ELEMENTARY SEARCH ALGORITHMS

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1 BB - GOIT TODY DT. OF OUT GIIG KUT D ymbol Tables I lementary implementations Ordered operations TY GOIT ar., cknowledgement:.the$course$slides$are$adapted$from$the$slides$prepared$by$.$edgewick$ and$k.$wayne$of$rinceton$university. I lementary implementations Ordered operations YBO TB ymbol tables Key-value pair abstraction. Insert a value with specified key. Given a key, search for the corresponding value. x. D lookup. Insert U with specified I address. Given U, find corresponding I address. U I address key value

2 ymbol table applications Basic symbol table I ssociative array abstraction. ssociate one value with each key. application purpose of search key value dictionary find definition word definition book index find relevant pages term list of page numbers file share find song to download name of song computer ID financial account process transactions account number transaction details web search find relevant web pages keyword list of page names compiler find properties of variables variable name type and value routing table route Internet packets destination best route D find I address given U U I address reverse D find U given I address I address U genomics find markers D string known positions file system find file on disk filename location on disk public class T<Key, Value> T() void put(key key, Value val) Value get(key key) create a symbol table put key-value pair into the table (remove key from table if value is null) value paired with key (null if key is absent) void delete(key key) remove key (and its value) from table boolean contains(key key) is there a value paired with key? boolean ismpty() is the table empty? int size() number of key-value pairs in the table Iterable<Key> keys() all the keys in the table I for a generic basic symbol table a[key] = val; a[key] onventions Values are not null. ethod get() returns null if key not present. ethod put() overwrites old value with new value. Intended consequences. asy to implement contains(). public boolean contains(key key) return get(key)!= null; an implement lazy version of delete(). public void delete(key key) put(key, null); Keys and values Value type. ny generic type. Key type: several natural assumptions. ssume keys are omparable, use compareto(). ssume keys are any generic type, use equals() to test equality. ssume keys are any generic type, use equals() to test equality; use hashode() to scramble key. built-in to Java (stay tuned) specify omparable in I. Best practices. Use immutable types for symbol table keys. Immutable in Java: tring, Integer, Double, java.io.file, utable in Java: tringbuilder, java.net.u, arrays,...

3 quality test ll Java classes inherit a method equals(). Java requirements. For any references x, y and z: eflexive: x.equals(x) is true. ymmetric: x.equals(y) iff y.equals(x). Transitive: if x.equals(y) and y.equals(z), then x.equals(z). on-null: x.equals(null) is false. do x and y refer to the same object? Default implementation. (x == y) ustomized implementations. Integer, Double, tring, File, U, User-defined implementations. ome care needed. equivalence relation Implementing equals for user-defined types eems easy. public class Date implements omparable<date> private final int month; private final int day; private final int year;... public boolean equals(date that) if (this.day!= that.day ) return false; if (this.month!= that.month) return false; if (this.year!= that.year ) return false; return true; check that all significant fields are the same Implementing equals for user-defined types eems easy, but requires some care. public final class Date implements omparable<date> private final int month; private final int day; private final int year;... public boolean equals(object y) if (y == this) return true; if (y == null) return false; if (y.getlass()!= this.getlass()) return false; Date that = (Date) y; if (this.day!= that.day ) return false; if (this.month!= that.month) return false; if (this.year!= that.year ) return false; return true; typically unsafe to use equals() with inheritance (would violate symmetry) must be Object. Why? xperts still debate. optimize for true object equality check for null objects must be in the same class (religion: getlass() vs. instanceof) cast is guaranteed to succeed check that all significant fields are the same quals design "tandard" recipe for user-defined types. Optimization for reference equality. heck against null. heck that two objects are of the same type and cast. ompare each significant field: - if field is a primitive type, use == - if field is an object, use equals() - if field is an array, apply to each entry apply rule recursively alternatively, use rrays.equals(a, b) or rrays.deepquals(a, b), but not a.equals(b) Best practices. o need to use calculated fields that depend on other fields. ompare fields mostly likely to differ first. ake compareto() consistent with equals(). x.equals(y) if and only if (x.compareto(y) == )

4 T test client for traces Build T by associating value i with i th string from standard input. T test client for analysis Frequency counter. ead a sequence of strings from standard input and print out one that occurs with highest frequency. public static void main(tring[] args) T<tring, Integer> st = new T<tring, Integer>(); for (int i = ;!tdin.ismpty(); i++) tring key = tdin.readtring(); st.put(key, i); for (tring s : st.keys()) tdout.println(s + " " + st.get(s)); keys values output % more 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 % java Frequencyounter < tinytale.txt it % java Frequencyounter < tale.txt business % java Frequencyounter < leipzig.txt government tiny example ( words, distinct) real example (, words,, distinct) real example (,, words,, distinct) Frequency counter implementation public class Frequencyounter public static void main(tring[] args) int minlen = Integer.parseInt(args[]); T<tring, Integer> st = new T<tring, Integer>(); while (!tdin.ismpty()) tring word = tdin.readtring(); ignore short strings if (word.length() < minlen) continue; if (!st.contains(word)) st.put(word, ); else st.put(word, st.get(word) + ); tring max = ""; st.put(max, ); for (tring word : st.keys()) if (st.get(word) > st.get(max)) max = word; tdout.println(max + " " + st.get(max)); create T read string and update frequency print a string with max freq I lementary implementations Ordered operations YBO TB

5 equential search in a linked list lementary T implementations: summary Data structure. aintain an (unordered) linked list of key-value pairs. earch. can through all keys until find a match. Insert. can through all keys until find a match; if no match add to front. key value first red nodes are new gray nodes are untouched worst-case cost search insert search hit insert / sequential search (unordered list) black nodes are accessed in search average case (after random inserts) (after inserts) T implementation ordered iteration? key interface no equals() circled entries are changed values hallenge. fficient implementations of both search and insert. Trace of linked-list T implementation for standard indexing client lementary T implementations: summary worst case Binary search Data structure. aintain an ordered array of key-value pairs. average case T implementation sequential search (unordered list) search insert search hit insert / ordered iteration? operations on keys ank helper function. ow many keys < k? no equals() compares operations keys[] successful search for lo hi m lo hi m entry in red is a[m] loop exits with keys[m] = : return unsuccessful search for Q osts for java Frequencyounter < tale.txt using equentialearcht entries in black are a[lo..hi] loop exits with lo > hi: return Trace of binary search for rank in an ordered array

6 Binary search: Java implementation Binary search: mathematical analysis roposition. Binary search uses ~ lg compares to search any array of size. public Value get(key key) if (ismpty()) return null; int i = rank(key); if (i < && keys[i].compareto(key) == ) return vals[i]; else return null; private int rank(key key) int lo =, hi = -; while (lo <= hi) int mid = lo + (hi - lo) / ; int cmp = key.compareto(keys[mid]); if (cmp < ) hi = mid - ; else if (cmp > ) lo = mid + ; else if (cmp == ) return mid; return lo; f. T() number of compares to binary search in a sorted array of size. left or right half T( / ) number of keys < key + ecall lecture. Binary search: trace of standard indexing client lementary T implementations: frequency counter roblem. To insert, need to shift all greater keys over. vals[] entries in gray did not move entries in red were inserted compares keys[] entries in black moved to the right operations osts for java Frequencyounter < tale.txt using equentialearcht circled entries are changed values cost key value Trace of ordered-array T implementation for standard indexing client operations osts for java Frequencyounter < tale.txt using BinaryearchT

7 lementary T implementations: summary YBO TB T implementation worst-case cost (after inserts) average case (after random inserts) ordered iteration? key interface I lementary implementations Ordered operations search insert search hit insert sequential search (unordered list) / no equals() binary search (ordered array) log log / yes compareto() hallenge. fficient implementations of both search and insert. Ordered symbol table I Ordered symbol table I min() get(::) floor(::) select() keys(::, ::) ceiling(::) max() size(::, ::) is rank(::) is keys values :: hicago :: hoenix :: ouston :: hicago :: ouston :: hicago :: eattle :: eattle :: hoenix :: hicago :: hicago :: hicago :: eattle :: eattle :: hicago :: hicago :: eattle :: hoenix public class T<Key extends omparable<key>, Value> void Value T() put(key key, Value val) get(key key) create an ordered symbol table put key-value pair into the table (remove key from table if value is null) value paired with key (null if key is absent) void delete(key key) remove key (and its value) from table boolean contains(key key) is there a value paired with key? boolean ismpty() is the table empty? int size() number of key-value pairs Key min() smallest key Key max() largest key Key floor(key key) largest key less than or equal to key Key ceiling(key key) smallest key greater than or equal to key int rank(key key) number of keys less than key Key select(int k) key of rank k void deletein() delete smallest key void deleteax() delete largest key int size(key lo, Key hi) number of keys in [lo..hi] xamples of ordered symbol-table operations Iterable<Key> keys(key lo, Key hi) keys in [lo..hi], in sorted order Iterable<Key> keys() all keys in the table, in sorted order I for a generic ordered symbol table

8 Binary search: ordered symbol table operations summary sequential search binary search search lg insert min / max floor / ceiling lg rank lg select ordered iteration log order of growth of the running time for ordered symbol table operations

ELEMENTARY SEARCH ALGORITHMS

ELEMENTARY SEARCH ALGORITHMS BB - GOIT TODY DT. OF OUT GIIG KUT D ymbol Tables I lementary implementations Ordered operations TY GOIT ar., cknowledgement: The course slides are adapted from the slides prepared by. edgewick and K.