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1 Algorithms Symbol table implementations: summary implementation guarantee average case search insert delete search hit insert delete ordered ops? interface.4 HASH TABLES sequential search (unordered list) N N N ½ N N ½ N equals() Algorithms F O U R T H E D I T I O N hash functions separate chaining linear probing context binary search (ordered array) lg N N N lg N ½ N ½ N compareto() BST N N N.9 lg N.9 lg N N compareto() red-black BST lg N lg N lg N. lg N. lg N. lg N compareto() Q. Can we do better? A. Yes, but with different access to the data. Hashing: basic plan Save items in a -indexed table (index is a function of the ). Hash function. Method for computing array index from. Issues. hash("times") = Computing the hash function. Equality test: Method for checking whether two s are equal. hash("it") = Collision resolution: Algorithm and data structure to handle two s that hash to the same array index.?? "it" 4 5 Algorithms.4 HASH TABLES hash functions separate chaining linear probing context Classic space-time tradeoff. No space limitation: trivial hash function with as index. No time limitation: trivial collision resolution with sequential search. Space and time limitations: hashing (the real world).

2 Computing the hash function Java s hash code conventions Idealistic goal. Scramble the s uniformly to produce a table index. Efficiently computable. Each table index equally likely for each. Ex. Phone numbers. Bad: first three digits. Better: last three digits. Ex. Social Security numbers. Bad: first three digits. Better: last three digits. thoroughly researched problem, still problematic in practical applications 57 = California, 574 = Alaska (assigned in chronological order within geographic region) Practical challenge. Need different approach for each type. table index All Java classes inherit a method, which returns a -bit int. Requirement. If x.equals(y), then (x. == y.). Highly desirable. If!x.equals(y), then (x.!= y.). x y x. y. Default implementation. Memory address of x. Legal (but poor) implementation. Always return 7. Customized implementations. Integer, Double, String, File, URL, Date, User-defined types. Users are on their own. 5 6 Implementing hash code: integers, booleans, and doubles Implementing hash code: strings Java library implementations public final class Integer private final int value; public int return value; public final class Boolean private final boolean value; public int if (value) return ; else return 7; public final class Double private final double value; public int long bits = doubletolongbits(value); return (int) (bits ^ (bits >>> )); convert to IEEE 64-bit representation; xor most significant -bits with least significant -bits Warning: -. and +. have different hash codes 7 Horner's method to hash string of length L: L multiplies/adds. Equivalent to h = s[] L + + s[l ] + s[l ] + s[l ]. Ex. Java library implementation public final class String private final char[] s; public int int hash = ; for (int i = ; i < length(); i++) hash = s[i] + ( * hash); return hash; i th character of s String s = "call"; int code = s.; 4598 = = 8 + (8 + (97 + (99))) (Horner's method) char Unicode 'a' 97 'b' 98 'c' 99 8

3 Implementing hash code: strings Implementing hash code: user-defined types Performance optimization. Cache the hash value in an instance variable. Return cached value. public final class String private int hash = ; private final char[] s; cache of hash code public final class Transaction implements Comparable<Transaction> private final String who; private final Date when; private final double amount; public Transaction(String who, Date when, double amount) /* as before */ public int int h = hash; if (h!= ) return h; for (int i = ; i < length(); i++) h = s[i] + ( * h); hash = h; return h; Q. What if of string is? return cached value store cache of hash code public boolean equals(object y) /* as before */ public int nonzero constant int hash = 7; hash = *hash + who.; hash = *hash + when.; hash = *hash + ((Double) amount).; return hash; typically a small prime for reference types, use for primitive types, use of wrapper type 9 Hash code design Modular hashing "Standard" recipe for user-defined types. Combine each significant field using the x + y rule. If field is a primitive type, use wrapper type. If field is null, return. If field is a reference type, use. If field is an array, apply to each entry. applies rule recursively or use Arrays.deepHashCode() Hash code. An int between - and -. Hash function. An int between and M - (for use as array index). private int hash(key ) return. % M; bug typically a prime or power of x x. In practice. Recipe works reasonably well; used in Java libraries. In theory. Keys are bitstring; "universal" hash functions exist. private int hash(key ) return Math.abs(.) % M; -in-a-billion bug of "polygenelubricants" is - hash(x) Basic rule. Need to use the whole to compute hash code; consult an expert for state-of-the-art hash codes. private int hash(key ) return (. & x7fffffff) % M; correct

4 Uniform hashing assumption Uniform hashing assumption Uniform hashing assumption. Each is equally likely to hash to an integer between and M -. Uniform hashing assumption. Each is equally likely to hash to an integer between and M -. Bins and balls. Throw balls uniformly at random into M bins. Bins and balls. Throw balls uniformly at random into M bins Birthday problem. Expect two balls in the same bin after ~ π M / tosses. Coupon collector. Expect every bin has ball after ~ M ln M tosses. Load balancing. After M tosses, expect most loaded bin has Θ ( log M / log log M ) balls. Hash value frequencies for words in Tale of Two Cities (M = 97) Java's String data uniformly distribute the s of Tale of Two Cities 4 Collisions.4 HASH TABLES Collision. Two distinct s hashing to same index. Birthday problem can't avoid collisions unless you have a ridiculous (quadratic) amount of memory. Coupon collector + load balancing collisions are evenly distributed. Algorithms hash functions separate chaining linear probing context hash("it") = hash("times") =?? "it" Challenge. Deal with collisions efficiently. 6

5 Separate-chaining symbol table Separate-chaining symbol table: Java implementation Use an array of M < N linked lists. [H. P. Luhn, IBM 95] Hash: map to integer i between and M -. Insert: put at front of ith chain (if not already there). Search: need to search only ith chain. hash value S E A R 4 C 4 4 H 4 5 E 6 X 7 A 8 M 4 9 P L 4 A 8 null X 7 L M 9 E S P H 5 C 4 R public class SeparateChainingHashST<Key, Value> private int M = 97; // number of chains private Node[] st = new Node[M]; // array of chains private static class Node private Object ; private Object val; private Node next; no generic array creation (declare and value of type Object) private int hash(key ) return (. & x7fffffff) % M; public Value get(key ) int i = hash(); for (Node x = st[i]; x!= null; x = x.next) if (.equals(x.)) return (Value) x.val; return null; array doubling and halving code omitted E 7 8 Separate-chaining symbol table: Java implementation Analysis of separate chaining public class SeparateChainingHashST<Key, Value> private int M = 97; // number of chains private Node[] st = new Node[M]; // array of chains private static class Node private Object ; private Object val; private Node next; private int hash(key ) return (. & x7fffffff) % M; Proposition. Under uniform hashing assumption, prob. that the number of s in a list is within a constant factor of N / M is extremely close to. Pf sketch. Distribution of list size obeys a binomial distribution. (,.5).5 Binomial distribution (N = 4, M =, = ) public void put(key, Value val) int i = hash(); for (Node x = st[i]; x!= null; x = x.next) if (.equals(x.)) x.val = val; return; st[i] = new Node(, val, st[i]); Consequence. Number of probes for search/insert is proportional to N / M. M too large too many empty chains. M too small chains too long. equals() and M times faster than sequential search Typical choice: M ~ N / 4 constant-time ops. 9

6 Resizing in a separate-chaining hash table Deletion in a separate-chaining hash table Goal. Average length of list N / M = constant. Double size of array M when N / M 8. Halve size of array M when N / M. Need to rehash all s when resizing. x. does not change but hash(x) can change Q. How to delete a (and its associated value)? A. Easy: need only consider chain containing. before deleting C after deleting C before resizing A B C D E F G H I J K L M N O P K I P N L K I P N L J F C B J F B after resizing K I O M O M P N L E A J F C B O M H G D Symbol table implementations: summary implementation guarantee average case search insert delete search hit insert delete ordered ops? interface sequential search (unordered list) N N N ½ N N ½ N equals().4 HASH TABLES binary search (ordered array) lg N N N lg N ½ N ½ N compareto() BST N N N.9 lg N.9 lg N N compareto() red-black BST lg N lg N lg N. lg N. lg N. lg N compareto() Algorithms hash functions separate chaining linear probing context separate chaining N N N -5 * -5 * -5 * equals() * under uniform hashing assumption

7 Collision resolution: open addressing Linear-probing hash table demo Open addressing. [Amdahl-Boehme-Rocherster-Samuel, IBM 95] When a new collides, find next empty slot, and put it there. Hash. Map to integer i between and M-. Insert. Put at table index i if free; if not try i+, i+, etc. st[] st[] jocularly null listen linear-probing hash table st[] suburban null st[] browsing M = 6 linear probing (M =, N = 5) 5 Linear-probing hash table demo Linear-probing hash table summary Hash. Map to integer i between and M-. Search. Search table index i; if occupied but no match, try i+, i+, etc. Hash. Map to integer i between and M-. Insert. Put at table index i if free; if not try i+, i+, etc. Search. Search table index i; if occupied but no match, try i+, i+, etc. Note. Array size M must be greater than number of -value pairs N. search K hash(k) = P M A C S H L E R X P M A C S H L E R X M = 6 K M = 6 search miss (return null) 8

Linear-probing symbol table: Java implementation Linear-probing symbol table: Java implementation public class LinearProbingHashST<Key, Value> private int M = ; private Value[] vals = (Value[]) new

8 Linear-probing symbol table: Java implementation Linear-probing symbol table: Java implementation public class LinearProbingHashST<Key, Value> private int M = ; private Value[] vals = (Value[]) new Object[M]; private Key[] s = (Key[]) new Object[M]; private int hash(key ) /* as before */ private void put(key, Value val) /* next slide */ public Value get(key ) for (int i = hash(); s[i]!= null; i = (i+) % M) if (.equals(s[i])) return vals[i]; return null; array doubling and halving code omitted public class LinearProbingHashST<Key, Value> private int M = ; private Value[] vals = (Value[]) new Object[M]; private Key[] s = (Key[]) new Object[M]; private int hash(key ) /* as before */ private Value get(key ) /* previous slide */ public void put(key, Value val) int i; for (i = hash(); s[i]!= null; i = (i+) % M) if (s[i].equals()) break; s[i] = ; vals[i] = val; 9 Clustering Knuth's parking problem Cluster. A contiguous block of items. Observation. New s likely to hash into middle of big clusters. Model. Cars arrive at one-way street with M parking spaces. Each desires a random space i : if space i is taken, try i +, i +, etc. Q. What is mean displacement of a car? displacement = Half-full. With M / cars, mean displacement is ~ /. Full. With M cars, mean displacement is ~ π M / 8.

9 Analysis of linear probing Resizing in a linear-probing hash table Proposition. Under uniform hashing assumption, the average # of probes in a linear probing hash table of size M that contains N = α M s is: + + ( ) Goal. Average length of list N / M ½. Double size of array M when N / M ½. Halve size of array M when N / M ⅛. Need to rehash all s when resizing. search hit search miss / insert Pf. before resizing s[] vals[] E S R A Parameters. M too large too many empty array entries. M too small search time blows up. Typical choice: α = N / M ~ ½. # probes for search hit is about / # probes for search miss is about 5/ after resizing s[] vals[] A S E R 4 Deletion in a linear-probing hash table ST implementations: summary Q. How to delete a (and its associated value)? A. Requires some care: can't just delete array entries. implementation guarantee average case search insert delete search hit insert delete ordered ops? interface before deleting S sequential search (unordered list) N N N ½ N N ½ N equals() s[] P M A C S H L E R X binary search (ordered array) lg N N N lg N ½ N ½ N compareto() vals[] BST N N N.9 lg N.9 lg N N compareto() after deleting S? doesn't work, e.g., if hash(h) = red-black BST lg N lg N lg N. lg N. lg N. lg N compareto() separate chaining N N N -5 * -5 * -5 * equals() s[] vals[] P M A C H L E R X linear probing N N N -5 * -5 * -5 * equals() 5 * under uniform hashing assumption 6

10 War story: algorithmic complexity attacks Q. Is the uniform hashing assumption important in practice? A. Obvious situations: aircraft control, nuclear reactor, pacemaker. A. Surprising situations: denial-of-service attacks..4 HASH TABLES Algorithms hash functions separate chaining linear probing context malicious adversary learns your hash function (e.g., by reading Java API) and causes a big pile-up in single slot that grinds performance to a halt Real-world exploits. [Crosby-Wallach ] Bro server: send carefully chosen packets to DOS the server, using less bandwidth than a dial-up modem. Perl 5.8.: insert carefully chosen strings into associative array. Linux.4. kernel: save files with carefully chosen names. 8 War story: algorithmic complexity attacks Algorithmic complexity attack on Java A Java bug report. Goal. Find family of strings with the same hash code. Solution. The base- hash code is part of Java's string API. Jan Lieskovsky -- ::47 EDT Description Julian Wälde and Alexander Klink reported that the String. hash function is not sufficiently collision resistant. value is used in the implementations of HashMap and Hashtable classes: A specially-crafted set of s could trigger hash function collisions, which can degrade performance of HashMap or Hashtable by changing hash table operations complexity from an expected/average O() to the worst case O(n). Reporters were able to find colliding strings efficiently using equivalent substrings and meet in the middle techniques. This problem can be used to start a denial of service attack against Java applications that use untrusted inputs as HashMap or Hashtable s. An example of such application is web application server (such as tomcat, see bug #755) that may fill hash tables with data from HTTP request (such as GET or POST parameters). A remote attack could use that to make JVM use excessive amount of CPU time by sending a POST request with large amount of parameters which hash to the same value. This problem is similar to the issue that was previously reported for and fixed in e.g. perl: "Aa" "BB" "AaAaAaAa" "AaAaAaBB" "AaAaBBAa" "AaAaBBBB" "AaBBAaAa" "AaBBAaBB" "AaBBBBAa" "BBAaAaAa" "BBAaAaBB" "BBAaBBAa" "BBAaBBBB" "BBBBAaAa" "BBBBAaBB" "BBBBBBAa" "AaBBBBBB" "BBBBBBBB" N strings of length N that hash to same value! 9 4

Diversion: one-way hash functions Separate chaining vs. linear probing One-way hash function. "Hard" to find a that will hash to a desired value (or two s that hash to same value). Ex.

11 Diversion: one-way hash functions Separate chaining vs. linear probing One-way hash function. "Hard" to find a that will hash to a desired value (or two s that hash to same value). Ex. MD4, MD5, SHA-, SHA-, SHA-, WHIRLPOOL, RIPEMD-6,. known to be insecure String password = args[]; MessageDigest sha = MessageDigest.getInstance("SHA"); byte[] bytes = sha.digest(password); /* prints bytes as hex string */ Separate chaining. Performance degrades gracefully. Clustering less sensitive to poorly-designed hash function. Linear probing. Less wasted space. Better cache performance. 4 A 8 null X 7 L E S P M 9 H 5 C 4 R Applications. Digital fingerprint, message digest, storing passwords. Caveat. Too expensive for use in ST implementations. s[] vals[] P M A C S H L E R X Hashing: variations on the theme Hash tables vs. balanced search trees Many improved versions have been studied. Two-probe hashing. [ separate-chaining variant ] Hash to two positions, insert in shorter of the two chains. Reduces expected length of the longest chain to log log N. Double hashing. [ linear-probing variant ] Use linear probing, but skip a variable amount, not just each time. Effectively eliminates clustering. Can allow table to become nearly full. More difficult to implement delete. Cuckoo hashing. [ linear-probing variant ] Hash to two positions; insert into either position; if occupied, reinsert displaced into its alternative position (and recur). Constant worst-case time for search. 4 Hash tables. Simpler to code. No effective alternative for unordered s. Faster for simple s (a few arithmetic ops versus log N compares). Better system support in Java for strings (e.g., cached hash code). Balanced search trees. Stronger performance guarantee. Support for ordered ST operations. Easier to implement compareto() correctly than equals() and. Java system includes both. Red-black BSTs: java.util.treemap, java.util.treeset. Hash tables: java.util.hashmap, java.util.identityhashmap. 44

Algorithms 3.4 HASH TABLES. hash functions separate chaining linear probing context ROBERT SEDGEWICK KEVIN WAYNE.

Algorithms 3.4 HASH TABLES. hash functions separate chaining linear probing context ROBERT SEDGEWICK KEVIN WAYNE. Algorithms ROBERT SEDGEWICK KEVIN WAYNE 3.4 HASH TABLES hash functions separate chaining linear probing context http://algs4.cs.princeton.edu Hashing: basic plan Save items in a key-indexed table (index