CS.15.A.SymbolTables.API. Alice

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15.Symbol Tables APIs and clients A design challenge Binary search trees Implementation Analysis 15. Symbol Tables Section 4.4 http://introcs.cs.princeton.edu CS.15.A.SymbolTables.

Unsupported operations Alice Change number associated wh a given name. Add a new name, associated wh a given number.

1 15.Symbol Tables APIs and clients A design challenge Binary search trees Implementation Analysis 15. Symbol Tables Section CS.15.A.SymbolTables.API FAQs about sorting and searching Why are telephone books obsolete? Hey, Alice. That whelist filter wh mergesort and binary search is working great. Right, but 's a pain sometimes. Why? Bob We have to sort whole list whenever we add new customers. Unsupported operations Alice Change number associated wh a given name. Add a new name, associated wh a given number. Remove a given name and associated number Also, we want to process transactions and associate all sorts information wh our customers. Observation. Mergesort + binary search has same problem wh add and remove. Bottom line. Need a more flexible API. 3 see Sorting and Searching lecture 4

Associative array abstraction Symbol table ADT Imagine using arrays whose indices are string values. A symbol table is an ADT whose values are sets key-value pairs, wh keys all different.

fundamental abstraction Use keys to access associated values. Keys and values could be any type data. Client code could not be simpler. Q. How to implement?

princeton.edu" URL["130.132.143.21"] = "www.yale.edu" URL["128.103.060.55"] = "www.harvard.edu" IPaddr["www.cs.princeton.edu"] = "128.112.136.11" IPaddr["www.princeton.edu"] = "128.112.128.15" IPaddr["www.

] Return value associated wh a given key. Test if a given key is in table. Iterate though keys. Useful addional assumptions Keys are comparable and eration is in order.

2 Associative array abstraction Symbol table ADT Imagine using arrays whose indices are string values. A symbol table is an ADT whose values are sets key-value pairs, wh keys all different. phonenumber["alice"] = "(212) " phonenumber["bob"] = "(609) " phonenumber["carl"] = "(800) " phonenumber["dave"] = "(888) " phonenumber["eve"] = "(999) " A fundamental abstraction Use keys to access associated values. Keys and values could be any type data. Client code could not be simpler. Q. How to implement? transactions["alice"] = "Dec 12 12:01AM $ Amazon, Dec 12 1:11 AM $ Ebay" legal code in some programming languages (not Java) URL[" "] = " URL[" "] = " URL[" "] = " URL[" "] = " IPaddr[" = " " IPaddr[" = " " IPaddr[" = " " Basic symbol-table operations Associate a given key wh a given value. [If key is not in table, add to table.] [If key is in table, change s value.] Return value associated wh a given key. Test if a given key is in table. Iterate though keys. Useful addional assumptions Keys are comparable and eration is in order. No lim on number key-value pairs. All keys not in table associate wh null. key: time+channel value: TV show key: name value: phone number key: word value: definion key: number value: function value IPaddr[" = " " 5 key: term value: article 6 Benchmark example symbol-table operations Parameterized API for symbol tables Application. Count frequency occurrence strings in StdIn. Goal. Simple, safe, and clear client code for symbol tables holding any type data. Keys. Strings from a sequence. Values. Integers. key times worst value symbol-table contents after operation change value worst 1 Java approach: Parameterized data types (generics) Use placeholder type names for both keys and values. Substute concrete types for placeholder in clients. Symbol Table API public class ST<Key extends Comparable<Key>, Value> ST<Key, Value>() void put(key key, Value val) Value get(key key) boolean contains(key key) Iterable<Key> keys() implements compareto() create a symbol table associate key wh val return value associated wh key, null if none is re a value associated wh key? all keys in table 7 8

println(s); Enhanced for loop. Useful for any collection. Iterate through each entry in collection. so you don't have to. create a stack objects, all type Item void push(item em) add em to stack A.

3 Aside: Iteration (client code) Aside: Iteration (implementation) Q. How to print contents a stack/queue? Q. How to "implement Iterable"? public class Stack<Item> implements Iterable<Item> Stack<Item>() Java foreach construct A. Use Java's foreach construct. Stack<String> stack = new Stack<String>(); for (String s : stack) StdOut.println(s); Enhanced for loop. Useful for any collection. Iterate through each entry in collection. so you don't have to. create a stack objects, all type Item void push(item em) add em to stack A. We did for Stack and Queue, Item pop() boolean isempty() int size() remove and return em most recently pushed is stack empty? # objects on stack Order determined by implementation. Substantially simplifies client code. A. Implement an Iterator (see text pp ) public class Stack<Item> implements Iterable<Item> Works when API "implements Iterable". Stack<Item>() create a stack objects, all type Item void push(item em) add em to stack Item pop() remove and return em most recently pushed boolean isempty() int size() is stack empty? Meets performance specification. Constant-time per entry. # objects on stack Performance specification. Constant-time per entry. Bottom line. Use eration in client code that uses collections. 9 Why ordered keys? Symbol table client example 1: Sort (wh dedup) Natural for many applications 10 Goal. Sort lines on standard input (and remove duplicates). Numeric types. Key type. String (line on standard input). Value type. (ignored). Strings. Date and time. Client-supplied types (Account numbers, ). Enables useful API extensions public class Sort public static void main(string[] args) // Sort lines on StdIn BST<String, Integer> st = new BST<String, Integer>(); while (StdIn.hasNextLine()) st.put(stdin.readline(), 0); for (String s : st.keys()) StdOut.println(s); foreach construct Provide keys in sorted order. Find k th largest key. Enables efficient implementations Mergesort. Binary search. BSTs (this lecture). thingsorganizedneatly.tumblr.com 11 % more tale.txt times worst times age wisdom age foolishness epoch belief epoch increduly season light season darkness spring hope winter despair % java Sort < tale.txt age foolishness age wisdom times epoch belief epoch increduly season darkness season light spring hope winter despair worst times 12

Symbol table client example 2: Frequency counter Symbol table client example 3: Index Goal. Compute frequencies words on standard input. Key type. String (word on standard input). Value type.

4 Symbol table client example 2: Frequency counter Symbol table client example 3: Index Goal. Compute frequencies words on standard input. Key type. String (word on standard input). Value type. Integer (frequency count). public class Freq public static void main(string[] args) // Frequency counter BST<String, Integer> st = new BST<String, Integer>(); while (!StdIn.isEmpty()) String key = StdIn.readString(); if (st.contains(key)) st.put(key, st.get(key) + 1); else st.put(key, 1); for (String s : st.keys()) StdOut.printf("%8d %s\n", st.get(s), s); % more tale.txt times worst times age wisdom age foolishness epoch belief % java Freq < tale.txt java Sort epoch increduly 1 belief season light 1 season darkness 1 darkness spring hope 1 despair winter despair 1 foolishness 1 hope 1 increduly 1 light 1 spring 1 winter 1 wisdom 1 worst 2 age 2 epoch 2 season 2 times Goal. Print index to words on standard input. Key type. String (word on standard input). Value type. Queue<Integer> (indices where word occurs). public class Index public static void main(string[] args) BST<String, Queue<Integer>> st; st = new BST<String, Queue<Integer>>(); int i = 0; while (!StdIn.isEmpty()) String key = StdIn.readString(); if (!st.contains(key)) st.put(key, new Queue<Integer>()); st.get(key).enqueue(i++); for (String s : st.keys()) StdOut.println(s + " " + st.get(s)); % more tale.txt times worst times age wisdom age foolishness epoch belief % java Index < tale.txt epoch increduly age season light belief 29 season darkness 3 spring hope darkness 47 winter despair despair 59 epoch foolishness 23 hope 53 increduly light season spring times winter 57 wisdom 17 worst 9 14 Symbol-table applications application key value Symbol tables are ubiquous in today's computational infrastructure. contacts name phone number, address cred card account number transaction details file share name song computer ID dictionary word definion web search keyword list web pages book index word list page numbers cloud storage file name file contents We're going to need a good symbol-table implementation! domain name service domain name IP address reverse DNS IP address domain name compiler variable name value and type internet routing destination route 15 CS.15.A.SymbolTables.API

5 Benchmark Application. Linguistic analysis Zipf's law (for a natural language corpus) 15.Symbol Tables APIs and clients A design challenge Binary search trees Implementation Analysis Suppose most frequent word occurs about t times. 2nd most frequent word occurs about t/2 times. 3rd most frequent word occurs about t/3 times. 4th most frequent word occurs about t/4 times. Goal. Validate Zipf's law for real natural language data. Method. % java Freq < data.txt java Sort Required. Efficient symbol-table implementation. % java Freq < mobydick.txt java Sort hyposis 1940 i observation his 2911 that 4037 in 4508 to 4583 a 6247 and CS.15.B.SymbolTables.Challenge 18 Benchmark statistics Strawman I: Ordered array Goal. Validate Zipf's law for real natural language data. Method. % java Freq < data.txt java Sort Idea keys values keys values Keep keys in order in an array. Keep values in a parallel array. alice 121 alice 121 bob 873 bob 873 carlos 884 carlos 884 carol 712 carol 712 dave 585 craig 999 erin 247 dave 585 Reasons (see "Sorting and Searching" lecture) file description words distinct Melville's Moby Dick 210,028 16,834 liepzig100k.txt 100K random sentences 2,121, ,256 liepzig200k.txt 200K random sentences 4,238,435 mobydick.txt liepzig1m.txt 1M random sentences 21,191,455 Takes advantage fast sort (mergesort). Enables fast search (binary search). eve 577 erin 247 oscar 675 eve ,515 peggy 895 oscar ,580 trent 557 peggy 895 trudy 926 trent 557 walter 51 trudy 926 wendy 152 walter 51 wendy 152 Known challenge. How big to make arrays? Fatal flaw. How to insert a new key? Reference: Wortschatz corpus, Universät Leipzig To keep key array in order, need to move larger entries à la insertion sort. Required. Efficient symbol-table implementation. Hyposis: Quadratic time for benchmark. 19 easy to validate wh experiments 20

6 Strawman II: Linked list Idea Keep keys in order in a linked list. Add a value to each node. Design challenge Implement scalable symbol tables. Goal. Simple, safe, clear, and efficient client code. Reason. Meets memory-use performance specification. Only slightly more costly than stacks or queues! alice 2 bob 7 carlos 1 carol 8 dave 2 erin 8 eve 1 oscar 8 peggy 2 Performance specifications Order growth running time for put(), get() and contains() is logarhmic. Memory use is proportional to size collection, when is nonempty. No lims whin code on collection size. Fatal flaw. How to search? Binary search requires indexed access. Example: How to access middle a linked list? Only choice: search sequentially through list. Hyposis: Quadratic time for benchmark. easy to validate wh experiments 21 Are such guarantees achievable?? Can we implement associative arrays wh just log-factor extra cost?? phonenumber["alice"] = "(212) " No way! This lecture. Yes way! Symbol Tables APIs and clients A design challenge Binary search trees Implementation Analysis CS.15.A.SymbolTables.API CS.15.B.SymbolTables.Challenge CS.15.C.SymbolTables.BSTs

7 Doubly-linked data structures A doubly-linked data structure: binary search tree Wh two links ( Doubly-linked list ) a wide variety data structures are possible. Binary tree (this lecture) Tree Binary search tree (BST) A recursive data structure containing distinct comparable keys that is ordered. Def. A BST is a null or a reference to a BST node ( root). Def. A BST node is a data type that contains references to a key, a value, and two BSTs, a left subtree and a right subtree. Ordered. All keys in left subtree each node are smaller than s key and all keys in right subtree each node are larger than s key. Doubly-linked circular list Maintenance can be complicated! General case From point view a particular object, all se structures look same. A BST private class Node private Key key; private Value val; private Node left; private Node right; left right BST processing code BST processing code: Search Standard operations for processing data structured as a binary search tree Search for value associated wh a given key. Add a new key-value pair. Traverse BST (vis every node, in order keys). Remove a given key and associated value (not addressed in this lecture). root Goal. Find value associated wh a given key in a BST. If less than key at current node, go left. If greater than key at current node, go right. Example. get("") public Value get(key key) return get(root, key); private Value get(node x, Key key) if (x == null) return null; int cmp = key.compareto(x.key); if (cmp < 0) return get(x.left, key); else if (cmp > 0) return get(x.right, key); else if (cmp == 0) return x.val; root GREATER go right? 2 SEARCH HIT return value 1 LESS go left

8 BST processing code: Associate a new value wh a key Goal. Associate a new value wh a given key in a BST. If less than key at current node, go left. If greater than key at current node, go right. BST processing code: Add a new key Goal. Add a new key-value pair to a BST. Search for key. Return link to new node when null reached. Example. put("", 2) public void put(key key, Value val) root = put(root, key, val); root GREATER go right? 2 private Node put(node x, Key key, Value val) if (x == null) return new Node(key, val); int cmp = key.compareto(x.key); if (cmp < 0) x.left = put(x.left, key, val); else if (cmp > 0) x.right = put(x.right, key, val); else x.val = val; return x; SEARCH HIT update value 21 LESS go left 2 29 Example. put("worst", 1) public void put(key key, Value val) root = put(root, key, val); root worst? GREATER go right 2 private Node put(node x, Key key, Value val) if (x == null) return new Node(key, val); int cmp = key.compareto(x.key); if (cmp < 0) x.left = put(x.left, key, val); else if (cmp > 0) x.right = put(x.right, key, val); else x.val = val; return x; 2 GREATER go right 2 worst 1 NULL add new node 30 BST processing code: Traverse BST Goal. Put keys in a BST on a queue, in sorted order. Do for left subtree. Put key at root on queue. Do for right subtree. root 2 public Iterable<Key> keys() Queue<Key> queue = new Queue<Key>(); inorder(root, queue); return queue; private void inorder(node x, Queue<Key> queue) if (x == null) return; inorder(x.left, queue); q.enqueue(x.key); inorder(x.right, queue); Queue 2 times 2 31 CS.15.A.SymbolTables.API CS.15.C.SymbolTables.BSTs

9 ADT for symbol tables: review A symbol table is an idealized model an associative storage mechanism. An ADT allows us to wre Java programs that use and manipulate symbol tables. 15.Symbol Tables APIs and clients A design challenge Binary search trees Implementation Analysis API Performance specifications public class ST<Key extends Comparable<Key>, Value> ST<Key, Value>() void put(key key, Value val) Value get(key key) boolean contains(key key) Iterable<Key> keys() create a symbol table associate key wh val return value associated wh key, null if none is re a value associated wh key? all keys in table Order growth running time for put(), get() and contains() is logarhmic. Memory use is proportional to size collection, when is nonempty. No lims whin code on collection size. CS.15.D.SymbolTables.Implementation 34 Symbol table implementation: Instance variables and constructor BST implementation: Test client (frequency counter) Data structure choice. Use a BST to hold collection. public class BST<Key extends Comparable<Key>, Value> private Node root = null; private class Node private Key key; private Value val; private Node left; private Node right; root times instance variables constructor methods test client public static void main(string[] args) BST<String, Integer> st = new BST<String, Integer>(); while (!StdIn.isEmpty()) String key = StdIn.readString(); if (st.contains(key)) st.put(key, st.get(key) + 1); else st.put(key, 1); for (String s : st.keys()) StdOut.printf("%8d %s\n", st.get(s), s); What we expect, once implementation is done. % java BST < tale.txt 2 age 1 belief 1 1 darkness 1 despair 2 epoch 1 foolishness 1 hope 1 increduly 10 1 light 10 2 season 1 spring 10 2 times 10 1 winter 1 wisdom 1 worst instance variables constructors methods test client 35 36

10 BST implementation: Methods BST implementation Methods define data-type operations (implement API). public class BST<Key extends Comparable<Key>, Value> public boolean isempty() return root == null; public void put(key key, Value value) /* See BST add slides and next slide. */ public Value get(key key) /* See BST search slide and next slide. */ public boolean contains(key key) return get(key)!= null; public Iterable<Key> keys() /* See BST traverse slide and next slide. */ instance variables constructors methods test client 37 public class BST<Key extends Comparable<Key>, Value> private Node root = null; instance variable private class Node private Key key; private Value val; private Node left; private Node right; public boolean isempty() return root == null; public void put(key key, Value val) root = put(root, key, val); public Value get(key key) return get(root, key); public boolean contains(key key) return get(key)!= null; public Iterable<Key> keys() Queue<Key> q = new Queue<Key>(); inorder(root, q); return q; nested class private methods public methods private Value get(node x, Key key) if (x == null) return null; int cmp = key.compareto(x.key); if (cmp < 0) return get(x.left, key); else if (cmp > 0) return get(x.right, key); else if (cmp == 0) return x.val; private Node put(node x, Key key, Value val) if (x == null) return new Node(key, val); int cmp = key.compareto(x.key); if (cmp < 0) x.left = put(x.left, key, val); else if (cmp > 0) x.right = put(x.right, key, val); else x.val = val; return x; private void inorder(node x, Queue<Key> q) if (x == null) return; inorder(x.left, q); q.enqueue(x.key); inorder(x.right, q); public static void main(string[] args) // Frequency counter test client 38 Trace BST construction times 39 CS.15.A.SymbolTables.API CS.15.D.SymbolTables.Implementation

11 BST analysis Costs depend on order key insertion. Best case Worst case 15.Symbol Tables times worst APIs and clients A design challenge Typical case times Binary search trees Implementation Analysis worst worst times CS.15.E.SymbolTables.Analysis 42 BST insertion: random order visualization BST analysis Running time depends on order key insertion. Insert keys in random order. Tree is roughly balanced. Tends to stay that way! Model. Insert keys in random order. Tree is roughly balanced. Tends to stay that way! Proposion. Building a BST by inserting N randomly ordered keys into an inially empty tree uses ~2 N ln N (about 1.39 N lg N ) compares. Pro. A very interesting exercise in discrete math. Interested in details? Take a course in algorhms

12 Benchmarking BST implementation Empirical tests BSTs BST implements associative-array abstraction for randomly ordered keys. Symbol table API public class ST<Key extends Comparable<Key>, Value> ST<Key, Value>() void put(key key, Value value) Value get(key key) boolean contains(key key) Iterable<Key> keys() create a symbol table associate key wh value return value associated wh key, null if none is re a value associated wh key? all keys in table (sorted) for random keys (but stay tuned) Count number words that appear more than once in StdIn. Frequency count whout output N TN (seconds) 1 million 5 TN/TN/2 2 million million million 34 2 % java Generator (5 seconds) % java Generator (9 seconds) % java Generator (17 seconds) % java Generator (34 seconds) % java Generator (72 seconds) Performance specifications Order growth running time for put(), get() and contains() is logarhmic. Memory use is proportional to size collection, when is nonempty. No lims whin code on collection size. 16 million BILLION = java DupsBST 6-dig integers Made possible by binary tree data structure. times worst 45 Confirms hyposis that order growth is N log N WILL scale Easy to process 21M word corpus NOT possible wh brute-force 46 Performance guarantees Red-black tree insertion: random order visualization Practical problem. Keys may not be randomly ordered. BST may be unbalanced. Running time may be quadratic. Happens in practice (insert keys in order). Remarkable resolution. Balanced tree algorhms perform simple transformations that guarantee balance. AVL trees (Adelson-Velskii and Landis, 1962) proved concept. Red-black trees (Guibas and Sedgewick, 1979) are implemented in many modern systems. Insert keys in random order. Same # black links on every path from root to leaf. No two red links in a row. Tree is roughly balanced. Guaranteed to stay that way! 47 48

ST implementation wh guaranteed logarhmic performance Summary import java.util.treemap; BSTs. Simple symbol-table implementation, usually efficient.

13 ST implementation wh guaranteed logarhmic performance Summary import java.util.treemap; BSTs. Simple symbol-table implementation, usually efficient. public class ST<Key extends Comparable<Key>, Value> private TreeMap<Key, Value> st = new TreeMap<Key, Value>(); public void put(key key, Value val) if (val == null) st.remove(key); else st.put(key, val); public Value get(key key) return public Value remove(key key) return public boolean contains(key key) return public Iterable<Key> keys() return Java's TreeMap library uses red-black trees. Red-black trees. More complicated variation, guaranteed to be efficient. Applications. Many, many, many things are enabled by efficient symbol tables. Example. Search among 1 trillion customers wh less than 80 compares! st.get(key); st.remove(key); st.containskey(key); st.keyset(); Example. Search among all atoms in universe wh less than 200 compares! YES! Proposion. In a red-black tree size N, put(), get() and contains() are guaranteed to use fewer than 2lg N compares. Pro. A fascinating exercise in algorhmics. Several or useful operations also available. Interested in details? Take a course in algorhms. Can we implement associative arrays wh just log-factor extra cost?? Symbol Tables Section CS.15.E A.SymbolTables.An Pa Ilysis

COMPUTER SCIENCE. 15. Symbol Tables. Section 4.4.

COMPUTER SCIENCE. 15. Symbol Tables. Section 4.4. COMPUTER SCIENCE S E D G E W I C K / W A Y N E 15. Symbol Tables Section 4.4 http://introcs.cs.princeton.edu COMPUTER SCIENCE S E D G E W I C K / W A Y N E 15.Symbol Tables APIs and clients A design challenge