vs. linear probing/double hashing. Space for links vs. empty table slots. Small table + linked allocation vs. big coherent array.

• •

Linear probing vs. double hashing. load factor 50% linear probing double hashing get put get put 1.5 2.5 1.4 1.5 66% 2.0 5.0 1.6 2.0 75% 3.0 8.5 1.8 3.0 90% 5.5 55.5 2.6 5.5

number of probes

32

Summary of symbol-table implementations

implementation unordered array ordered array unordered list ordered list BST randomized BST red-black tree hashing

guarantee search N lg N N N N 7 lg N 2 lg N 1* insert N N N N N 7 lg N 2 lg N 1* delete N N N N N 7 lg N 2 lg N 1* search N/2 lg N N/2 N/2 1.38 lg N 1.38 lg N lg N 1*

average case insert N/2 N/2 N N/2 1.38 lg N 1.38 lg N lg N 1* delete N/2 N/2 N/2 N/2 ? 1.38 lg N lg N 1*

ordered iteration? no yes no yes yes yes yes no

operations on keys equals() compareTo() equals() compareTo() compareTo() compareTo() compareTo() equals() hashCode()

* assumes random hash code

33

Hashing versus balanced trees

Hashing simpler to code no effective alternative for unordered keys faster for simple keys (a few arithmetic ops versus lg N compares) (Java) better system support for strings [cached hashcode] does your hash function produce random values for your key type??

• • • • • • • • • •

Balanced trees stronger performance guarantee can support many more operations for ordered keys easier to implement compareTo() correctly than equals() and hashCode()

Java system includes both red-black trees: java.util.TreeMap, java.util.TreeSet hashing: java.util.HashMap, java.util.IdentityHashMap
34

Typical “full” ST API public class *ST *ST() void Value boolean Key Key Key Key void Iterator put(Key key, Value val) get(Key key) contains(Key key) min() max() next(Key key) prev(Key key) remove(Key key) iterator() create a symbol table put key-value pair into the table return value paired with key (null if key is not in table) is there a value paired with key? smallest key largest key next largest key (null if key is max) next smallest key (null if key is min) remove key-value pair from table iterator through keys in table

Hashing is not suitable for implementing such an API (no order) BSTs are easy to extend to support such an API (basic tree ops)
Ex: Can use LLRB trees implement priority queues for distinct keys
35

hash functions collision resolution applications

36

Set ADT Set. Collection of distinct keys. public class *SET SET() void boolean void Iterator add(Key key) contains(Key key) remove(Key key) iterator() create a set put key into the set is there a value paired with key? remove key from the set iterator through all keys in the set

Normal mathematical assumption: collection is unordered Typical (eventual) client expectation: ordered iteration Q. How to implement? A0. Hashing (our ST code [value removed] or java.util.HashSet) A1. Red-black BST (our ST code [value removed] or java.util.TreeSet) ordered iterator O(log N) search unordered iterator O(1) search

37

SET client example 1: dedup filter Remove duplicates from strings in standard input Read a key. If key is not in set, insert and print it.

• •

No iterator needed. Output is in same order as input with dups removed.

public class DeDup { public static void main(String[] args) { SET set = new SET(); while (!StdIn.isEmpty()) { String key = StdIn.readString(); if (!set.contains(key)) { set.add(key); StdOut.println(key); } } } }

% more tale.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 ... % java Dedup < tale.txt it was the best of times worst age wisdom foolishness ...

Simplified version of FrequencyCount (no iterator needed)

38

SET client example 2A: lookup filter Print words from standard input that are found in a list Read in a list of words from one file. Print out all words from standard input that are in the list.

• •

public class LookupFilter { public static void main(String[] args) { SET set = new SET(); In in = new In(args[0]); while (!in.isEmpty()) set.add(in.readString()); while (!StdIn.isEmpty()) { String word = StdIn.readString(); if (set.contains(word)) StdOut.println(word); } } }

create SET

process list

print words that are not in list

39

SET client example 2B: exception filter Print words from standard input that are not found in a list Read in a list of words from one file. Print out all words from standard input that are not in the list.

• •

public class LookupFilter { public static void main(String[] args) { SET set = new SET(); In in = new In(args[0]); while (!in.isEmpty()) set.add(in.readString()); while (!StdIn.isEmpty()) { String word = StdIn.readString(); if (!set.contains(word)) StdOut.println(word); } } }

create SET

process list

print words that are not in list

40

SET filter applications

application dedup spell checker browser chess spam filter trusty filter credit cards

purpose eliminate duplicates find misspelled words mark visited pages detect draw eliminate spam allow trusted mail check for stolen cards

key

type dedup

in list duplicates dictionary visited pages positions spam good mail stolen cards

not in list unique keys misspelled words

word URL board IP addr URL number

exception lookup lookup exception lookup exception

good mail

good cards

41

Searching challenge: Problem: Index for a PC or the web Assumptions: 1 billion++ words to index Which searching method to use? 1) hashing implementation of SET 2) hashing implementation of ST 3) red-black-tree implementation of ST 4) red-black-tree implementation of SET 5) doesn’t matter much

42

Index for search in a PC
ST st = new ST(); for (File f: filesystem) { In in = new In(f); String[] words = in.readAll().split("\\s+"); for (int i = 0; i < words.length; i++) { String s = words[i]; if (!st.contains(s)) st.put(s, new SET()); SET files = st.get(s); files.add(f); } }

build index

SET files = st.get(s); for (File f: files) ...

process lookup request

43

Searching challenge: Problem: Index for a book Assumptions: book has 100,000+ words

Which searching method to use? 1) hashing implementation of SET 2) hashing implementation of ST 3) red-black-tree implementation of ST 4) red-black-tree implementation of SET 5) doesn’t matter much

44

Index for a book

public class Index { public static void main(String[] args) { String[] words = StdIn.readAll().split("\\s+"); ST st; st = new ST(); for (int i = 0; i < words.length; i++) { String s = words[i]; if (!st.contains(s)) st.put(s, new SET()); SET pages = st.get(s); pages.add(page(i)); } for (String s : st) StdOut.println(s + ": " + st.get(s)); } }

read book and create ST

process all words

print index!

Requires ordered iterators (not hashing)
45

Hashing in the wild: Java implementations Java has built-in libraries for hash tables. • java.util.HashMap = separate chaining implementation. • java.util.IdentityHashMap = linear probing implementation. import java.util.HashMap; public class HashMapDemo { public static void main(String[] args) { HashMap st = new HashMap (); st.put("www.cs.princeton.edu", "128.112.136.11"); "128.112.128.15"); st.put("www.princeton.edu", StdOut.println(st.get("www.cs.princeton.edu")); } }

Null value policy. Java HashMap allows null values. Our implementation forbids null values.

• •

46

Using HashMap Implementation of our API with java.util.HashMap. import java.util.HashMap; import java.util.Iterator; public class ST implements Iterable { private HashMap st = new HashMap(); public void put(Key key, Value val) { if (val == null) st.remove(key); else st.put(key, val); } public Value get(Key key) { public Value remove(Key key) { public boolean contains(Key key) { public int size() contains(Key key) { public Iterator iterator() { }

return return return return return

st.get(key); st.remove(key); st.contains(key); st.size(); st.keySet().iterator();

} } } } }

47

Hashing in the wild: algorithmic complexity attacks Is the random hash map assumption important in practice? Obvious situations: aircraft control, nuclear reactor, pacemaker. Surprising situations: denial-of-service attacks.

• •

malicious adversary learns your ad hoc hash function (e.g., by reading Java API) and causes a big pile-up in single address that grinds performance to a halt

Real-world exploits. [Crosby-Wallach 2003] Bro server: send carefully chosen packets to DOS the server, using less bandwidth than a dial-up modem Perl 5.8.0: insert carefully chosen strings into associative array. Linux 2.4.20 kernel: save files with carefully chosen names.

• • •

Reference:

http://www.cs.rice.edu/~scrosby/hash

48

Algorithmic complexity attack on the Java Library Goal. Find strings with the same hash code. Solution. The base-31 hash code is part of Java's string API.
Key
Aa BB

hashCode()
2112 2112

Key
AaAaAaAa AaAaAaBB AaAaBBAa AaAaBBBB AaBBAaAa AaBBAaBB AaBBBBAa AaBBBBBB BBAaAaAa BBAaAaBB BBAaBBAa BBAaBBBB

hashCode()
-540425984 -540425984 -540425984 -540425984 -540425984 -540425984 -540425984 -540425984 -540425984 -540425984 -540425984 -540425984 -540425984 -540425984 -540425984 -540425984
49

2N strings of length 2N that hash to same value!

Does your hash function produce random values for your key type??

BBBBAaAa BBBBAaBB BBBBBBAa BBBBBBBB

One-Way Hash Functions One-way hash function. Hard to find a key that will hash to a desired value, or to find two keys that hash to same value. Ex. MD4, MD5, SHA-0, SHA-1, SHA-2, WHIRLPOOL, RIPEMD-160. insecure String password = args[0]; MessageDigest sha1 = MessageDigest.getInstance("SHA1"); byte[] bytes = sha1.digest(password); // prints bytes as hex string

Applications. Digital fingerprint, message digest, storing passwords. Too expensive for use in ST implementations (use balanced trees)
50